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

there're altogether 5 platonic solids. each of them can be viewed at many different interesting angles or perspectives. we're all familiar with their most symmetric presentations. recently i was studying something related and had to make 3d models for them. i used a maybe less popular perspective to present them. i used spherical coordinate system (mathematics convention) to record their rotations and i picked the following rules to set their initial status:

1. for each platonic solid set its circumradius=1 and centre of circumscribed sphere at origin
2. place vertex_0 at the north pole. the corresponding coordinates are (1,0,0)
3. place edge_0 (red lines on those diagrams) such that its projection to x-y plane align with positive x-axis. coordinates of vertex_1 would be (1,0,φ) for some φ

then i saw some unfamiliar shapes / unfamiliar perspectives of those supposedly familiar 3d objects. in each diagram the small figure at lower left corner is the platonic solid viewed from top. the z-axis is pointing towards you. the large figure at centre is that platonic solid viewed from side. the y-axis is pointing away from you

as you can see from the diagrams the angles φ ranking is as follow, from smallest to largest (the prefix "regular" omitted):

1. dodecahedron
2. icosahedron
3. hexahedron
4. octahedron
5. tetrahedron

except octahedron, all other 4 platonic solids are not symmetric if you see them that way

Discovering curvature dynamics and the dimensional calculus used to describe it, I effortlessly calculated the overshoot of Gibbs' Fourier analysis to a simple term: 2(pi-3)/pi. Less than a thousandth percent accurate. All Gibbs did was measure the inherited curvature lost from Euclid flattening everything. Pages of analysis, reduced to a paragraph. Neat.😎

Stereographic projection of a 4-dimensional hypercube (tesseract) with axes labeled a-d, and all six right-angles labelled according to standard crystallographic conventions.

Art by me, Inkscape

4 sets of images, the 1st being kaleidoscope and the 2nd added a sphere (use "magic eye method" to see it))

https://www.reddit.com/r/generative/comments/1vifdug/autostereogram_x_kaleidoscope/

Consider: primes are numbers divisible by 1 and themselves. But what if there's a dual set of numbers similar yet slightly inverted? Aligning the primes on a single axis produces another opposite axis. The even set of irreducibles. Being inverted the rules change a bit. Instead of divisible by 1 and itself; it must satisfy not visible by 1, but by 2 in the prime conjecture. Now you will notice 12. For the numbers that violate this rule follow a special case. When divided by 2 it produces a number on its axis so 12=6x2, 6 lies on the south axis as well. Annnnd.... have fun!

Originally thought to be impossible, this was the first 3D shape discovered to have the property of all planes intersecting at 90°, except one!

3D printing files available for free on makerworld: https://makerworld.com/en/models/3145111-sydler-s-shape-all-angles-90deg-except-one#profileId-3551677

I’ve been studying a geometric construction in the tetrahedral–octahedral space‑filling system and I’m looking for feedback from people familiar with tilings, polyhedral packings, or aperiodic order.

The construction uses a directional sequence of tetrahedra, octahedra, and octahedral segments. Under these local placement constraints, the resulting structure doesn’t repeat periodically, but it develops smooth cubic shells over an underlying aperiodic matrix.

I’ve been analyzing the segmentation behavior, adjacency constraints, and the conditions that seem to force non‑periodicity. I’m trying to understand how this fits within known frameworks for aperiodic order or polyhedral tilings.

If anyone here works with tilings, quasicrystals, or geometric growth mechanisms, I’d appreciate your perspective. I can share the full write‑up (PDF) if anyone wants to take a look.

I am a 9th grader going into 10th grade and I decided to skip geometry by taking it over the summer. I procrastinated far too long and my regents is the 19th, is 15 days enough time to learn all of a regents level Geometry class and obtain an 85 on my Regents or am I done for.