# Compounds of Tetrahedra

## 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

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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.