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My Favorite Shapes!

Duocylinder

There aren't many good visualizations of this shape as it is incredibly tricky to render. It's smooth but also has solid parts. Most visualizations you can find online look like a clifford torus or just a cylinder. I think the best way to understand this shape is to look at slices of it, so I have embedded a desmos graph I made of the duocylinder. The duocylinder is the cartesian product of two discs. So its slices look like a cylinder that goes up and down in height like a circle. If you rotate it, it'll smoothly and satisfyingly morph into a cylinder on its side. Basically my favorite shape.



24-cell / Icositetrachoron / {3, 4, 3}

The 24-cell is a very common favorite amongst 4D addicts. It has octahedral sides, the only platonic solid to do so. It's the 4D equivelant of a rhombic dodecahedron, the shape formed by taking a cube and attatching pyramids to its sides and setting their height to make the sides of the pyramids co-planar. That shape is n-dimensional, and 4 dimensions is the only dimension where the shape is regular which is super cool. It's vertex figure is the cube and its self dual. All rhombic dodecahedrons tile space, how cool is that?!

It's also a 4D cuboctahedron. This is because its the rectified 16 cell, and has cuboctahedron slices. What's interesting is that the tesseract, the dual of the 16 cell, doesn't rectify to a 24 cell. In 3D, a shape and its dual rectify to the same thing. Ie, if you rectify a cube and rectify an octahedron, you get the cuboctahedron. But in 4D this is not true. The reason it is true in 3D is that when you rectify a shape, its sides become the rectified version and the verticies turn into sides as the vertex figure. But the rectified version of a 2D regular polygon is the same polygon but rotated. So a cube still has square faces, but gains triangular faces after rectification. And an octahedron still has triangular faces, but gains square faces after a rectification. I hope I explained that well. So why doesn't this hold true for 4D? Well obviously a 3D shape isn't the same after rectification, so the sides of the 4D polychoron aren't the same. So lets look at the tesseract. When we rectify it, its cube sides turn to rectified cubes (cuboctahedrons), and its vertex figure becomes some of the sides. So the rectified tesseract has cuboctahedron and tetrahedron sides. Remember, when a shape rectifies, all of its sides rectify. So with the 16 cell (the dual of the tesseract) the tetrahedral sides rectify into octahedrons, and the 16-cell's vertex figure (the octahedron) appears. So I hope you now understand why the 24 cell is a rectified 16 cell.

Simple rotating 24 cell - Double rotating 24 cell - Rectified Tesseract/8-cell - Rhombic dodecahedron tiling - 24 cell tiling (incomprehensible)




90 Degree Pentagon

In hyperbolic geometry, regular polygons can have any internal angle. So a pentagon can be made from 5 right angles, and as such it tiles the hyperbolic plane. So its kind of the hyperbolic equivelant of a square, though that's a silly thing to say as squares exist in hyperbolic space. (Though their internal angles are always less than 90 degrees, so if you care more about internal angles, then yes this pentagon is truly the closest to a square you're gonna get in hyperbolic space.)

16-cell / Hexadecachoron / {3, 3, 4}

The 16-cell used to be my favorite polychoron before I learned about the 24-cell. The 16 cell is an n dimensional shape, meaning it exists in every dimension. It's called the "orthoplex" or "cross-polytope". The 3-orthoplex is the octahedron. I think octahedrons are cool looking (probably because of ramiel) and this is the 4D octahedron so I like this one even more. The 16 cell has 16 tetrahedral sides and has 8 points, positioned at plus and negative one along each axis. It is the dual of the tesseract.



Torinder

The torinder is a cylinder revolved with an offset and a torus prism. It has cool slices. My favorite being the one where it rotates, flipping between a square torus and a circle torus. So satisfying. By far the simplest 4D torus. Closest 3D analogue is the square torus, though this is different as it is still kind of circular, just not fully like the spheritorus. The closest 4D analogue to the square torus is of course the cube torus. That's the thing with analogues, there's more shapes in higher dimensions so things don't translate too well and shapes often fork into multiple higher dimensional interpretations.



Cube Pyramid

The cube pyramid is an irregular polychoron formed by taking a cube and extruding it to a vertex. It has 6 square pyramid cells and 1 cube cell. It has two properties, base edge length and pyramid height. It's my favorite irregular polytope.



Dodecahedron Prism

I love dodecahedra. The fact that pentagons can form a polyhedron is super cool, and its regular no less. This shape is awesome and when it gets extruded into 4D, its even cooler. I used my 4D wireframe renderer to make the gif below. I even added it to the official dodecahedral prism wikipedia article! I don't have much more to say about this shape. I just like it.



Summary

Shapes are cool. As a geometry addicted nerd, I can't get enough of them. Thanks for reading my silly webpage, I put a lot of time and effort into creating it. I will definitely update this page with more cool shapes.









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