This page describes the 25 classes of isohedra with a finite number of sides.
Many of these don't have standard names, though specific examples of a particular class may.
So instead I use a simple notation to describe each isohedral class.
 First the number of sides: 4, 6, 8, 12, 20, 24, 30, 48, 60 or 120
 Second, a description of the sides:
 if it's a Platonic solid or it's made from raising triangles on the sides of a Platonic solid, an upper case letter representing the base shape  T: tetrahedron, O: octahedron, C: cube, D: dodecahedron, I: icosahedron
 otherwise, a lower case letter representing the face shape  i: isosceles triangle, s: scalene triangle, r: rhombus, k: kite, q: quadrilateral with two adjacent equal sides, y: "pyrite" pentagon with 4 equal sides and bilateral symmetry, p: pentagon with two pairs of adjacent equal sides
Not described here are the 5 classes of isohedra with an infinite number of possible sides:
 The dipyramid (aka bipyramid), with 2n triangular sides
 The trapezohedron, with 2n quadrilateral sides
 The skewed trapezoidal dihedron (aka trigonal trapezohedron), with 2n quadrilateral sides
 The triangular dihedron skewed up/down (aka scalenohedron), with 4n triangular sides
 The triangular dihedron skewed in/out, with 4n triangular sides
4T
Known as: 
tetrahedron

Categories: 
Platonic solid, crystallography

Specific form: 
tetrahedron made from 4 equilateral triangles

transform: 
4t(0.333, 0.577), 4t(3, 1.732), 4q(0.333, 0.5), 4q(1, 0.5), 4p(a, 0), 4p(0.5, a)



General form: 
none

4i
Known as: 
isosceles tetrahedron, tetragonal disphenoid

Categories: 
crystallography

Specific form: 
4T is a special case

General form: 
tetrahedron with isosceles triangular faces where each edge is the diagonal of a cuboid

transform: 
4d(a, a), 4d(1, a), 4d(a, 1)



4s
Known as: 
scalene tetrahedron, rhombic disphenoid

Categories: 
crystallography

Specific form: 
4i is a special case

General form: 
tetrahedron with scalene triangular faces where each edge is the diagonal of a cuboid

transform: 
4d(a, b)



6C
Known as: 
cube, hexahedron, dual of the octahedron

Categories: 
Platonic solid, crystallography

Specific form: 
cube made from 6 squares

transform: 
4t(1, 0.577), 4p(a, 1.636a), 8t(1.732, 1.414), 8q(0.5, 0.5), 8p(0.5, a)



General form: 
none

8O
Known as: 
octahedron, dual of the cube

Categories: 
Platonic solid, crystallography

Specific form: 
octahedron made from 8 equilateral triangles

transform: 
8t(0.577, 0.707), 8q(1, a), 8p(a, 0)



General form: 
none

12T
Known as: 
triakis tetrahedron, tristetrahedron, trigonal tristetrahedron, cumulation of the tetrahedron, dual of the truncated tetrahedron

Categories: 
Catalan solid, Archimedean dual, crystallography

Specific form: 
dual of the truncated tetrahedron

transform: 
4t(0.675, 0.577), 4t(1.481, 0.854)



General form: 
made by raising or lowering the center of each face of a tetrahedron an equal distance

transform: 
4t(a,0.577), 4t(a, 0.577a)



Interesting examples: 
equilateral stellated tetrahedron: 4t(1.667, 0.571).
All edges have the same length.



12r
Known as: 
rhombic dodecahedron, dual of the cuboctahedron

Categories: 
Catalan solid, Archimedean dual, crystallography

Specific form: 
made from 12 rhombi

transform: 
4t(1, 1.155), 4q(0.5, 0.5), 8t(0.866, 0.707), 8q(a, 1a) as a>0, 8p(a, a)



General form: 
none

12k
Known as: 
trapezoidal dodecahedron, trapezohedral tristetrahedron

Categories: 
crystallography

Specific form: 
none

General form: 
made from 12 kite shaped quadrilaterals

transform: 
4t(a, f(a)), 4q(a, 0.5)



