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My favorite shapes

There are a lot of shapes. Uncountably many in fact. These are some of my favorites.

Inspired by STOLEN from Tessimal's My Favorite Shapes!

Click to enlarge the images.

Duocylinder

Duocylinders and rotatopes in general are pretty cool, really like their swirly symmetry. The more complex examples can only exist above 3D (the 4D duocylinder being the simplest one). There is one such 5D equivalent, the cylspherinder (circle × sphere), and three 6D equivalents, the triocylinder (circle × circle × circle), duospherinder (sphere × sphere) and the cylglomerinder (circle × glome).

Cross sections of a duocylinder, read from left-right, up-down
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The single Clifford torus face of the duocylinder
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Rotating Clifford torus
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Duocone

The duocone is a VERY underrated shape. It is the pyramid product of two circles, being the result if you place two circles fully perpendicular to each other and then one above the other, therefore being 5D. Its slices are a duocylinder with one circle factor getting smaller and the other getting larger (circle -> duocylinder -> biperpendicular circle). Just like the non-trivial rotatopes, the non-trivial pyramid products (pyratopes?) are also pretty cool.

Diagram of the duocone

Poke sections of the duocone
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Great dirhombicosidodecahedron

The great dirhombicosidodecahedron, or just gidrid for short, is a very complex looking uniform polyhedron. In fact, it is the only non-Wythoffian (cannot be constructed with a standard Coxeter diagram. Under a specific type of filling (binary filling I think?), it is hollow.

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Quasitruncated great stellated dodecahedron

The quasitruncated great stellated dodecahedron, or quit gissid for short, is in my opinion the prettiest uniform polyhedron, even though it is the conjugate (abstractly identical) to the truncated dodecahedron (which I'd say is one of the ugliest, very unequal faces). It has the perfect amount of simplicity and complexity.

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Quartic icositetrahedron and quartic pentecontahexahedron

These two, the quartic icositetrahedron and its dual the quartic pentecontahexahedron, are abstract regular polyhedra. That means that in some weird geometric space (the "best one" being the Klein quartic), they have equal vertices, equal edges and equal faces (well, equal flags). They are a quotient of {7,3} and {3,7}, respectively. That means the former has heptagonal faces and triangular vertex figures while the other has triangular faces and heptagonal vertex figures. Interestingly, they form neat analogs of the dodecahedron and icosahedron, with more stellations ({7,7/2}, {7/2,7}, {4,7/3}, {7/3,4}) and many uniforms under its symmetry group.

Quartic icositetrahedron fundamental domain
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Quartic pentecontahexahedron fundamental domain
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Pseudoheptagon highlighted

Pseudosquare highlighted

Quintic triacosipentecontachoron and quintic tetrachilienacosichoron

Just as the dodecahedron can form the hecatonicosachoron, the quartic icositetrahedron can form its 4D analog: the quintic triacosipentecontachoron, with 350 cells in a tetrahedral vertex figure. Its dual is the quintic tetrachilienacosichoron, with 4900 tetrahedral cells, which is the analog of the hexacosichoron. These have their own stellations and a certainly VERY large set of uniforms. No images here; they're pretty much unvisualizable.

Icositetrachoron

The icositetrachoron is a 4D platonic solid made of 24 octahedra, joined in a cubic vertex figure. It is cool because it is the only non-trivial self-dual regular polytope (that is, neither a simplex nor a polygon). It is also the rectified hexadecachoron.

It has various analogs in other dimensions: it can be considered the birectified cube, rectified orthoplex, rectified demicube and their duals.

Wireframe model
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Half of octahedron-first slices, read from left-right, up-down
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Grand antiprism

In my opinion, the grand antiprism is the most unique convex uniform polytope of any dimension, having no way to construct a uniform variant in a higher dimension. It is one of the two exceptional convex uniform polychora, the other being the snub icositetrachoron, and is also non-Wythoffian. Despite its name, it is neither a usual antiprism nor a stellation of something. It has two rings of 10 pentagonal antiprisms each, joined by 300 tetrahedra. Its symmetry is related to the duocylinder.

A higher-celled analog of this are the double antiprismatoids, but they cannot be made uniform. There might be 6D and 8D versions of this, but they're probably not uniform.

3D wireframe projection, with the pentagonal antiprismatic cells coloured.
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Half of pentagonal-antiprism-first slices
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