User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells

David Brooks Christie dc@samizdat.org June 2024

"Grünbaum and Coxeter independently discovered the 11-cell, a regular 4-polytope with cells that are the hemi-icosahedron, a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the 120-cell, the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the 120-vertex regular 4-polytope { $5⁄2$, 5, 3} with 11 rhombicosidodecahedron cells. Eleven 11-cells (plural) are the 137-cell, the 600-vertex regular 4-polytope { $5⁄2$, 3, 3} with 137 triacontahedron cells."

Introduction
Branko Grünbaum discovered the 11-cell around 1970, about a decade before H.S.M. Coxeter extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying. Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, eleven of them.

The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-simplex (called the 5-cell, because it is built from 5 tetrahedra), and the 8-point 4-orthoplex (called the 16-cell, because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (except the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point (8-cell) tesseract, the 4-cubic, which is a compound of two 16-cells in exact dimensional analogy to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 120-cell, the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its tetrahedral symmetry group $$A_4$$ that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

5-cells and hemi-icosahedra in the 11-cell
The regular 5-cell is not only the regular simplex 4-polytope with 5 tetrahedron facets, it is also the regular decahedron 3-polytope with 10 triangle facets. The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally involved with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron containing pentads in its interior, most handsomely illustrated by Séquin. The 11-cell and 5-cell of radius $\sqrt{2}$ both have edge length $\sqrt{5}$, so their triangular faces are congruent. The 11-cell's 11 hemi-icosahedral cells have 10 triangle faces each, and a 5-cell is behind each face. Only 3 of the 5-cell's 5 vertices are visible on the surface of the hemi-icosahedron as the exposed face, but its other two vertices are visible on the opposite side of the hemi-icosahedron as an exposed edge. The other parts of the 5-cell (its other 6 edges, its other 9 faces and its 5 tetrahedral cells) are hidden inside the hemi-icosahedron.

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face. Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

We shall see in the next section §Compounds in the 120-cell that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's quaternion s to render the polyhedra which are found in the interior of n-dimensional polytopes. Hamilton was the first wise child to discover a 4-dimensional building block, in his flash of genius on Broom bridge in 1843, though he didn't think of his quaternion formula $i^{2} = j^{2} = k^{2} = ijk = −1$ as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian coordinates of Euclidean 4-space. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are radially equilateral like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his "Hull # = 8 with 60 vertices", lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space $$S^3$$, embedded in Euclidean 4-space $$R^4$$. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be 600-cell faces, separated from each other by rectangles.

The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves. There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell faces do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point rhombicosidodecahedron from Kepler's 1619 Harmonices Mundi, which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares or the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what Alicia Boole Stott named a contraction operation. In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the e2 expansion of the icosahedron (or equivalently of its dual polyhedron the dodecahedron). But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is apparently Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell parts in its interior to make 20 5-cell tetrahedral cells, but it does not contain any whole 5-cells in its interior. The 20 interior (5-cell tetrahedral cells) are disjoint, and belong individually to neighboring 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells. This is the way 4-polytopes bond together, by sharing a 3-polytope cell, and indeed in this context the rhombicosadodecahedra and 5-cells are both 4-polytopes which contain tetrahedral cells of edge-length $\sqrt{5}$. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a $\sqrt{6}$ triangle which lies in a central plane. The 120 rhombicosadodecahedra are also bonded directly together at all their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point pentagonal prism cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior $\sqrt{6}$ triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior $\sqrt{5}$ triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at $\sqrt{6}$ 5-cell faces) and to each other (at $\sqrt{6}$ central triangle faces). In the section §What's in the box, below, we will look closely at the relation between the $\sqrt{5}$ chord and the $\sqrt{6}$ chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in §Cell rings of the 11-cells we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point (pentagonal prism cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron.

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames all the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910, and those wise young friends Coxeter & Petrie, building together with polyhedral blocks, rediscovered before 1938. Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden $\sqrt{6}$ chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 $\sqrt{5}$ chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, $\sqrt{6}$ apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the Jessen's icosahedron, inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the Borromean rings of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in §The perfection of Fuller's cyclic design we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

120 of them
The largest regular convex 4-polytope is the 120-cell, the convex hull of a regular compound of 120 disjoint regular 5-cells, with 600 vertices, 1200 $\sqrt{5}$ chords and 1200 $\sqrt{6}$ interior triangles (in a $\sqrt{5}$ radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

How many building blocks, how many ways
The 120-cell is the maximally complex regular 4-polytope, which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 non-disjoint instances of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by $$5^2$$ non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points.

