Compounds of Tetrahedra

Contents

Introduction

In [HVerh00] Verheyen introduces a theory to derive a complete set of compounds for polyhedra. In the book he uses the orbit theory to derive the complete set of compounds of cubes. In [HVerh01] Verheyen applies this theory on all uniform polyhedra with the exception of the prisms and anti-prisms. This page summarises the results for the tetrahedron by means of interactive VRML models, for which a VRML player is required.1

Compounds of Tetrahedra with Central Freedom

The following table contains compounds for which the descriptive can rotate freely. The interactive models have 2 slidebars to rotate the descriptive around 2 different axes. The animations are examples of varying the angle around these two axes. The models with symmetries that are cyclic or dihedral use cylinders to model the edges. For the other compounds, in the lower part of the table, lines are used when the amount of constituents is bigger than 12 to improve performance. As a consequence the you will notice some stitching effects.
Compound2 Static Example Interactive Model Animated Model
n | Cn / E for n=3
for n=4 		for n=3
for n=4 		for n=3
for n=4
2n | C2nCn / E for n=3
for n=4 		for n=3
for n=4 		for n=3
for n=4
2n | Cn x I / E for n=3
for n=4 		for n=3
for n=4 		for n=3
for n=4
2n | DnCn / E for n=3
for n=4 		for n=3
for n=4 		for n=3
for n=4
2n | Dn / E for n=3
for n=4 		for n=3
for n=4 		for n=3
for n=4
4n | D2nDn / E for n=2
for n=3 		for n=2
for n=3 		for n=2
for n=3
4n | Dn x I / E for n=2
for n=3
for n=4 		for n=2
for n=3
for n=4 		for n=2
for n=3
for n=4
12 | A4 / E Example Interactive Model Animation
24 | A4 x I / E Example Interactive Model Animation
24 | S4A4 / E Example Interactive Model Animation
24 | S4 / E Example Interactive Model Animation
48 | S4 x I / E Example Interactive Model Animation
60 | A5 / E Example Interactive Model Animation
120 | A5 x I / E Example Interactive Model Animation

Compounds of Tetrahedra with Rotational Freedom

The list in the previous section is far from complete, but all the remaining compounds can be derived from the compounds in the above list, by using special positions for the descriptive. For instance special compounds occur when an n-fold symmetry axes of the descriptive are shared with m*n-fold symmetry axes of the whole compound. To keep that property the descriptive can only be rotated around one axis (being the axis that is shared).

The descriptions explains how these models can be obtained from one compound with central freedom. The description gives a sufficient requirement how to obtain a compound with rotational freedom, which means that it might not describe all properties. E.g. according the description 6 | S4A4 / C4C2 is obtained from 24 | S4A4 / E by sharing an 2-fold axis between the descriptive and the compound. This is sufficient, but as a bonus both share a rotary inversion (where the rotation is a half-turn) as well. Note as well that some compounds with rotational freedom might be derived from mode than just one compound with central freedom.

The description also lists special angles. These are only the domain angles, which usually result in rigid compounds.

The following table contains all compounds (with roational freedom) that have a dihedral and cyclic symmetry:

