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.
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.
medusa |
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medusa has 22 links between fullspots. But in this diagram the wall between cells 0 and 4 is missing. |
uniqueness of segments
[edit | edit source]Each segment should have exactly one contiguous representation in the Euler diagram. Especially spots and links.
medusa |
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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 | ||
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In the left diagram of gilera the crossing of borders A and B appears twice. |
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.)
kukobo |
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A gapspot must be inserted to connect the three fullspots. |
XOR | ||
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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
[edit | edit source]All links corresponding to the same atom should form one connected surface.
sopuda |
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In this diagram the border of D is disconnected. (Anyway, this is probably the most practical way to represent this function.) |
gufaro |
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In this diagram of gufaro all borders except E, B and D are disconnected. |
torus | |||||||||
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Each border on this toroidal surface has two disconnected halves. |
other segments
[edit | edit source]nosafu |
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This diagram of nosafu has two disconnected crossings of B and C. |
incrementality
[edit | edit source]Every link changes exactly one bit. (Thus every link corresponds to one atom.)
gapspots between bundles | ||
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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 foravo (hexagon) or logota (octagon).
(They could be drawn with incrementality between all segments, but that would require arbitrary gapspots.)
rudege with multicrossing |
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This diagram of rudege contains a multicrossing of borders A, C and D. |
non-arbitrariness
[edit | edit source]Arbitrary choices should generally be avoided.
cells and border crossings
[edit | edit source]enlarged multicrossings | |||
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Crossing of multiple borders could be enlarged into arbitrary gapspots.
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vidita | ||||||
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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. |
symmetry
[edit | edit source]Ideally, non-arbitrariness should extend not only to what is shown, but also to how it is shown.
niliko | ||
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The diagram of niliko should have mirror symmetry, but the one on the left has no symmetry. |
farofe | ||
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The 3D diagram of farofe is topologically more sound, but lacks the rotational symmetry of the 2D diagram. |
burito | ||
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sopuda | ||||
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On the left all sets are represented by the area within a circle, resulting in a diagram with only mirror symmetry.
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veneto | ||||
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The diagram on the left shows only the mirror symmetries
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