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times $$5^2$$ (75) disjoint instances of itself into the 600-point, which then magically contains $$3^2$$ times $$5^2$$ (225) instances of the 24-point, and  $$3^3$$ times $$5^2$$ (675) instances of the original 8-point.

They will be rare wise children who figure all this out for themselves, and even wiser who can see why it is so. Schoute was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s 10 different ways, and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.

What's in the box
The picture on the back of the box of building blocks, showing its contents: That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.

The pentad building block is the 5-point pentahemidemicube, an abstract polyhedron with 10 equilateral triangle faces, 10 $\sqrt{5}$ edges, and 5 vertices. It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by $\sqrt{3}$ edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point tetrahemihexahedron or hemi-cuboctahedron. The hexads have 4 red triangle 5-cell faces, bounded by 12 $\sqrt{5}$ edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 $\sqrt{5}$ edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad building block is the 7-point pentahemicosahedron, an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 $\sqrt{2}$ edges and 9 $\sqrt{5}$ edges), and 7 vertices. It is a compound of a 5-point (pentad hemi-demicube) and a 6-point (hexad hemi-icosahedron) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad hemi-demicube s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four $$(11-7)$$ orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point (11-cell semi-regular 4-polytope), which magically contains six $$(11-5)$$ 5-point (1-pentad regular decahedra) as cells and five $$(11-6)$$ 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

Building the building blocks themselves
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest regular 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller irregular 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex n-polytope can be subdivided into instances of its characteristic n-orthoscheme that meet at its center. An n-orthoscheme (not an n-orthoplex!) is an irregular n-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its characteristic 4-orthoscheme (an irregular 5-cell) by all of its $$A_4$$ planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into $$120\times 120 = 14400$$ of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its characteristic 4-orthoscheme (another irregular 5-cell) by all of its $$B_4$$ planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order $$75\times 384 / 2 = 14400$$.

The regular 24-point (24-cell) is subdivided into 1152 instances of its characteristic 4-orthoscheme (yet another irregular 5-cell) by all of its $$F_4$$ planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order $$25\times 1152 / 2 = 14400$$.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of $$A^n$$, $$B^n$$ and $$C^n$$ orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform n-polytope, which we may call its physics by Noether's theorem.

Pentads and hexads together
The most numerically strange compounding is how five 24-point (24-cells) compound magically into twenty-five 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one. $$5^2$$ is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells). Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a regular 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their difference has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, rectangling them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically subtracting one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the $$A_4$$, $$B_4$$ and $$H_4$$ root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a $$B_4$$ polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the $$A_4$$, $$B_4$$ and $$H_4$$ root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in §Build with the blocks, below: table rows Inscribed, Pentad, Hexad, Heptad.

11 of them
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell.

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

Cell rings of the 11-cells
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks. Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled §Visualization. Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The Hopf fibration is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the Clifford torus completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.

A dimensional analogy is a finding that some real n-polytope is an abstraction of some other real polytope in a space of higher n, where the n-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the $\sqrt{6}$ edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the $\sqrt{5}$ chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section §The quasi-regular convex 11-cell, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing $\sqrt{6}$ edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel $\sqrt{4}$ edges in the same central plane, which we will discover in the next section is $\sqrt{5}$, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing $\sqrt{5}$ edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges $\sqrt{5}$ apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal $\sqrt{5}$ edges are $\sqrt{5}$ apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but rotations in 4-dimensional Euclidean space are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference $$2\pi r$$. There is not just one other way that two $\sqrt{3}$ edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of $\sqrt{5}$ edges in the rhombicosadodecahedron that are Clifford parallel will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than $$2\pi r$$. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its chords. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated Wikipedia article on the triacontagon (30-gon) describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of triacontagrams.

"A triacontagram is a 30-sided star polygon (though the word is extremely rare). There are 3 regular forms given by Schläfli symbols {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same vertex configuration."