Compound2 VRML1 model Description
n | DnCn / C2C1 | μ
for n>1
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ 		Examples:
for n=2
for n=3
for n=4
for n=6
for n=8
Animations:
for n=2
for n=3
for n=4
for n=6
for n=8
Interactive Models:
for n=2
for n=3
for n=4
for n=6
for n=8 		2n | DnCn / E, for which the descriptive shares a mirror with a mirror in DnCn.
The special angles are:
μ0 = 0
μ1 = atan(√2)
μ2 = π/2
μ3 = atan(1/√2)
with μ = μ0
and n is odd 		3 | D12D6 / D4D2 n | D4nD2n / D4D2
with μ = μ0
and n = 2 		1 | S4A4 / S4A4 -
with μ = μ0
and n = 4 		2 | S4 x I / S4A4 -
with μ = μ0
and n = 2m, m > 1, and m odd 		3 | D12D6 / D4D2 m | D4mD2m / D4D2
with μ = μ0
and n = 4m, m > 1 		4 | D8 x I / D4D2 2m | D4m x I / D4D2
with μ = μ1
and n is coprime with 3 		2 | D6C6 / D3C3 n | D3nC3n / D3C3
with μ = μ1
and n = 3 		1 | S4A4 / S4A4 -
with μ = μ1
and n = 3m, where m > 1 		2 | D6C6 / D3C3 m | D3mC3m / D3C3
with μ = μ2
and n = 2 		2 | S4 x I / S4A4 -
with μ = μ2
and n is odd 		3 | D6D3 / D2C2 n | D2nDn / D2C2
with μ = μ2
and n = 2m and m > 1 		4 | D4 x I / D2C2 2m | D2m x I / D2C2
with μ = μ3
and n = 2 		2 | D6D3 / D3C3 2 | D6D3 / D3C3
2n | D3nC3n / C3 | μ
μ ∈ ]μ0, μ1[ 		Examples:
for n=1
for n=2
for n=3
Animations:
for n=1
for n=2
for n=3
Interactive Models:
for n=1
for n=2
for n=3 		2m | DmCm / E, for which the descriptive shares a 3-fold axis with an m-fold axis in DmCm, where m = 3*n.
The special angles are:
μ0 = 0
μ1 = π/6n
with μ = μ0
and n = 1 		1 | S4A4 / S4A4 -
with μ = μ0
and n > 1 		2 | D6C6 / D3C3 n | D3nC3n / D3C3
with μ = μ1
and m = 2n 		2 | D6C6 / D3C3 m | D3mC3m / D3C3
n | Dn / C2 | μ
for n>2
μ ∈ ]μ0, μ1[ 		Examples:
for n=3
for n=4
for n=6
for n=8
Animations:
for n=3
for n=4
for n=6
for n=8
Interactive Models:
for n=3
for n=4
for n=6
for n=8 		2n | Dn / E, for which the descriptive shares a 2-fold axis with a half turn in Dn (i.e. not with the principal axis).
If n = 2 then 2 | D2 / C2 becomes a 2k | D4kD2k / C4C2(with k=1)
The special angles are:
μ0 = 0
μ1 = π/4
with μ = μ0
and n = 4 		2 | S4 x I / S4A4 -
with μ = μ0
and n is odd 		3 | D12D6 / D4D2 n | D4nD2n / D4D2
with μ = μ0
and n = 2m, m > 1, and m odd 		3 | D12D6 / D4D2 m | D4mD2m / D4D2
with μ = μ0
and n = 4m, m > 1 		4 | D8 x I / D4D2 2m | D4m x I / D4D2
with μ = μ1
and n is odd 		3 | D6D3 / D2C2 n | D2nDn / D2C2
with μ = μ1
and n = 2m 		6 | D6 x I / D2C2 2m | D2m x I / D2C2
2n | D3n / C3 | μ
μ ∈ ]μ0, μ1[ 		Examples:
for n=1
for n=2
for n=3
Animations:
for n=1
for n=2
for n=3
Interactive Models:
for n=1
for n=2
for n=3 		2m | Dm / E, for which the descriptive shares a 3-fold axis with an m-fold axis in Dm, where m = 3*n.
The special angles are:
μ0 = 0
μ1 = π/6n
with μ = μ0
and n is odd 		2 | D6D3 / D3C3 2n | D6nD3n / D3C3
with μ = μ0
and n = 2m 		4 | D6 x I / D3C3 2n | D3n x I / D3C3
with μ = μ1
and n = 1 		2 | S4 x I / S4A4 -
with μ = μ0
and n is odd and n > 1 		6 | D9 x I / D3C3 2n | D3n x I / D3C3
with μ = μ0
and n = 2m 		4 | D12D6 / D3C3 2n | D6nD3n / D3C3
2n | D2nDn / C2 | μ
for n>1
μ ∈ ]μ0, μ1[ 		Examples:
for n=2
for n=3
for n=4
Animations:
for n=2
for n=3
for n=4
Interactive Models:
