# Uniform Polyhedra and Other Three Dimensional Shapes

Welcome to my new polyhedron web "subsite". This subsite will deal with uniform polyhedra, uniform compound polyhedra, 3-D dice, semi-uniform polyhedra, and any other interesting 3-D shapes. Last updated June 10, 2012.

2012 June 10 - Polyhedron hub page is published online.

2012 January 24 - Polyhedron hub page is created.

## Basic Three Dimensional Shapes These three shapes can be considered as the basic 3-D shapes, they are either flat in each dimension or they join them in a uniform curve. The three shapes are:

Cube - which can be generalized as a variety of blocks when the dimensions are of different length. The most symmetric is the cube of the dyad (line segment), then there are the square prisms (dyad squared time dyad), finally there are the blocks (dyad times dyad times dyad). This shape is usually used for the shape of blocks, bricks, and boxes. It could be represented by |||.

Cylinder - flat versions can be called coins and long ones called rods. Commmonly used as the shape for a can. It is flat in one dimension, the other two form a curve and it can roll forwards or backwards. It is the product of a dyad and a disk (solid circle), this could be represented as |().

Sphere - no flat dimensions, all three dimensions are curved together. It can roll all over a 2-D surface. This shape is commonly used for balls, planets, marbles, etc. It could be represented as (|). Sometimes the word "sphere" refers to the hollow version where the solid version is called a ball.

## Uniform Polyhedra A polyhedron is uniform when all of its vertices are congruent and all of its faces are regular. Below are the 75 uniform polyhedra plus the two infinite groups divided up into categories.

Category A: Prisms - This is the infinite set of prisms. For every polygon there is a prism which is basically the polygon extended into the third dimension.

Category B: Antiprisms - The infinite set of antiprisms has triangles for their lateral faces instead of squares.

Category 1: Regulars - (Polyhedra 1 - 9) The regular polyhedra include the five Platonic solids and the four Kepler-Poinsot solids. Their faces are all regular and congruent as well as their vertices and edges. Their verfs are regular also.

Category 2: Truncates - (Polyhedra 10 - 19) These include the 7 truncates and the three quasitruncates, they have isosceles triangle verfs.

Category 3: Quasiregulars - (Polyhedra 20 - 35) These include the polyhedra with rectangular vertex figures along with their regiments. Also included are the three ditrigonaries.

Category 4: Trapeziverts - (Polyhedra 36 - 56) These include the polyhedra with trapezoid vertex figures along with their regiments.

Category 5: Omnitruncates - (Polyhedra 57 - 63) These contain scalene triangles for their verfs. They also have the maximum number of vertices for their symmetry group.

Category 6: Snubs - (Polyhedra 64 - 75) These have faces that are off the main symmetry axises, many of these are chiral.

The following picture pages show the uniform polyhedra with their short names: 1, 2, 3, 4, 5, 6, 7.

## Uniform Compound Polyhedra A compound polyhedron is uniform when all of its component polyhedra are uniform and all of its vertices are congruent. Some of them have a continuum of morphs. Below are the uniform compound polyhedra plus the two infinite groups divided up into categories.

Category CA: Compound Prisms - These are the prisms of the regular compound polygons and form an infinite group.

Category CB: Compound Antiprisms - These are the anitprisms of the regular compound polygons including the antiprisms of compounds of "digons". This is an infinite category.

Category C1: Compound Regulars - (Polyhedron compound 1 - 6) These include the regular compounds along with a few others that are nearly regular.

Category C2: Compound Truncates - (Polyhedron compound 7 - 11) These are the truncates of the regulars.

Category C3: Fivers - (Polyhedron compound 12 - 20) These are the compounds of 5 Xs, where X is a cubic symmetric polyhedron. They are related to rhom.

Category C4: Ikers - (Polyhedron compound 21 - 28) These are the compounds of ike and their relatives.

Category C5: Tets and Cubes - (Polyhedron compound 29 - 33) These are the remaining compounds of tets and cubes.

Category C6: Prismatics - (Polyhedron compound 34 - 37) These are compounds of n prisms, where n is an even number.

Category C7: Chiral and Doubled Prismatics - (Polyhedron compound 38 - 45) These are the compounds of n prisms where n is odd, these form chiral compounds and their doubled forms.

Category C8: Antiprismatics - (Polyhedron compound 46 - 53) These are the compounds of square, pentagonal, and star based antiprisms.

