r/Geometry

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).

As you can see, my answer was SQ with an arrow above it pointing to the left, and SR with an arrow above it pointing to the left. My understanding is that S must be included in every ray because it's at the end of the line. RQ with an arrow of the left would not be a solution because S is not included. But the book I'm using, Geometry: A Self-Teaching Guide, says that the solution is SQ with an arrow above it pointing to the left, and RQ with an arrow above it pointing to the left. Is the book wrong? Thanks.

[supprimé]

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 discovered a new class of shapes while developing my new generative lamp sculpture technique

Hey everyone!

I have a unique design technique that I have been cultivating for four years now. The technique involves specific types of patterns, but it is basically extruded patterns profiled to a certain volume (shape). The effect delivers very pleasing light diffusion, and it is quite dynamic in terms of its potential.

Well, I seem to have broken through on that potential!

To make a long story digestible, after i graduated I decided to try and build my design workflows for different assembly techniques (I had only done stacking variations previously). I wanted to use polyhedron to make pendants, and because my technique builds profiles off of surfaces, I made it so the base of each module is the face on a polyhedron, with the profile of the extrusions creating the new shape, once all of the modules are assembled. Essentially, the new shape is determined by the existing properties of the seed shape I use. Each face's neighboring faces determine the kleetope that is produced, using an algorithm i coded which employs ray-point averaging.

It is similar to stellateing or greatening in terms of geometric terminology, but it is much more dynamic. The transformation works on every single convex polyhedron, and produces many incredible results. A few of them match the stellated versions of the seed shape, like the dodecahedron for example, but the majority of convex polyhedron get transformed into a brand new shape when using my algorithm.

Now, to be clear, the video is not the transformed shapes themselves. They have the profile of the shapes, but are artistic abstractions that employ my detailing technique. Also, my script allows for me to customize the designs with a lot of control, so I can actually stray away from the default shape for the sake of my artistic practice. Only two of the images shown do that though, as the rest match the default transformed profiles of their seed shapes.

The last thing I will say is, I have not made an official publication to any journal yet, so i technically cannot claim any discovery yet. However, that is because there is a team of mathematicians at Georgia Tech building a comprehensive publication piece. I am in communication with a professor who is having a group of PhD students develop the publication over this summer, but they have told me that they have confirmed it is a new transformation that creates a genuinely important new class of shapes! If you want a little proof, here is the doc i sent the professor that started this all. BTW, the more unique shapes are on the way. I started off with some simpler shapes to hone in on the assembly process.

I honestly don't know what impact this will have on me, but I hope the publication can bring some attention to my work. I really want to keep designing full time, and I am having to work part-time restaurant jobs to fund this passion.

If you want to support me, or print some of these yourself, check out the links on my page. I hope you guys appreciate my work!

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

Prove that:

The surface of A is equal to the surface of B, C, and D.

And

The surface of A, B, C and D can be written as x² cm/inches with x greater than 0.

To make it easier, you can send me pictures of your math and I will tell you if it is correct.

OC by Kiwimagobluwe. If you really want to repost, at least upvote.