for n=2
for n=3
for n=4 		4n | D2nDn / E, for which the descriptive shares a 2-fold axis with a half turn in D2nDn (i.e. not with the principal axis).
If n = 1 then 2 | D2D1 / C2 becomes a 2k | D4kD2k / C4C2(with k=1)
The special angles are:
μ0 = 0
μ1 = π/4
with μ = μ0
and n = 2 		1 | S4A4 / S4A4 -
with μ = μ0
and n is odd 		6 | D12 x I / D4D2 2n | D4n x I / D4D2
with μ = μ0
and n = 2m and m > 1 		4 | D8 x I / D4D2 2m | D4m x I / D4D2
with μ = μ1
and n is odd 		3 | D6D3 / D2C2 n | D2nDn / D2C2
with μ = μ1
and n = 2m 		4 | D4 x I / D2C2 2n | D2n x I / D2C2
2n | D2nDn / C2C1 | μ
for n>1
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ 		Examples:
for n=2
for n=3
for n=4
Animations:
for n=2
for n=3
for n=4
Interactive Models:
for n=2
for n=3
for n=4 		4n | D2nDn / E, for which the descriptive shares a mirror with a mirror (through the pricipal axis) in D2nDn.
If n = 1 then 2 | D2D1 / C2C1 equals to a k | DkCk / C2C1(with k=2)
The special angles are:
μ0 = 0
μ1 = atan(√2)
μ2 = π/2
with μ = μ0
and n = 2 		1 | S4A4 / S4A4 -
with μ = μ0
and n is odd 		6 | D12 x I / D4D2 2n | D4n x I / D4D2
with μ = μ0
and n = 2m and m > 1 		4 | D8 x I / D4D2 2m | D4m x I / D4D2
with μ = μ1
and n is coprime with 3 		4 | D12D6 / D3C3 2n | D6nD3n / D3C3
with μ = μ1
and n = 3m 		2 | D6D3 / D3C3 2m | D6mD3m / D3C3
with μ = μ2
and n is odd 		3 | D6D3 / D2C2 n | D2nDn / D2C2
with μ = μ2
and n = 2m 		4 | D4 x I / D2C2 2n | D2n x I / D2C2
2n | D2nDn / D1C1 | μ
for n>1 and n is odd
μ ∈ ]μ0, μ1[ 		Examples:
for n=3
for n=5
Animations:
for n=3
for n=5
Interactive Models:
for n=3
for n=5 		4n | D2nDn / E, for which the descriptive shares a mirror with a mirror that is perpendicular to the pricipal axis in D2nDn (thus requiring that n is odd).
If n = 1 then 2 | D2D1 / D1C1 equals to a k | DkCk / C2C1(with k=2)
The special angles are:
μ0 = 0
μ1 = π/2n
with μ = μ0 3 | D6D3 / D2C2 n | D2nDn / D2C2
with μ = μ1 6 | D6 x I / D2C2 2n | D2n x I / D2C2
2n | D4nD2n / C4C2 | μ
for n is odd
μ ∈ ]μ0, μ1[ 		Examples:
for n=1
for n=3
Animations:
for n=1
for n=3
Interactive Models:
for n=1
for n=3 		4m | D2mDm / E, for which the descriptive shares a half turn with an m-fold axis in D2mDm, where m = 2*n.
The special angles are:
μ0 = 0
μ1 = π/4n
with μ = μ0
and n = 1 		2 | S4 x I / S4A4 -
with μ = μ0 6 | D12 x I / D4D2 2n | D4n x I / D4D2
with μ = μ1
and n = 1 		1 | S4A4 / S4A4 -
with μ = μ1 3 | D12D6 / D4D2 n | D4nD2n / D4D2
4n | D6nD3n / C3 | μ
μ ∈ ]μ0, μ1[ 		Examples:
for n=1
for n=2
for n=3
Animations:
for n=1
for n=2
for n=3
Interactive Models:
for n=1
for n=2
for n=3 		4m | D2mDm / E, for which the descriptive shares a 3-fold axis with an m-fold axis in D2mDm, where m = 3*n.
The special angles are:
μ0 = 0
μ1 = π/6n
with μ = μ0 4 | D12D6 / D3C3 2n | D6nD3n / D3C3
with μ = μ1
and m = 2n 		8 | D12 x I / D3C3 2m | D3m x I / D3C3
2n | Dn x I / C2 | μ
for n>2
μ ∈ ]μ0, μ1[ 		Examples:
for n=3
for n=4
for n=6
for n=8
Animations:
for n=3
for n=4
for n=6
for n=8
Interactive Models:
for n=3
for n=4
for n=6
for n=8 		4n | Dn x I / E, for which the descriptive shares a 2-fold axis with a half turn in DnxI (i.e. not with the principal axis).
