Studies of Euler diagrams/criteria

This project aims to automatically find s that meet the criteria listed below.

When the dimension required for the perfect Euler diagram is too high, one can relinquish some criteria to find a useful diagram.

Only the completeness of spots and links should be considered essential.

The following terms are used below:


 * spot: cell of an Euler diagram, can be a fullspot (true) or a gapspot (false)
 * link: connection between neighboring spots (in an Euler diagram the wall between two cells)
 * border: all links belonging to the same set (all walls of the same color)
 * segment: generalization of spots and links, including crossings of borders

completeness of spots and links
Every fullspot must be represented by a cell. Every pair of fullspots with a of 1 must be represented by a wall between cells.

has 22 links between fullspots. But in this diagram the wall between cells 0 and 4 is missing.

uniqueness of segments
Each segment should have exactly one contiguous representation in the Euler diagram. Especially spots and links.

In this diagram of the spot 0 is represented by two cells. Each of the links (0, 1) and (0, 8) is represented by two walls.

In the left diagram of the crossing of borders A and B appears twice.

spots
All fullspots must be connected by links. This can require the insertion of gapspots. (The of the Euler diagram must be a .)

A gapspot must be inserted to connect the three fullspots.

Functions of this kind have no links between fullspots. Gapspots must be inserted, to make sure that they are all connected.

links in the same border
All links corresponding to the same atom should form one connected surface.

In this diagram the border of D is disconnected. (Anyway, this is probably the most practical way to represent this function.)

In this diagram of all borders except E, B and D are disconnected.

Each border on this surface has two disconnected halves.

other segments
This diagram of has two disconnected crossings of B and C. If A and B were represented by 3D paraboloids instead of 2D parabolas, they would intersect in a ring, whose edges would be connected. That would create gapspots (outside of the ring, between planes A and B), so this would be the 3D Euler diagram of.

incrementality
Every link changes exactly one bit. (Thus every link corresponds to one atom.)

This means that sometimes gapspots must be inserted between fullspots, to make sure they are all connected.

Between other segments incrementality is not required. This way multicrossings are allowed, which are desireable for functions like (hexagon) or  (octagon). (They could be drawn with incrementality between all segments, but that would require arbitrary gapspots.)

This diagram of contains a multicrossing of borders A, C and D. This means that e.g. the two blue walls meeting in this point differ both in A and in D.

non-arbitrariness
Arbitrary choices should generally be avoided.

cells and border crossings
Crossing of multiple borders could be enlarged into arbitrary gapspots.

The middle image below shows a 2D diagram of with crossings of multiple borders. (There is also a 3D version without.) Just like in the examples above, they can be arbitrarily enlarged. Moving A and D to the outside creates the gapspots 1 and 8. Moving them to the inside creates a separated cell 6.

The representation with gapspots is not too bad — although somewhat arbitrary, as shown above. Cropping a diagram in a way that takes away symmetry is a much worse arbitrary choice. The asymmetrical diagrams below are crops of their symmetric equivalents. That with one gapspot is a crop of the diagram with both gapspots. That with one multicrossing is a crop of the diagram with both multicrossings.

symmetry
Ideally, non-arbitrariness should extend not only to what is shown, but also to how it is shown.

The diagram of should have mirror symmetry, but the one on the left has no symmetry.

The 3D diagram of is topologically more sound, but lacks the rotational symmetry of the 2D diagram.

On the left all sets are represented by the area within a circle, resulting in a diagram with only mirror symmetry. On the right the blue set is represented by the area outside of the red circle, allowing a representation with symmetry between inside and outside.


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The diagram on the left shows only the mirror symmetries  and. The full symmetry is shown in a rare example of an Euler diagram with dihedral and.


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