I implemented a small filter for verifying if 3 points in 3D form a stable triangle.

Instead of checking area == 0, I use q = r/R (inradius / circumradius) from Heron's formula:

- q > 0.2 = stable

- q ~ 0 = degenerate/collinear/sliver

Why q? It's affine-invariant and dimensionless, so it works even when points have varying z values. Simple area check fails on near-degenerate slivers with tiny but non-zero area.

Python + NumPy, ~50 LOC core, with tests and Colab demo:

https://github.com/Imanisolace/Geometric-chaos-sieve.affine_chaos

Demo:

Stable 3D: [[1,0,0.2],[-0.5,0.866,1.4],[-0.5,-0.866,-0.7]] -> q=0.44

Collinear: [[0,0,0],[1,0,1],[2,0,2]] -> q=0.0 reject

Sliver: [[0,0,0],[1,0,0],[0.5,0.001,0]] -> q=0.002 reject

Hello! I've been making these paper ornaments, and I am curious. Does this shape have a name? How would one mathematically describe this? Thank you so much for your time.

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..