If n = 1 then 2 | D1 x I / C2 becomes a 2 | S4 x I / S4A4
If n = 2 then 4 | D2 x I / C2 becomes a 4k | D4k x I / C4C2(with k=1)
The special angles are:
μ0 = 0
μ1 = π/2
with μ = μ0
and n = 4 		2 | S4 x I / S4A4 -
with μ = μ0
and n is odd 		6 | D12 x I / D4D2 2n | D4n x I / D4D2
with μ = μ0
and n = 2m and m odd 		6 | D12 x I / D4D2 2m | D4m x I / D4D2
with μ = μ0
and n = 4m and m > 1 		4 | D8 x I / D4D2 2m | D4m x I / D4D2
with μ = μ1
and n is odd 		6 | D6 x I / D2C2 2n | D2n x I / D2C2
with μ = μ1
and n = 2m 		4 | D4 x I / D2C2 2m | D2m x I / D2C2
2n | Dn x I / C2C1 | μ
for n>1
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ 		Examples:
for n=2
for n=3
for n=4
for n=6
for n=8
Animations:
for n=2
for n=3
for n=4
for n=6
for n=8
Interactive Models:
for n=2
for n=3
for n=4
for n=6
for n=8 		4n | Dn x I / E, for which the descriptive shares a mirror with a mirror (through the pricipal axis) in DnxI.
If n = 1 then 2 | D1 x I / C2 becomes a 2 | S4 x I / S4A4
The special angles are:
μ0 = 0
μ1 = atan(√2)
μ2 = π/2
with μ = μ0
and n = 2 or n = 4 		2 | S4 x I / S4A4 -
with μ = μ0
and n is odd 		6 | D12 x I / D4D2 2n | D4n x I / D4D2
with μ = μ0
and n = 2m and m odd 		6 | D12 x I / D4D2 2m | D4m x I / D4D2
with μ = μ0
and n = 4m and m > 1 		4 | D8 x I / D4D2 2m | D4m x I / D4D2
with μ = μ1
and n = 3 		2 | S4 x I / S4A4 -
with μ = μ1
and n is coprime with 3 		4 | D6 x I / D3C3 2n | D3n x I / D3C3
with μ = μ1
and n = 3m and m > 1 		4 | D6 x I / D3C3 2m | D3m x I / D3C3
with μ = μ2
and n = 2 		2 | S4 x I / S4A4 -
with μ = μ2
and n is odd 		6 | D6 x I / D2C2 2n | D2n x I / D2C2
with μ = μ2
and n = 2m 		4 | D4 x I / D2C2 2m | D2m x I / D2C2
4n | D2n x I / D1C1 | μ
for n>1
μ ∈ ]μ0, μ1[ 		Examples:
for n=2
for n=3
Animations:
for n=2
for n=3
Interactive Models:
for n=2
for n=3 		4m | Dm x I / E, for which the descriptive shares a mirror with a mirror that is perpendicular to the pricipal axis in DmxI, where m = 2*n.
If n = 1 then 4 | D2 x I / D1C1 equals to a 2k | Dk x I / C2C1(with k=2)
The special angles are:
μ0 = 0
μ1 = π/2n
with μ = μ0 4 | D4 x I / D2C2 2n | D2n x I / D2C2
with μ = μ1
and m = 2n 		8 | D8 x I / D2C2 2m | D2m x I / D2C2
4n | D3n x I / C3 | μ
μ ∈ ]μ0, μ1[ 		Examples:
for n=1
for n=2
Animations:
for n=1
for n=2
Interactive Models:
for n=1
for n=2 		4m | Dm x I / E, for which the descriptive shares a 3-fold axis with an m-fold axis in DmxI, where m = 3*n.
The special angles are:
μ0 = 0
μ1 = π/6n
with μ = μ0
and n = 1 		2 | S4 x I / S4A4 -
with μ = μ0
and n > 1 		4 | D6 x I / D3C3 2n | D3n x I / D3C3
with μ = μ1
and m = 2n 		8 | D12 x I / D3C3 2m | D3m x I / D3C3
4n | D4n x I / C4C2 | μ
μ ∈ ]μ0, μ1[ 		Examples:
for n=1
for n=2
for n=3
Animations:
for n=1
for n=2
for n=3
Interactive Models:
for n=1
for n=2
for n=3 		4m | Dm x I / E, for which the descriptive shares a half turn with an m-fold axis in DmxI, where m = 4*n.
The special angles are:
μ0 = 0
μ1 = π/8n
with μ = μ0
and n = 1 		2 | S4 x I / S4A4 -
with μ = μ0
and n > 1 		6 | D12 x I / D4D2 2n | D4n x I / D4D2
with μ = μ1
and m = 2n 		4 | D8 x I / D4D2 2m | D4m x I / D4D2

