Studies of Euler diagrams/criteria

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This project aims to automatically find Euler diagrams 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.

completeness of spots

A fullspot is a true bit of the Boolean function.
Every fullspot must be represented by a cell.

completeness of links

A link is the connection between neighboring spots. In an Euler diagram it is a wall between two cells.
Every pair of fullspots with a Hamming distance of 1 must be represented by a wall between cells.

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

contiguousness of spots and links

A fullspot should be represented by exactly one contiguous cell. A link should be represented by exactly one contiguous wall between cells.

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

contiguousness of borders

All links corresponding to the same atom should form one contiguous border.

verote
In this diagram of verote the border of D is not contiguous.   (Anyway, this is probably the most practical way to represent this function.)

gufaro
In this diagram of gufaro all borders except E, B and D are not contiguous.

torus
The borders on this torus surround eight contiguous cells, but are themselves not contiguous.

connectedness

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

kukobo
A gapspot must be inserted to connect the three fullspots. kukobo

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

incrementality

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

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

non-arbitrariness

Arbitrary choices should generally be avoided. Spots that have the same properties in a Venn diagram or hypercube should also have the same properties in the Euler diagram.
This is mostly about the choice of gapspots, but not only (as the vidita example shows.)

enlarged multicrossings
Crossing of multiple borders could be enlarged into arbitrary gapspots. foravo rudege logota

vidita

The middle image below shows a 2D diagram of vidita 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. gapspots multicrossings 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. gapspots   (same as the image above) only one gapspot only one multicrossing

niliko
Ideally non-arbitrariness should extend not only to what is shown, but also to how it is shown. These two diagrams of niliko show exactly the same, but the arbitrary layout on the left hides the symmetry seen on the right.