Category C9: Octahedral Continuums - (Polyhedron compound 54 - 67) There are two continuums of octahedral compounds where the octahedra act as trigon antiprisms, there's also a compound of 20 thahs.

Category C10: Disnubs - (Polyhedron compound 68 - 75) These are the compounds of the right and left handed snub polyhedra.

The following picture pages show the uniform compounds with their short names: 1, 2, 3, 4, 5, 6, 7.

## Three Dimensional Dice

In a geometric sense, dice are normally defined as convex polytopes with congruent sides. We could allow curved objects into the mix by changing the definition so that a die is any convex shape with congruent "contact regions". Contact regions can be thought of as the part that can contact a surface when the object is setting on one. The contact region of a cube is a square face (creases such as edges or vertices don't count as contact regions), the contact region of a cylinder (which is not a die) is either a disc or a dyad (line-segment) depending on how it is set, and the contact region of a sphere is a point. Three dimensional dice can be put into five groups:

Platonic Solids - These five polyhedra has been known since antiquity.

Catalan Solids - These 13 are the duals of the Archimedean solids.

Gems - AKA the bipyramids or tegums, there is an infinite group of these.

Crystals - AKA the antibipyramids or antitegums, this is also an infinite group.

Curved - only two exist in three space, they are the sphere and the spindle.

## Semi-Uniform Polyhedra A semi-uniform polyhedron has congruent vertices and its faces must all have congruent vertices as well, but could have different edge lengths. All uniforms can be considered to be semi-uniforms also.

I started searching for these in the early 1990's, at first I found hundreds - it now appears there could be millions! In order to narrow them down, I decided to look for the tame ones, calling them "Tasu" for short (tame semi uniform). The tasus were easy to number when there was a low vertex count, like 24 - but even these become unwieldy when they have larger vertex counts like 60 or 120. So I narrowed the search even more to the "primary" tame semi-uniforms - or "Petsu". Primary polyhedra can't have snub faces or any element off the main symmetry axises. I finally came up with a count (7078 petsus). Each petsu forms a "teepee" that depends on the number of degrees of variance that the semi-uniform can morph, for unlike their uniform counterparts - they sometimes form a continuum of polyhedra (uncountably infinite number). For example here are several versions of tut using symbols - oxy, oxz, oyx, ozx, oyz, ozy. where o represents edge length 0, x represents 1, y and z can be any positive value you wish. The verf of sidtaxhi and dattady are two such tuts - symboled as oxy and oyx respectively where y = 1/tau.

A teepee is a piece of this continuum where all of the polyhedra within it has the same vertex, edge and face arrangement, but allows for the vertices to freely morph from one end to the other - for example: the grid based teepees have two degrees of variance. The teepee allows the vertices of grid to move anywhere between the icosahedron, icosidodecahedron, and the dodecahedron and all in between forming a triangle shaped piece of "polyhedron space" where each point is an individual polyhedron. If we take a teepee and continue to morph past its boundaries, it will lead into other teepees forming a large set of teepees called a "tribe". The grid tribe consist of any polyhedron with the symbol ab^c, where a, b, and c represent edge lengths (they can also be negative). Tribes can also be divided up into clans which further divide up into teepees. For example, the quitdid clan contains all semi-uniforms with the symbol x^y*'z, where x,y, and z are positive values. To get the rest of the tribe, turn some of the values negative, this will lead to the following clans: x^'y*'z, x^y*z, and x^'y*z which form the quitdid tribe. More info on teepees, tribes, clans, and other terms can be seen in the glossary.

Below are the petsu count results described under their convex hulls:

Tet - Only one semi-uniform, tet itself which is petsu.

Oct - Only two petsus, oct and thah. There are two other semiuniforms which resemble the verfs of duhd and phud.

Cube - Only two petsus, the cube and the compound stella octangula (short name "so"). There are several other semi-uniforms, they form the verfs of the ico regiment members.

Ike - Only four petsus: ike, gad, sissid, and gike. There are four non-petsu semi-uniforms that have pyritic symmetry (one is a compound).

Doe - Nine petsus: doe, sidtid, ditdid, gidtid, rhom, e, ki, gissid, and "ditti" (verf of idhi from sishi regiment). Rhom, e, and ki are compounds. There are many other semi-uniforms, many of which are verfs of sishi members.