The following table contains all compounds (with roational freedom) that do not have a dihedral and cyclic symmetry:

Compound2 VRML1 model Description
4 | A4 / C3 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		12 | A4 / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in A4.
The special angles are:
μ0 = 0
μ1 = π/3
with μ = μ0 1 | S4A4 / S4A4 -
with μ = μ1 4 | S4A4 / D3C3 -
12 | A4 x I / C2C1 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		24 | A4 x I / E, for which the descriptive shares a mirror with a mirror in A4xI.
The special angles are:
μ0 = 0
μ1 = π/4
with μ = μ0 6 | S4 x I / D4D2 -
with μ = μ1 12 | S4 x I / D2C2 -
8 | A4 x I / C3 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		24 | A4 x I / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in A4xI.
The special angles are:
μ0 = 0
μ1 = π/3
with μ = μ0 2 | S4 x I / S4A4 -
with μ = μ1 8 | S4 x I / D3C3 -
12 | S4A4 / C2C1 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		24 | S4A4 / E, for which the descriptive shares a mirror with a mirror in S4A4.
The special angles are:
μ0 = 0
μ1 = π/2
with μ = μ0 1 | S4A4 / S4A4 -
with μ = μ1 12 | S4 x I / D2C2 -
8 | S4A4 / C3 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		24 | S4A4 / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in S4A4.
The special angles are:
μ0 = 0
μ1 = π/3
with μ = μ0 1 | S4A4 / S4A4 -
with μ = μ1 4 | S4A4 / D3C3 -
6 | S4A4 / C4C2 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		24 | S4A4 / E, for which the descriptive shares a half-turn with a half-turn in S4A4.
The special angles are:
μ0 = 0
μ1 = π/4
with μ = μ0 1 | S4A4 / S4A4 -
with μ = μ1 6 | S4 x I / D4D2 -
12 | S4 / C2 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		24 | S4 / E, for which the descriptive shares a half-turn with a half-turn in S4.
The special angles are:
μ0 = 0
μ1 = π/4
with μ = μ0 12 | S4 x I / D2C2 -
with μ = μ1 6 | S4 x I / D4D2 -
8 | S4 / C3 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		24 | S4 / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in S4.
The special angles are:
μ0 = 0
μ1 = π/3
with μ = μ0 2 | S4 x I / S4A4 -
with μ = μ1 8 | S4 x I / D3C3 -
24 | S4 x I / D1C1 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		48 | S4 x I / E, for which the descriptive shares a mirror with a mirror in S4xI, where the normal of mirror plane shares a 4-fold axis of the compound.
The special angles are:
μ0 = 0
μ1 = π/4
with μ = μ0 6 | S4 x I / D4D2 -
with μ = μ1 12 | S4 x I / D2C2 -
24 | S4 x I / C2C1 | μ
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ 		Example
Animation
Interactive Model 		48 | S4 x I / E, for which the descriptive shares a mirror with a mirror in S4xI, where the normal of mirror plane shares a 2-fold axis of the compound.
The special angles are:
μ0 = 0
μ1 = asin(2√2/3)
μ2 = π/2
with μ = μ0 2 | S4 x I / S4A4 -
with μ = μ1 8 | S4 x I / D3C3 -
with μ = μ2 12 | S4 x I / D2C2 -
24 | S4 x I / C2 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		48 | S4 x I / E, for which the descriptive shares a half-turn with a half-turn in S4xI.
The special angles are:
μ0 = 0
μ1 = π/4
with μ = μ0 6 | S4 x I / D4D2 -
with μ = μ1 12 | S4 x I / D2C2 -
16 | S4 x I / C3 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		48 | S4 x I / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in S4xI.
The special angles are:
μ0 = 0
μ1 = π/3
with μ = μ0 2 | S4 x I / S4A4 -
with μ = μ1 8 | S4 x I / D3C3 -
12 | S4 x I / C4C2 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		48 | S4 x I / E, for which the descriptive shares a half-turn with a 4-fold axis in S4xI.
The special angles are:
μ0 = 0
μ1 = π/4
with μ = μ0 2 | S4 x I / S4A4 -
with μ = μ1 6 | S4 x I / D4D2 -
30 | A5 / C2 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		60 | A5 / E, for which the descriptive shares a half-turn with a half-turn in A5.
The special angles are:
μ0 = 0
μ1 = π/4
with μ = μ0 30 | A5 x I / D2C2 -
with μ = μ1 5 | A5 / A4 -
20 | A5 / C3 | μ
μ ∈ ]μ2, μ0[ ∪ ]μ0, μ1[ 		Example
Animation
Interactive Model 		60 | A5 / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in A5.
The special angles are:
μ0 = 0
μ1 = acos(√10/4)
μ2 = -acos(√2(3+√5)/8)
with μ = μ0 5 | A5 / A4 -
with μ = μ1 20A | A5 x I / D3C3 -
with μ = μ2 20B | A5 x I / D3C3 -
60 | A5 x I / C2C1 | μ
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ ∪ ]μ2, μ3[ 		Example
Animation
Interactive Model 		120 | A5 x I / E, for which the descriptive shares a mirror with a mirror in A5xI.
The special angles are:
μ0 = 0
μ1 = acos((√2+1)√5 + √2-1/6)
μ2 = acos((√2-1)√5 + √2+1/6)
μ3 = π/2
with μ = μ0 30 | A5 x I / D2C2 -
with μ = μ1 20A | A5 x I / D3C3 -
with μ = μ2 20B | A5 x I / D3C3 -
with μ = μ3 30 | A5 x I / D2C2 -
60 | A5 x I / C2 | μ
μ ∈ ]μ0, μ1[ 		Example
Animation
Interactive Model 		120 | A5 x I / E, for which the descriptive shares a half-turn with a half-turn in A5xI.
The special angles are:
μ0 = 0
μ1 = π/4
with μ = μ0 30 | A5 x I / D2C2 -
with μ = μ1 10 | A5 x I / A4 -
40 | A5 x I / C3 | μ
μ ∈ ]μ2, μ0[ ∪ ]μ0, μ1[ 		Example
Animation
Interactive Model 		120 | A5 x I / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in A5xI.
The special angles are:
μ0 = 0
μ1 = acos(√10/4)
μ2 = -acos(√2(3+√5)/8)
with μ = μ0 10 | A5 x I / A4 -
with μ = μ1 20A | A5 x I / D3C3 -
with μ = μ2 20B | A5 x I / D3C3 -

