Studies of Euler diagrams/criteria
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[edit  edit source]
A fullspot is a true bit of the Boolean function.
Every fullspot must be represented by a cell.
completeness of links[edit  edit source]
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[edit  edit source]
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[edit  edit source]
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[edit  edit source]
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. 
XOR  

Functions of this kind have no links between fullspots. Gapspots must be inserted, to make sure that they are all connected. 
incrementality[edit  edit source]
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. 
nonarbitrariness[edit  edit source]
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.

vidita  

The middle image below shows a 2D diagram of vidita with crossings of multiple borders. (There is also a 3D version without.) The representation with gapspots is not too bad — although somewhat arbitrary, as shown above. 
niliko  

Ideally nonarbitrariness should extend not only to what is shown, but also to how it is shown. 