12D
Known as: 
dodecahedron, regular dodecahedron, dual of the icosahedron

Categories: 
Platonic solid

Specific form: 
made from 12 regular pentagons

transform: 
4p(0.405, 0.156), 8q(0.722, 0.278), 20t(1.258, 1.176), 20q(0.7236, 0.5), 20p(0.5, a)



General form: 
none

12y
Known as: 
octahedral pentagonal dodecahedron, pentagonal dodecahedron, pyritohedron

Categories: 
crystallography

Specific form: 
the regular dodecahedron is a special case

General form: 
made from 12 pentagons with 4 equal sides and bilateral symmetry

transform: 
4p(a, 0.818(12a)), 8q(a, 1a)



Interesting examples: 
nonconvex equilateral pyritohedron: 4p(0.09549150, 0.6605596).
This is interesting because all the edges have the same length.



12p
Known as: 
tetragonal pentagonal dodecahedron, tetartoid

Categories: 
crystallography

Specific form: 
12y is a special case

General form: 
made from 12 pentagons that have two pairs of equal adjacent sides

transform: 
4p(a, b)



20I
Known as: 
icosahedron, dual of the dodecahedron

Categories: 
Platonic solid

Specific form: 
made from 20 equilateral triangles

transform: 
20t(0.795, 0.851), 20q(1, 0.5), 20p(a, 0)



General form: 
none

24O
Known as: 
triakis octahedron, small triakis octahedron, trisoctahedron, trigonal trisoctahedron, cumulation of the octahedron, dual of the truncated cube

Categories: 
Catalan solid, Archimedean dual, crystallography

Specific form: 
dual of the truncated cube

transform: 
8t(0.717, 0.707)



General form: 
made by raising or lowering the center of each face of a octahedron an equal distance

transform: 
8t(a, 0.707)



Interesting examples: 
equilateral triakis octahedron, stellated octahedron, stella octangula: 8t(1.732, 0.707).
All edges have the same length.



24C
Known as: 
tetrakis hexahedron, tetrahexahedron, cumulation of the cube, dual of the truncated octahedron

Categories: 
Catalan solid, Archimedean dual, crystallography

Specific form: 
dual of the truncated octahedron

transform: 
4t(1, 0.866), 8t(1.155, 0.943)



General form: 
made by raising or lowering the center of each face of a cube an equal distance

transform: 
4t(1, a), 8t(a, a*sqrt(2/3))



Interesting examples: 
equilateral tetrakis hexahedron, stellated cube, a Mobius deltahedron: 8t(0.717, 0.586).
All edges have the same length.



24T
Known as: 
hexakis tetrahedron, hextetrahedron

Categories: 
crystallography

Specific form: 
none

General form: 
made by changing the length of the face axes and edgemidpoint axes of a tetrahedron

transform: 
4t(a, b)



Interesting examples: 
equilateral hexakis tetrahedron, a Mobius deltahedron: 4t(0.2251485, 1.061010).
All edges have the same length.



24k
Known as: 
trapezoidal icositetrahedron, strombic icositetrahedron, trapezohedral trisoctahedron, trapezohedron, dual of the rhombicuboctahedron

Categories: 
Catalan solid, Archimedean dual, crystallography

Specific form: 
dual of the rhombicuboctahedron

transform: 
8t(0.947, 1), 8q(0.707, 0.5)



General form: 
made from 24 kite shaped quadrilaterals

transform: 
8t(a, f(a)), 8q(a, 0.5)



24q
Known as: 
dyakis dodecahedron, didodecahedron, diploid

Categories: 
crystallography

Specific form: 
none

General form: 
made from 24 quadrilaterals with only two equal and adjacent sides

transform: 
8q(a, b)



24p
Known as: 
pentagonal icositetrahedron, pentagon trioctahedron, gyroid, dual of the snub cube

Categories: 
Catalan solid, Archimedean dual, crystallography

Specific form: 
dual of the snub cube

transform: 
8p(0.419643, 0.124064)



General form: 
made from 24 pentagons that have two pairs of equal adjacent sides

transform: 
8p(a, b)



Interesting examples: 
bilateral pentagonal icositetrahedron: 8p(0.4301597, 0.2451223).
This one is interesting in that the faces have bilateral symmetry,
though it's different from the pentagonal icositetrahedron Catalan solid, which also has faces with bilateral symmetry.


equilateral pentagonal icositetrahedron: 8p(0.3456397, 0.03187700).
All the edges are the same length.


nonconvex equilateral pentagonal icositetrahedron: 8p(0.05625923, 0.42629483).
This one also has edges that are all the same length.