Rigid Compounds of Tetrahedra

For some special angles the compounds with rotational freedom obtain a higher order symmetry. These are refered to as rigid compounds of cubes.

The following table contains all (rigid) compounds that have a dihedral and cyclic symmetry:

Compound2 Special Case of
n | D3nC3n / D3C3
for n>1, e.g.
2 | D6C6 / D3C3
3 | D9C9 / D3C3 		n | DnCn / C2C1 | μ1
2n | D3nC3n / C3 | μ0
2n | D3nC3n / C3 | μ1
2n | D6nD3n / D3C3, e.g.
2 | D6D3 / D3C3
4 | D12D6 / D3C3 		n | DnCn / C2C1 | μ3
2n | D3n / C3 | μ0
2n | D2nDn / C2C1 | μ1
4n | D6nD3n / C3 | μ0
n | D4nD2n / D4D2
for n is odd
and n>1, e.g.
3 | D12D6 / D4D2
5 | D20D10 / D4D2 		n | DnCn / C2C1 | μ0
n | Dn / C2 | μ0
2n | D4nD2n / C4C2 | μ1
n | D2nDn / D2C2
for n is odd
and n>1, e.g.
3 | D6D3 / D2C2
5 | D10D5 / D2C2 		n | DnCn / C2C1 | μ2
n | Dn / C2 | μ1
2n | D2nDn / C2 | μ1
2n | D2nDn / C2C1 | μ2
2n | D2nDn / D1C1 | μ0
2n | D4n x I / D4D2
for n>1, e.g.
4 | D8 x I / D4D2
6 | D12 x I / D4D2 		n | DnCn / C2C1 | μ0
n | Dn / C2 | μ0
2n | D2nDn / C2 | μ0
2n | D2nDn / C2C1 | μ0
2n | D4nD2n / C4C2 | μ0
2n | Dn x I / C2 | μ0
2n | Dn x I / C2C1 | μ0
4n | D4n x I / C4C2 | μ0
4n | D4n x I / C4C2 | μ1
2n | D3n x I / D3C3
for n>1, e.g.
4 | D6 x I / D3C3
6 | D9 x I / D3C3
8 | D12 x I / D3C3 		2n | D3n / C3 | μ0
4n | D6nD3n / C3 | μ1
2n | Dn x I / C2C1 | μ1
4n | D3n x I / C3 | μ0
4n | D3n x I / C3 | μ1
2n | D2n x I / D2C2
for n>1, e.g.
4 | D4 x I / D2C2
6 | D6 x I / D2C2
8 | D8 x I / D2C2 		n | DnCn / C2C1 | μ2
n | Dn / C2 | μ1
2n | D2nDn / C2 | μ1
2n | D2nDn / C2C1 | μ2
2n | D2nDn / D1C1 | μ1
2n | Dn x I / C2 | μ1
2n | Dn x I / C2C1 | μ2
4n | D2n x I / D1C1 | μ0
4n | D2n x I / D1C1 | μ1

