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.

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[edit | edit source]

Every fullspot must be represented by a cell. Every pair of fullspots with a Hamming distance of 1 must be represented by a wall between cells.

uniqueness of segments[edit | edit source]

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

connectedness[edit | edit source]

spots[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.)

links in the same border[edit | edit source]

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

other segments[edit | edit source]

incrementality[edit | edit source]

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

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

non-arbitrariness[edit | edit source]

Arbitrary choices should generally be avoided.

cells and border crossings[edit | edit source]

enlarged multicrossings

Crossing of multiple borders could be enlarged into arbitrary gapspots.

symmetry[edit | edit source]

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

niliko

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