Studies of Euler diagrams/criteria

dummy

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

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.

medusa
medusa 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.

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.

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

connectedness

spots

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

links in the same border

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

sopuda
In this diagram the border of D is disconnected.   (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 disconnected.

torus
Each border on this toroidal surface has two disconnected halves.

other segments

nosafu
This diagram of nosafu 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 medusa.

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

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.)

rudege with multicrossing
This diagram of rudege 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

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

symmetry

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.

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

burito

sopuda

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. This code uses the Python library discrete helpers. from discretehelpers.boolf.examples import sopuda assert sopuda.symmetric_spots.blocks_with_singletons() == [[0, 1, 2, 5, 6, 7], [3, 4], [11, 12]]

veneto

The diagram on the left shows only the mirror symmetries [1, 2] and [13, 14]. The full symmetry is shown in a rare example of an Euler diagram with dihedral and translational symmetry. This code uses the Python library discrete helpers. from discretehelpers.boolf.examples import veneto assert veneto.symmetric_spots.blocks_with_singletons() == [[0, 3], [1, 2], [4, 7, 8, 11], [5, 6, 9, 10], [13, 14]]