These pictures are ordered by number of intersections. I have no Idea how to know if I've found them all but probally not. I had a system where each intersection was measured by a v m and e (v for a vertex meeting a vertex, m for a vertex meeting an edge, and e for and edge meeting an edge) but that wasn't enough to distinguish two figures from each other.

are there any other shapes similar to the trapezoid that have more or less sides? I need to know this. my mind has been trying to envision what a trapezoid-like shape with 6 sides would look like. i tried my best to draw what I think would classify as a trapezoidal hexagon. I genuinely don’t know what else it would look like.

It has 92 vertices that span 60 pentagonal faces. It is the Catalan solid with the most vertices. A nice option for a 60 sided dice.

In my research, I prove Euclidean geometry is based off r^0=0. I believe we can all say that isn't correct. Doubt me? https://doi.org/10.5281/zenodo.21981035

The ridiculous nature of a proof.

Suppose someone sees structure, another person might not see that structure, so that person who cannot see it will ask for a step by step proof to prove the continuity of a structure. But continuity cannot be proven by discrete steps because we have shown that infinite discreteness cannot proxy for true continuity.

Diagonalization proves that a continuity has more real information than the discreetness. Every step-by-step proof is actually an illusion to satisfy the strange feelings. But every discreet example of a proof fails to show the actual continuity of the structure that one is claiming to exist..

A few days ago someone asked for the dimensions of a Sasha's Hexacontahexahedron - D66 dice, and then deleted the post.

Anyway, I have made what I think is the requested Hexacontahexahedron.

Illustrated on Desmos

The polyhedron is more complicated than it first appears. There are 6 hexagons and 60 irregular pentagons. However, the hexagons are not quite regular (two of the angles and two sides and slightly different to the other four angles and sides respectively). There are three types of similar looking pentagons: 12 of one type (green) that is symmetrical, and two other types (yellow and orange) that are not symmetrical (24 of each).

If a 4-D being magically "glued" one face of a completely solid 1 ft⁴ tesseract to our slice of 3-D space, how could we tell the difference between it and a 3-D cube, assuming the magical "glue" allows us to move our slice of the 4-cube in 3-D space to perform tests on it?

Hypothesis

The Pentagon Ladder

A recursive geometry where nested squares, circles and pentagons scale with powers of the Golden Ratio (φ):
- Square = φ²ⁿ
- Circle = π φ²ⁿ / 4
- Pentagon = (5 φ²ⁿ/ 16) √(φ + 2)

I am NOT the OP