Isohedra

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
4 6 8 12 20 24 30 48 60 120
triangleequilateral 4T8O 20I
isosceles 4i 12T 24O 24C 60I 60D
scalene 4s 24T 48O 120I
quadsquare 6C
rhombus 12r 30r
kite 12k 24k 60k
2 equal sides 24q
pentaregular 12D
2-fold symm. 12y
no symmetry 12p 24p 60p

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)
tetrahedron
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)
tetragonal disphenoid tetragonal disphenoid

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)
rhombic disphenoid rhombic disphenoid

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)
cube
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)
octahedron
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)
12T, triakis tetrahedron
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)
12T, triakis tetrahedron 12T, triakis tetrahedron 12T, triakis tetrahedron 12T, triakis tetrahedron 12T, triakis tetrahedron
Interesting examples:
equilateral stellated tetrahedron: 4t(1.667, 0.571).
All edges have the same length. 		12T, equilateral stellated tetrahedron

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, 1-a) as a->0, 8p(a, a)
12r, rhombic dodecahedron
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)
12k, trapezoidal dodecahedron 12k, trapezoidal dodecahedron 12k, trapezoidal dodecahedron 12k, trapezoidal dodecahedron

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)
dodecahedron
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(1-2a)), 8q(a, 1-a)
12y, octahedral pentagonal dodecahedron, pyritohedron 12y, octahedral pentagonal dodecahedron, pyritohedron 12y, octahedral pentagonal dodecahedron, pyritohedron 12y, octahedral pentagonal dodecahedron, pyritohedron 12y, octahedral pentagonal dodecahedron, pyritohedron
12y, octahedral pentagonal dodecahedron, pyritohedron 12y, octahedral pentagonal dodecahedron, pyritohedron 12y, octahedral pentagonal dodecahedron, pyritohedron 12y, octahedral pentagonal dodecahedron, pyritohedron 12y, octahedral pentagonal dodecahedron, pyritohedron
Interesting examples:
non-convex equilateral pyritohedron: 4p(0.09549150, 0.6605596).
This is interesting because all the edges have the same length. 		12y, non-convex equilateral pyritohedron

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)
12p, tetragonal pentagonal dodecahedron, tetartoid 12p, tetragonal pentagonal dodecahedron, tetartoid 12p, tetragonal pentagonal dodecahedron, tetartoid 12p, tetragonal pentagonal dodecahedron, tetartoid 12p, tetragonal pentagonal dodecahedron, tetartoid 12p, tetragonal pentagonal dodecahedron, tetartoid

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)
icosahedron
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)
24O, triakis octahedron
General form: made by raising or lowering the center of each face of a octahedron an equal distance
 transform: 8t(a, 0.707)
24O, triakis octahedron 24O, triakis octahedron 24O, triakis octahedron 24O, triakis octahedron 24O, triakis octahedron, stella octangula
Interesting examples:
equilateral triakis octahedron, stellated octahedron, stella octangula: 8t(1.732, 0.707).
All edges have the same length. 		24O, triakis octahedron, stella octangula

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)
24C, tetrakis hexahedron
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))
24C, equilateral tetrakis hexahedron 24C, tetrakis hexahedron 24C, tetrakis hexahedron 24C, tetrakis hexahedron 24C, tetrakis hexahedron
Interesting examples:
equilateral tetrakis hexahedron, stellated cube, a Mobius deltahedron: 8t(0.717, 0.586).
All edges have the same length. 		24C, equilateral tetrakis hexahedron

24T

Known as: hexakis tetrahedron, hextetrahedron
Categories: crystallography
Specific form: none
General form: made by changing the length of the face axes and edge-midpoint axes of a tetrahedron
 transform: 4t(a, b)
24T, hexakis tetrahedron 24T, hexakis tetrahedron 24T, hexakis tetrahedron 24T, hexakis tetrahedron 24T, hexakis tetrahedron 24T, hexakis tetrahedron
Interesting examples:
equilateral hexakis tetrahedron, a Mobius deltahedron: 4t(0.2251485, 1.061010).
All edges have the same length. 		24T, equilateral hexakis tetrahedron

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)
24k, trapezoidal icositetrahedron
General form: made from 24 kite shaped quadrilaterals
 transform: 8t(a, f(a)), 8q(a, 0.5)
24k, trapezoidal icositetrahedron 24k, trapezoidal icositetrahedron 24k, trapezoidal icositetrahedron 24k, trapezoidal icositetrahedron 24k, trapezoidal icositetrahedron