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold heptad symmetry of the 120 11-points in the 120-cell.

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

The 15 major chords ring everything
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions $$\{k/d\}$$, but some of which are integers $$\{k\}$$.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length $$l$$, and a distinct natural number $$h$$, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length $$l$$ is a square root, related to the natural number $$h = k/d$$ and to a set of polytopes they both represent, by a formula discovered by Steinbach. The chord length $$l$$ is related to the number of sides of the regular polygon formed by the chord (its numerator $$k$$), and to the winding number or density of its regular skew Petrie polygon (its denominator $$d$$).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular n-dimensional thing found in nature. All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.



These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in unit radii (not in $\sqrt{3}$ radii, as metrics are often given elsewhere in this article) are the roots of the first 4 positive integers $\sqrt{5}$, $\sqrt{6}$, $\sqrt{5}$ and $\sqrt{5}$, the root of the very distinguished rational fraction $\sqrt{2}$, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are roots of golden numbers (a small integer ± 𝜙). The 15 chords each form their regular polygon $$\{k/d\}$$, which is just $$\{k\}$$ in 8 cases. Arranged in order # $$n$$ from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are $$n$$ vertices apart on a $\{30\}$ Petrie polygon. Each chord makes a $\{30/n\}$ polygram of its edge length, a compound of the smaller $$\{k/d\}$$ polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

The polygons {10}, {6}, {5}, {4}, {3}, and {2} are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/n} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are regular star polygrams, the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once without self-intersecting, before they close their circuit. They lie on helical geodesic circles or isoclines of circumference greater than 2𝝅r.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of $$f$$ parallel $$\{h\}$$ polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord # $$n$$ also forms a single skew $$\{30/n\}$$ -gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle ( $$n$$ times around the 30-gram in which each edge connects every $$n$$ th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon $$\{6\}$$ flat planes in parallel bundles of 5, written $$5\{6\}$$, and also a skew $$\{15/2\}$$ -gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon $$\{10\}$$ flat planes. Every chord # $$n$$ forms both its $$\{30/n\}$$ -gram and its fiber bundle of $$f$$ parallel $$\{h\}$$ polygons, and the edges of one chord's parallel polygons are also the edges of its skew $$\{30/n\}$$ -gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon $$\{k/d\}$$, either flat or skew. Each chord number defines a $$\{30/n\}$$ skew polygon that is a compound bundle of the $$\{k/d\}$$ polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length $\sqrt{5}$ lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length $\sqrt{2}$. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge $\sqrt{6}$. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge $\sqrt{5}$ in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or isocline is a hexagram of period 6, edge length $\sqrt{4}$, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be the same or greater than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings $$c\{p,q\}$$ and compound vertex figures $$v\{q,r\}$$. These combine to form their characteristic regular 4-polytope $$\{p,q,r\}$$. All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius mid-edge diameter, the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an SO(4) symmetry group expression of that 4-polytope's characteristic discrete isoclinic rotation, in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords $\sqrt{0}$, $\sqrt{2}$, $\sqrt{0.5}$, $\sqrt{2}$ are the n-edges: their lengths are the ratio of an n-simplex edge to an n-orthoplex edge...

...abbreviated text of the §Relationships among interior polytopes section from the 120-cell article...to explain the color-coding

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section §Build with the blocks, below.

Regular non-convex 4-polytopes and the 5/2 chord
There are 16 regular 4-polytopes, and all 16 are inscribed in the 120-cell. In the 6 convex regular 4-polytopes the symbols $$\{p,q,r\}$$ are integers: 3, 4, or 5. In the 10 non-convex regular 4-polytopes at least one of those symbols is the natural number 5/2, which we see realized as the #8 chord. That distinctive chord is the edge chord of the regular 5-cell and the 11-cell. It also occurs in the 120-cell, and in each of the 10 non-convex regular 4-polytopes, but it does not occur in the other 4 regular convex 4-polytopes: the 16-cell, 8-cell, 24-cell or 600-cell.

