Studies of Euler diagrams/medusa

This Boolean function was initially found by adding another intersection to the inflated function (shown on the right), creating an Euler diagram with an unusual geometry (which could be described as a torus). Each pair of circles overlap, so no two sets are disjoint or in a subset relationship. It turned out to be difficult to verify, that this function does not require gapspots.

2D
This Boolean function does not have a good 2-dimensional Euler diagram. Both attempts shown have duplicates of cell 0 and its yellow and red borders.

3D
The representation with spheres is not really useful. It is shown here as a step in the search for a good Euler diagram - which started with the circles, and ended with the curved planes. (Only the front side of the spheres is shown, because the full version contains duplicates of the two vertices and the edge between them.) This object has 13 cells, 22 faces, 12 edges and 2 vertices. The edge between red and yellow is separate, and the others are.

filtrates
Each of the 6 pairs is a 2-split (with all 4 quadrants). 2 of the 4 triples are 3-splits (with all 8 octants). (See also here.)

gap variants
Spot 3 is on its own an octant of the lower 3-split. Removing it causes this 3-split to disappear. The resulting function Studies_of_Euler_diagrams/farofe still has the other one, and thus also requires an Euler diagram with 3 dimensions. Removing spot 2 as well, causes also the upper 3-split to lose its bottom front octant, so the resulting function has a 2-dimensional Euler diagram. Removing spots 0 and 2 creates the new gapspot 11, which is on its own an octant of the new 3-split ABD. Theoretically the resulting function has a 3D diagram, but no properties are lost in the 2D projection shown.

gapspots
One could perceive 11, 13 and 15 as gapspots - which would mean, that the Euler diagram is the Venn diagram with all 16 intersections.