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

Building the building blocks themselves
And here at last with the pentad and hexad orthoschemes we must be able to find formulae describing each polytope that relate $$pentads^4 = hexads^4$$, the two equivalent constructions from root systems for every 4-polytope, even for polytopes which are obviously constructed one way, and not so obviously the other. One possible construction is always by pentad orthoschemes, since it is only necessary to construct the polytope's characteristic 4-orthoscheme, and 4-orthoschemes are pentads themselves. The other is by the hexads of the 16-cell, since 16-cells compound into everything larger. Surely every uniform 4-polytope can be constructed by some function of either of these root systems, however indirect the recipe. The expressions to do so then, with an equal sign between them, make a conservation law defining the 4-polytope, which we may call its physics by Noether's theorem. Dc.samizdat (discuss • contribs) 07:11, 26 April 2024 (UTC)

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


 * - Carl Jung

"Even the wise cannot see all ends."


 * - Gandalf

Dc.samizdat (discuss • contribs) 16:46, 29 April 2024 (UTC)

Shlafli's criterion for the regular polytopes
Coxeter discovered and explored the 11-cell and the 57-cell, leading us up to the 137-cell. He would have inevitably discovered that too, if he had lived longer, and I am sure that if he had, this article would be just another Wikipedia article I have edited popularizing Coxeter's Regular Polytopes on the web. I see no possible scenario in which I could have conceived the 137-cell myself, built the first physical model, and presented it to Coxeter even two minutes before he saw it; rather, my sight of it comes to me directly from him, two decades after him. Dc.samizdat (discuss • contribs) 17:14, 1 May 2024 (UTC)

The 137-point
The Great Celebrity arrives at the party, famous for guarding the mystery of her affiliations. She was invited, of course, and we hoped she would put in an appearance, but never dreamed she would arrive so soon. Dc.samizdat (discuss • contribs) 19:17, 1 May 2024 (UTC)

Acknowledgements
I am indebted to Tom Ruen for the original illustration of the Coxeter plane of the 137-cell, which he made during pre-publication review of this paper. Further, this research would never have occurred if Ruen had not originated and illustrated a large series of Wikipedia articles on polytopes. The triacontagon article that he illustrated so completely is the beating heart of this subject, and it was the lavishly illustrated Wikipedia articles on regular 4-polytopes that led me to study them in the first place, and attempt to explain them to myself by adding to the articles. Dc.samizdat (discuss • contribs) 07:17, 4 May 2024 (UTC)

I am deeply indebted to J. Gregory Moxness, first of course for his discovery of Moxness's 60-point (Hull #8) in the 120-cell, and his prior publication of the first image of it captured in the wild, but also personally for his illuminating pre-publication review of this paper, in which he saved me from several egregious errors, about which the less said the better. Second, I am beyond grateful for his splendid transparent renderings of the 11-cell and the 137-cell that he generously made for inclusion in this paper. Moxness and the quaternion graphics software he built are responsible, in two ways, for the gift that we can now see these objects with our own two eyes. Dc.samizdat (discuss • contribs) 19:31, 3 May 2024 (UTC)

Heptad workspace architecture
Since long before covid I have worked at home in my study, a square outbuilding that has one concave corner. I no longer go in to an office to work, but my ideal imaginary office cubicals are irregular hexagons abutting a square closet or structural pillar, and opening onto a shared work area with a round table. Dc.samizdat (discuss • contribs) 13:48, 4 May 2024 (UTC)

Pentads and hexads together
In all three building cycles, in each step where 4-polytope building blocks are joined together to make a larger 4-polytope, an expansion of some 3-polytope embedded at the center of the 3-sphere is taking place, in which the 3-polytope's points are split into 2, 3, 4, or 5 points each and moved apart uniformly, creating a dimensionally analogous 4-polytope. Alicia Boole Stott invented these uniform expansion operations, and explored various distinct forms of them which turn points into edges, edges into faces, faces into cells, and do combinations of more than one of these things. Her goal was to discover new semi-regular 4-polytopes, by starting with the six regular 4-polytopes which were known and performing expansion and contraction operations on them. She began by studying the five dimensional analogies between the five regular polyhedra and their regular 4-polytope analogues, and working out the 3D sections of the regular 4D polytopes, but found many more ways to perform expansions and contractions that paired a regular polytope with a semi-regular analogue. There was even a sixth regular 4-polytope which had no regular polyhedron it was an expansion of: the 24-cell. She found the 24-cell was an expansion of its central section, which is a semi-regular 3-polytope: the cuboctahedron. By working in both directions, she found all 15 Archimedean semi-regular 3-polytopes as contractions of regular 4-polytopes, and discovered their 45 Archimedean semi-regular 4-polytope analogues.