The following table contains all (rigid) compounds that do not have a dihedral and cyclic symmetry:

Compound2 Special Case of
4 | S4A4 / D3C3 4 | A4 / C3 | μ1
8 | S4A4 / C3 | μ1
1 | S4A4 / S4A4
Trivial "compound" of 1 tetrahedron 		n | DnCn / C2C1 | μ0
n | DnCn / C2C1 | μ1
2n | D3nC3n / C3 | μ0
2n | D2nDn / C2 | μ0
2n | D2nDn / C2C1 | μ0
2n | D4nD2n / C4C2 | μ1
4 | A4 / C3 | μ0
12 | S4A4 / C2C1 | μ0
8 | S4A4 / C3 | μ0
6 | S4A4 / C4C2 | μ0
12 | S4 x I / D2C2 12 | A4 x I / C2C1 | μ1
12 | S4A4 / C2C1 | μ1
12 | S4 / C2 | μ0
24 | S4 x I / D1C1 | μ1
24 | S4 x I / C2C1 | μ2
24 | S4 x I / C2 | μ1
8 | S4 x I / D3C3
= 2 x 4 | S4A4 / D3C3 		8 | A4 x I / C3 | μ1
8 | S4 / C3 | μ1
24 | S4 x I / C2C1 | μ1
16 | S4 x I / C3 | μ1
6 | S4 x I / D4D2 12 | A4 x I / C2C1 | μ0
6 | S4A4 / C4C2 | μ1
12 | S4 / C2 | μ1
24 | S4 x I / D1C1 | μ0
24 | S4 x I / C2 | μ0
12 | S4 x I / C4C2 | μ1
2 | S4 x I / S4A4
Stella Octangula 		n | DnCn / C2C1 | μ0
n | DnCn / C2C1 | μ2
n | Dn / C2 | μ0
2n | D3n / C3 | μ1
2n | D4nD2n / C4C2 | μ0
2n | Dn x I / C2 | μ0
2n | Dn x I / C2C1 | μ0
2n | Dn x I / C2C1 | μ1
2n | Dn x I / C2C1 | μ2
4n | D3n x I / C3 | μ0
4n | D4n x I / C4C2 | μ0
8 | A4 x I / C3 | μ0
8 | S4 / C3 | μ0
24 | S4 x I / C2C1 | μ0
16 | S4 x I / C3 | μ0
12 | S4 x I / C4C2 | μ0
5 | A5 / A4 30 | A5 / C2 | μ1
20 | A5 / C3 | μ0
30 | A5 x I / D2C2 30 | A5 / C2 | μ0
60 | A5 x I / C2C1 | μ0
60 | A5 x I / C2C1 | μ3
60 | A5 x I / C2 | μ0
20A | A5 x I / D3C3 20 | A5 / C3 | μ1
60 | A5 x I / C2C1 | μ1
40 | A5 x I / C3 | μ1
20B | A5 x I / D3C3 20 | A5 / C3 | μ2
60 | A5 x I / C2C1 | μ2
40 | A5 x I / C3 | μ2
10 | A5 x I / A4
= 2 x 5 | A5 / A4 		60 | A5 x I / C2 | μ1
40 | A5 x I / C3 | μ0

Notes

1 For available X3D / VRML plugins for your browser and OS check here.
2 The table uses the notation as introduced in [HVerh00], where n | G / S means that the compound consists of n consituents and belongs to the symmetry group G. S is the symmetry group of the stabilizer, which means that the isometries from the subgroup S of G leave one constituent invariant.

References

[HVerh00] Verheyen, Hugo F: Symmetry Orbits, Birkhauser; 1 edition (January 26, 1996)
[HVerh01] Verheyen, Hugo F: Compound Lines of Polyhedra, unpublished

Links

Last Updated

2007-03-12, 12:17