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)
24q, dyakis dodecahedron, diploid 24q, dyakis dodecahedron, diploid 24q, dyakis dodecahedron, diploid 24q, dyakis dodecahedron, diploid 24q, dyakis dodecahedron, diploid 24q, dyakis dodecahedron, diploid

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)
24p, pentagonal icositetrahedron, gyroid
General form: made from 24 pentagons that have two pairs of equal adjacent sides
 transform: 8p(a, b)
24p, pentagonal icositetrahedron, gyroid 24p, pentagonal icositetrahedron, gyroid 24p, pentagonal icositetrahedron, gyroid 24p, pentagonal icositetrahedron, gyroid 24p, bilateral pentagonal icositetrahedron, gyroid 24p, pentagonal icositetrahedron, gyroid
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. 		24p, bilateral pentagonal icositetrahedron
equilateral pentagonal icositetrahedron: 8p(0.3456397, 0.03187700).
All the edges are the same length. 		24p, equilateral pentagonal icositetrahedron
non-convex equilateral pentagonal icositetrahedron: 8p(0.05625923, 0.42629483).
This one also has edges that are all the same length. 		24p, non-convex equilateral pentagonal icositetrahedron

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)
30r, rhombic triacontahedron
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)
48O, hexakis octahedron
General form: made by changing the length of the face axes and edge-midpoint axes of an octahedron
 transform: 8t(a, b)
48O, hexakis octahedron 48O, hexakis octahedron 48O, hexakis octahedron 48O, hexakis octahedron 48O, hexakis octahedron 48O, hexakis octahedron
Interesting examples:
equilateral hexakis octahedron, a Mobius deltahedron: 8t(0.4279334, 1.141839).
All edges have the same length. 		48O, equilateral hexakis octahedron
equilateral hexakis octahedron, a Mobius deltahedron: 8t(1.013122, 0.109977).
All edges have the same length. 		48O, equilateral hexakis octahedron

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)
60I, triakis icosahedron
General form: made by raising or lowering the center of each face of an icosahedron an equal distance
 transform: 20t(a, 0.851)
60I, triakis icosahedron 60I, triakis icosahedron 60I, triakis icosahedron 60I, triakis icosahedron 60I, triakis icosahedron

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)
60D, pentakis dodecahedron
General form: made by raising or lowering the center of each face of a dodecahedron an equal distance
 transform: 20t(a, a*0.934)
60D, pentakis dodecahedron 60D, pentakis dodecahedron 60D, pentakis dodecahedron 60D, pentakis dodecahedron 60D, pentakis dodecahedron

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)
60k, trapezoidal hexecontahedron
General form: made from 60 kite shaped quadrilaterals
 transform: 20t(a, f(a)), 20q(a, 0.5)
60k, trapezoidal hexecontahedron, rhombic hexecontahedron 60k, trapezoidal hexecontahedron 60k, trapezoidal hexecontahedron 60k, trapezoidal hexecontahedron 60k, trapezoidal hexecontahedron
Interesting examples:
rhombic hexecontahedron: 20q(0.5, 0.5).
Each face is a rhombus. 		60k, trapezoidal hexecontahedron, rhombic hexecontahedron

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)
60p, pentagonal hexecontahedron
General form: made from 60 pentagons that have two pairs of equal adjacent sides
 transform: 20p(a, b)
60p, pentagonal hexecontahedron 60p, pentagonal hexecontahedron 60p, pentagonal hexecontahedron 60p, pentagonal hexecontahedron 60p, bilateral pentagonal hexecontahedron 60p, pentagonal hexecontahedron
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. 		60p, bilateral pentagonal hexecontahedron
non-convex equilateral pentagonal hexecontahedron: 20p(0.3140921, -0.04663939).
All the edges are the same length. 		60, non-convex equilateral pentagonal hexecontahedron
non-convex equilateral pentagonal hexecontahedron: 20p(0.03413652, 0.2976625).
This one also has edges that are all the same length. 		60p, non-convex equilateral pentagonal hexecontahedron

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)
120I, hexakis icosahedron
General form: made by changing the length of the face axes and edge-midpoint axes of an icosahedron
 transform: 20t(a, b)
120I, hexakis icosahedron 120I, hexakis icosahedron 120I, hexakis icosahedron 120I, hexakis icosahedron 120I, hexakis icosahedron 120I, hexakis icosahedron
Interesting examples:
equilateral hexakis icosahedron, a Mobius deltahedron: 20t(0.650328, 1.18675).
All edges have the same length. 		120I, equilateral hexakis icosahedron
equilateral hexakis icosahedron, a Mobius deltahedron: 20t(1.06090, 0.446900).
All edges have the same length. 		120I, equilateral hexakis icosahedron

Fair dice
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