Heinz Hopf discovered the general form of these dimensional analogy expansion-contraction operations, which is that every point in the original 3-polytope expands into a linked great circle polygon in the resulting 4-polytope. Each uniform expansion operation determines a distinct dimensional analogy between some 3-polytope Hopf map and 4-polytope Hopf fiber bundle of linked great circle polygon fibers, but every uniform expansion operation turns every point in the 3-polytope into a polygon in a dimensionally analogous 4-polytope, where the polygons are all disjoint but linked together, each passing through all the others. That, it turns out, is the general nature of uniform expansion operations on the 3-sphere. Of course they can be run backwards, as contraction operations, and sometimes a construction is not possible as a straightforward expansion operation (there is no way to get there by expansion alone) but the dimensionally analogous result can be reached by expanding beyond it, and then contracting to it. There is a complexity that arises in doing so, however, because expansion and contraction operations can be paired in two chiral ways (left-handed and right-handed), since they do not commute. Expanding first and then contracting produces a different result than contracting first and then expanding, for the exact same pair of operations on the same object.

Sequencing the 4-polytopes by complementary chord pairs
See also the elements, properties and metrics of the sequence of convex 4-polytopes summarized in tabular form in the section §Build with the blocks, below. Dc.samizdat (discuss • contribs) 17:33, 23 May 2024 (UTC)

Progression of the natural numbers as polytopes
The natural numbers are described as unit-radius regular 4-polytopes inscribed concentric to the 3-sphere. Except where explicitly described otherwise, polygons are regular great polygons in a central plane, not face polygons; regular polyhedra are inscribed at the center of the 3-sphere, not as cells or vertex figures; and 4-polytopes are one torus of the face-bonded cells of a regular convex 4-polytope, not the complete honeycomb that includes all its cells.

The 0-point {0} has no facets, and no extent. The 1-point {1} unigon has 1 non-facet a $\sqrt{}$ 0-point, and no 0-extent. The 2-point {2} digon has 2 0-facets each a $\sqrt{}$ unigon, and 1-extent $\sqrt{}$ = 2. The 3-point trigon {3} has 3 1-facets each a $\sqrt{}$ digon, and 2-extent $\sqrt{}$ = $\sqrt{}$ = 1.299. The 4-point quadragon {4} has 4 1-facets each a $\sqrt{}$ digon and 2-extent $\sqrt{2}$ = $\sqrt{6}$ = 2. It also has 2 diagonals each a unit-radius digon (an axis diameter) whose 1-extent is equal to that 2-extent. Thus it has 6 digons making an irregular flat tetrahedron with 4 non-disjoint right-triangle faces lying in one central plane.

The 4-point tetrahedron has 4 2-facets each a $\sqrt{}$ trigon and 3-extent $\sqrt{}$; 6 1-facets each a $\sqrt{4}$ digon. The 5-point 5-cell has .... Dc.samizdat (discuss • contribs) 17:44, 28 May 2024 (UTC)

Alicia Boole Stott's original formulation of dimensional analogy
...

If we apply the Boole Stott expansion and contraction operations more generally, by relaxing the requirement that only a single edge length be involved, we can also describe the relationship between polytopes of different edge lengths as instances of expansion or contraction. For example, a unit-edge-length 24-cell can be reached as an expansion of a 16-cell of edge length $\sqrt{}$, where the expansion distance between edges is also $\sqrt{1}$. In 3-sphere space this expansion can be performed in two chiral ways: left- or right-handed. Three concentric 16-cells result, inscribed in a $\sqrt{2}$ edge-length 24-cell, if the expansion is done both ways. Corresponding pairs of their vertices are $\sqrt{2^{2}/1^{4}}$ apart, corresponding pairs of their edges are $\sqrt{3}$ apart, corresponding pairs of their cell centers are $\sqrt{3^{3}/2^{4}}$ apart, and the 16-cell centers are $\sqrt{27/16}$ apart.

Dc.samizdat (discuss • contribs) 17:12, 13 July 2024 (UTC)