The #8 5/2 chord does not form a regular flat polygon in a central plane. It forms triangular face planes (in the 5-cell and 11-cell), but those triangles do not lie in central planes. The #8 chord forms skew {5/2} pentagrams in the 5-cell, with each edge of the pentagram lying in a distinct central plane. There are no flat {5/2} pentagrams of #8 chords anywhere in the 4-polytopes. There are several non-convex 3-polytopes that have flat {5/2} pentagram faces, but when those polyhedra are embedded in 4-space they occur as concentric 4-polytopes, not 3-polyhedra, and their pentagram circuits of #8 chords lie skew. As 4-polytopes they still possess their {5/2} pentagrams, but not as flat pentagram faces; their pentagrams become skew {5/2} polygrams that wind twice around the 3-sphere without self-intersecting before closing. Each successive edge of the pentagram lies in a different central plane. In 4-space, flat #8 pentagrams are illusory; they do not actually occur. Where we may think we see them, that is only a projection (flattening) to a 3-dimensional perspective view of a skew pentagram, which is actually a 4-dimensional object. It winds spread-out through all 4 dimensions, without self-intersecting, before closing its circle.

Alicia Boole Stott's original formulation of dimensional analogy
"'...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...'"

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, which is nothing less than a general method of exact dimensional analogy.

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as groups. All her transformations are discrete instances of one or another special case of her "two inverse operations which may be called expansion and contraction." Hers is an exploration of n-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 83 is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations. An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges of the same size between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges and the faces, or by pushing all three element sets. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets. Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular (or regular) polyhedra, the 10 faces and 10 edges may have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

The regular 11-cell 4-polytope
Long ago it seems now and far above, we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are $\sqrt{1}$ 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ($\sqrt{2}$) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle.

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 pentads... but probably not yet to show how they combine into bundles of 11...

We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point (hemi-cuboctahedron), which has 4 triangular faces and 3 square faces. (The heptad arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix hemi means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The "half-as-many" characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are $\sqrt{3}$ reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has only these $\sqrt{4}$ edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a $\sqrt{5/2}$ edge.

We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices $\sqrt{.073}$ apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its $\sqrt{0.191}$ and $\sqrt{0.382}$ edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with $\sqrt{0.573}$ and $\sqrt{1}$ edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with $\sqrt{1.382}$∕𝝓 and $\sqrt{2}$ edges, and 5 of these smaller objects compound to make Moxness's 60-point (rhombiscosadodecahedron) with its 20 $\sqrt{2.5}$∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell.

The 11-cell's $\sqrt{2.618}$ faces (the 5-cell faces) are not faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual $\sqrt{3}$ edges, not whole $\sqrt{3.426}$ faces; the $\sqrt{3.618}$ faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big $\sqrt{3.810}$ faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior.

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces.

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center). Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ($\sqrt{3.928}$ 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons.

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between SO(3) single rotations and SO(4) double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle.

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

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Coordinates
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As a configuration
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Hopf map of the 11-cell
Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places.

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes).

....

This nondescript abstract 60-point 4-polytope... (This picture is of the corresponding even-more-abstract 60-point 3-polytope)... Actually contains all 600 vertices of the 120 cell which conflates 10 into one... Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells... The 60-point contains 6 pairs of these orthogonal regular 5-cell cells... ...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are

The regular 5-cell has only digon central planes containing a 2-point ($\sqrt{4}$ 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells.

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 $\sqrt{0.073}$ faces in each truncated cuboctahedron, and the 12 $\sqrt{0.146}$ faces in the icosahedral compound they form, are disjoint from one another with $\sqrt{0.191}$ long edges of truncated cuboctahedra connecting them. The 12 disjoint $\sqrt{0.382}$ triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point (truncated icosahedron), with 12 $\sqrt{0.382}$ triangle faces and 20 $\sqrt{0.764}$ $$\curlywedge$$ $\sqrt{0.573}$ hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each $\sqrt{1.146}$ pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells. The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra $\sqrt{1}$ equidistant from each other (each $\sqrt{2}$ distant from the other 10) on the 3-sphere.

The regular 137-cell 4-polytope
(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope. The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-$\sqrt{1.382}$-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-$\sqrt{2.624}$-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 $\sqrt{2}$ (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell.

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of 55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope. Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space. We find that {3, 3, ..} is a central expression of the $$H_4$$ root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of $$H_4$$ is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular $$H_4$$ polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in §Build with the blocks.