Co - There are 16 semi-uniforms, where four are petsu: co, oho, cho, and it (inflected tetrahedron, which looks like a tetrahedron with four more dangling off the corners). "It" has a variant that looks like the 3-D version of the tripod, we could call the variant "quatut" for quasitruncated tetrahedron. Co's other semi-uniforms have either cubic or tetrahedral symmetry.

Id - Nine petsus: id, seihid, sidhid, did, sidhei, gidhei, gid, gidhid, and geihid. There are lots more semi-uniforms that are not tame, nor primary, many having strange 8 sided verfs.

Tut - Five petsus with one degree of variance each: tut, hittut (hypertruncated tet), tiddit (tetradistetrahedron), qradit (quasirhombitetrahedron), and ret (rhombitetrahedron) - the last three form a regiment. There's one more semi-uniform, a non-petsu called tret (tetrarhombitetrahedron).

Tic - There are 16 petsus, and atleast 32 more tasus. Six of the petsus are compounds and there are several of the tasus that are as well. Tic based petsus have one degree of variance.

Sirco - There are also 16 petsus here also with atleast 32 more tasus. Six of the petsus are compounds. These have one degree of variance.

Toe - There are 23 petsus, two being compounds. These have two degrees of variance, but six of them will lose their "petsuness" if they morph into a tetrahedral only symmetry - this is because toe can either be xx'o (cube symmetric) or xxx (tet symmetric). In summary, six petsus are cubic only (2 are compounds), nine can be both cubic or tetrahedral, and eight are strictly tetrahedral.

Snic - Snic is semi-uniform, but there are no petsus here.

Girco - Girco has 185 petsus: 68 have trigonal verfs (32 are compounds), and there are 39 tetragonal verfed regiments of three (two of the regiments are compounds - this leads to 117 total, 6 being compounds. All together there are 185 with 38 being compounds. These have two degrees of variance which means that they form triangle shaped teepees, where each point in the teepee is an individual variant of each petsu.

Tid - Tid has 153 petsus with one degree of variance (teepees have linear shapes). 12 have trigonal verfs, 6 have either a kite or a dart shaped verf, 13 regiments of three (39 total) are based off of trapezoid verfs, 12 pentagonal verfed regiments of four (48 total), two hexagonal verfed regiments of four (8 total), and finally two hexagonal verfed regiments of twenty (40 total) where there's one compound and three fissaries in both regiments.

Ti - Ti also has 153 petsus with one degree of variance. They are sub-divided in the same way tid's petsus are.

Srid - Srid also has 153 petsus which are subdivided in the same way as tid and ti.

Snid - Snid is semi-uniform, but there are no petsus here.

Grid - Grid has two degrees of variance and there are 6343 petsu teepees, 385 of these are spitsu (simple prime semi-uniforms), 5 of these are compounds. The spitsus have trigonal vertex figures. There are also 614 tetragonal verfed regiments of three (1842 petsus total), 248 pentagonal verfed regiments of 12 (2976 total), and 19 hexagonal verfed regiments of 60 (1140 total). The 385 spitsus can be divided up into the following tribes: xy^z - grid tribe with four clans and 60 teepees, x^y*z - quitdid tribe with four clans and 60 teepees, xy*z - gaquatid tribe with four clans and 60 teepees, (x^y*'z) - idtid tribe with four clans and 60 teepees, (xy*z) - becada tribe with three clans and 30 teepees, (x^y^z,) - fabeca tribe with three clans and 30 teepees, (xy^'z) - cafeta tribe with three clans and 30 teepees, (x*y*z,) - mocaba tribe with three clans and 30 teepees, (x*y*z*) - vameza tribe with two clans and 10 teepees, (x^'y^'z^') - jefari tribe with two clans and 10 teepees, and (x y z) - broza tribe with one clan and five teepees - this tribe contains compounds of 15 blocks.

Prisms - The only petsus here are prisms of the petsus of two dimensions, however there are many other semi-uniforms when the prism is based on an even numbered polygon.

Antiprisms - No petsus here, but there are three semi-uniforms under each odd numbered antiprism as well as its star versions. Even numbered antiprisms have no semi-uniforms other than the uniform antiprisms found within them.

Special thanks to Andrew Weimholt, who has let me use his polytope.net domain to store my polychoron pics. Without the domain there wouldn't of been enough room on my site to store all the pics. ```Page created by Jonathan Bowers, 2012
e-mail = hedrondude at suddenlink dot net ```