30r
Known as: 
rhombic triacontahedron, dual of the icosidodecahedron

Categories: 
Catalan solid, Archimedean dual

Specific form: 
made from 30 rhombi

transform: 
20t(0.911, 0.851), 20p(a, a * 0.65)



General form: 
none

48O
Known as: 
hexakis octahedron, disdyakis dodecahedron, hexoctahedron, octakis hexahedron, dual of the truncated cuboctahedron

Categories: 
Catalan solid, Archimedean dual, crystallography

Specific form: 
dual of the truncated cuboctahedron

transform: 
8t(0.916, 0.867)



General form: 
made by changing the length of the face axes and edgemidpoint axes of an octahedron

transform: 
8t(a, b)



Interesting examples: 
equilateral hexakis octahedron, a Mobius deltahedron: 8t(0.4279334, 1.141839).
All edges have the same length.


equilateral hexakis octahedron, a Mobius deltahedron: 8t(1.013122, 0.109977).
All edges have the same length.



60I
Known as: 
triakis icosahedron, cumulation of the icosahedron, dual of the truncated dodecahedron

Categories: 
Catalan solid, Archimedean dual

Specific form: 
dual of the truncated dodecahedron

transform: 
20t(0.855, 0.851)



General form: 
made by raising or lowering the center of each face of an icosahedron an equal distance

transform: 
20t(a, 0.851)



60D
Known as: 
pentakis dodecahedron, cumulation of the dodecahedron, dual of the truncated icosahedron

Categories: 
Catalan solid, Archimedean dual

Specific form: 
dual of the truncated icosahedron

transform: 
20t(1.027, 0.959)



General form: 
made by raising or lowering the center of each face of a dodecahedron an equal distance

transform: 
20t(a, a*0.934)



60k
Known as: 
trapezoidal hexecontahedron, strombic hexecontahedron, dual of the rhombicosidodecahedron

Categories: 
Catalan solid, Archimedean dual

Specific form: 
dual of the rhombicosidodecahedron

transform: 
20t(0.957, 0.975), 20q(0.873, 0.5)



General form: 
made from 60 kite shaped quadrilaterals

transform: 
20t(a, f(a)), 20q(a, 0.5)



Interesting examples: 
rhombic hexecontahedron: 20q(0.5, 0.5).
Each face is a rhombus.



60p
Known as: 
pentagonal hexecontahedron, dual of the snub dodecahedron

Categories: 
Catalan solid, Archimedean dual

Specific form: 
dual of the snub dodecahedron

transform: 
20p(0.425584, 0.0886258)



General form: 
made from 60 pentagons that have two pairs of equal adjacent sides

transform: 
20p(a, b)



Interesting examples: 
bilateral pentagonal hexecontahedron: 20p(0.4647861, 0.2504700).
The faces have bilateral symmetry,
though it's different from the pentagonal hexecontahedron Catalan solid, which also has faces with bilateral symmetry.


nonconvex equilateral pentagonal hexecontahedron: 20p(0.3140921, 0.04663939).
All the edges are the same length.


nonconvex equilateral pentagonal hexecontahedron: 20p(0.03413652, 0.2976625).
This one also has edges that are all the same length.



120I
Known as: 
hexakis icosahedron, disdyakis triacontahedron, dual of the truncated icosidodecahedron

Categories: 
Catalan solid, Archimedean dual

Specific form: 
dual of the truncated icosidodecahedron

transform: 
20t(0.938, 0.921)



General form: 
made by changing the length of the face axes and edgemidpoint axes of an icosahedron

transform: 
20t(a, b)



Interesting examples: 
equilateral hexakis icosahedron, a Mobius deltahedron: 20t(0.650328, 1.18675).
All edges have the same length.


equilateral hexakis icosahedron, a Mobius deltahedron: 20t(1.06090, 0.446900).
All edges have the same length.



Fair dice