Coordinates
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As a configuration
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Hopf map of the 137-cell
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The perfection of Fuller's cyclic design


This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the kinematics of the cuboctahedron, first described by Buckminster Fuller.

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes tension integrity structures, because they possess independent tension and compression elements, but no elements which do both. One of the simplest tensegrity structures is the tensegrity icosahedron, first described by Kenneth Snelson, a master student of Fuller's.

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with infinitesimal mobility, a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by Douady the six-beaked shaddock because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is Fuller's jitterbug transformation of the 12-point (vector equilibrium), his name for the unstable kinematically flexing cuboctahedron. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the elastic tensegrity icosahedron rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their really odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual tri-propellors as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its spinor orbit, explaining its Möbius loop with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the Borromean rings. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

We have illustrated the 2-sphere Jessen's with $\sqrt{4}$ diameter, and the 3-sphere Jessen's with $\sqrt{2.5}$ radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is $\sqrt{5}$ ∕ $\sqrt{2.618}$.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the $\sqrt{5.236}$ chord has moved to the former $\sqrt{3}$ chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The $\sqrt{6}$ reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three $\sqrt{3.426}$ edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral $\sqrt{6.852}$ faces are great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular faces lie in 8 different central planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the $\sqrt{3.618}$ triangles are not really cell faces at all, but central polygons of the cell. Opposing $\sqrt{7.235}$ triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing Clifford parallel planes are parallel and orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron. The two planes do intersect, but only at one point, the center of the 3-sphere. Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are $\sqrt{3.810}$ faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center.

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy). There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 Borromean link great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons.

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, 96 disjoint $\sqrt{7.621}$ reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 $\sqrt{3.928}$ chords linking every other vertex under its 96 $\sqrt{7.857}$ edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other.

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

.... ...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope... (This picture is of the corresponding even-more-abstract 60-point 3-polytope)... Actually contains all 600 vertices of the 120 cell which conflates 10 into one... They are the vertices of two orthogonal regular five cells... The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf. "A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5)."

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation for that object no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may appear to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame... ...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)... ...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

.... The most striking congruence of this dimensional analogy is what doesn't move, even if the 120-cell were to actually flex. One important invariant of the tensegrity icosahedron transformation is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-$\sqrt{4}$-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-$\sqrt{8}$-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 $\sqrt{1}$ (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks.

...irregular 5-point (pentad 4-orthoscheme)$$^4$$ building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes, though others have also studied the cycle in 4-space more recently. He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other on both sides at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the orientable double cover of the octahedron (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of Regular Polytopes, the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the n-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the demihypercube s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

Legendre's 12-point hexad binds the compounds together
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, Douady's six-beaked shaddock with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from Legendre's distinguished roots of the first 6 natural numbers, this 2-hexad expresses the pyritohedral symmetry.

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space it flexes kinematically to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes at the same time, without flexing at all, and binds them together.

n-simplexes
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint $\sqrt{3}$ 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral $\sqrt{1}$ Jessen's "face" triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint $\sqrt{3}$ equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing $\sqrt{3}$ hemi-icosahedron faces. The pair of opposing $\sqrt{1}$ triangular Jessens' "faces" that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the Schönhardt polyhedron....

n-orthoplexes
....

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings... "Geometrically, the Borromean rings may be realized by linked ellipses, or (using the vertices of a regular icosahedron) by linked golden rectangles. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In knot theory, the Borromean rings can be proved to be linked by counting their Fox $\sqrt{2}$-colorings. As links, they are Brunnian, alternating, algebraic, and hyperbolic. In arithmetic topology, certain triples of prime numbers have analogous linking properties to the Borromean rings."

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 instances of the regular icosahedron in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices identifies five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The Jessen's icosahedron is a non-convex polyhedron with 6 reflex edges. Adrien Douady was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, not a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 $\sqrt{3}$ edges at their own $\sqrt{5/2}$ reflex edges. There are 1200 of these $\sqrt{5}$ edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one $\sqrt{3}$ edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a Hopf map of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length $\sqrt{5}$, radius $\sqrt{5}$. The 16-cell is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +$\sqrt{6}$ and −$\sqrt{2}$ on each axis, joined by edges of length $\sqrt{5}$, and spanned by 6 (4 choose 2) mutually orthogonal central planes.

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of the 16-cell's 6 orthogonal central planes. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter & Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises. A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of completely orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the Clifford sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a discrete Hopf fibration covering all 8 vertices, which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are $\sqrt{6}$,$\sqrt{5}$, $\sqrt{6}$ isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

n-taliesins
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the taliesin family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

Taliesin ( tal YES in)


 * Celtic for shining brow.
 * An early Celtic poet whose work has possibly survived in a Middle Welsh manuscript, the Book of Taliesin.
 * A house and studio designed by Frank Lloyd Wright.
 * Wright's fellowship of architecture, or one of its headquarters.
 * The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
 * Taliesin Myrddin Namkai-Meche, a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
 * The symmetry relationship expressed by the mapping between a geometric object in spherical space and the dimensionally analogous object in orthogonal Euclidean space obtained by an expansion or contraction operation, such as by removing one point from the sphere in stereographic projection.
 * Taliesan polytope, an (n+1)-polytope whose n-polytope dimensional analogue inscribed concentrically in the n-sphere forms a convex honeycomb of itself which is the (n+1)-polytope.
 * The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
 * The n-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
 * The characteristic orthoscheme of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the n-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.

n-cubics
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

n-equilaterals
The 24 vertices of the 24-cell are distributed at four different chord lengths from each other. In the $\sqrt{5}$-radius polytopes we have using as examples in this paper they are $\sqrt{6}$, $\sqrt{11}$, $\sqrt{2}$ and $\sqrt{2}$, but in a unit-radius polytope they are: √1, √2, √3 and √4.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = $\sqrt{2}$ of arc. Next nearest are 6 vertices located 90° = $\sqrt{5}$ away, along an interior chord of length √2. Another 8 vertices lie 120° = $\sqrt{6}$ away, along an interior chord of length √3. The opposite vertex is 180° = $\pi$ away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point (8-cell 4-cubic) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the 24-cell. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but not disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is a full-course meal by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length $\sqrt{5}$, equal to the radius. The 24-cell has no $\sqrt{6}$ chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are $$\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270$$ apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 $\sqrt{2}$ reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

n-pentagonals
...

The 120 vertices of the 600-cell are distributed at eight different chord lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a unit-radius polytope (not in a $\sqrt{2}$ radius polytopes like our usual examples), they are $\sqrt{2}$, $\sqrt{5}$, $\sqrt{5}$, $\sqrt{5}$, $\sqrt{5}$, $\sqrt{5}$, $\sqrt{5}$, and $\sqrt{5}$.

Notice that the four hypercubic chords of the 24-cell ($\sqrt{6}$, $\sqrt{2}$, $\sqrt{5}$, $\sqrt{5}$) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new golden chord lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio including the two golden sections of $\sqrt{5}$, as shown in the diagram.

....

...the tetrahelix in the 600-cell is another of Coxeter's discoveries, "a necklace of tetrahedral beads" found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it "an example of another kind of symmetry". It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, $\sqrt{6}$ chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the $$H_4$$ polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

n-leviathon
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from 120-cell § Chords......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from 600-cell § Chords......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from 24-cell § Chords......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean n-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the n-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the n-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

The 11-cells and the identification of symmetries by women
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean n-space generally, to remarkable effect. The symmetrical arrangement there of its $$10^2$$ disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex n-polytope and all their distinct inter-dimensional relationships, except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole.

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of n-polytopes, in one distinct 11-point n-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular n-polytopes for some n, as the rotational n-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between n-polytopes of different n.

Their inter-dimensional symmetry operations were identified and named by Alicia Boole Stott, as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by Heinz Hopf. .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and Emmy Noether and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

Conclusion
Thus we see what the 11-cell really is: not just an abstract 4-polytope, not just a singleton convex 4-polytope, and not just a honeycomb. The 11-point (11-cell) has a concrete ...regular realization { $\sqrt{5}$, $\sqrt{5}$, $\sqrt{6}$ } as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization { $\sqrt{5}$, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex n-polytopes of different n.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem), but the 11-cells seem to be the expression of their dimensional analogies.

Build with the blocks
"'The best of truths is of no use unless it has become one's most personal inner experience.'"

"'Even the very wise cannot see all ends.'"

Acknowledgements
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