Click here if you want to watch the models with a X3D player instead of a VRML player.
In the book Symmetry Orbits Mr. Verheyen derives all possible groups of cube compounds. This page is based on his book. A rough description of a method to obtain a certain cube compound can be as follows: Choose one of the finite groups of isometries, define an orientation for this group and define a position of a cube (refered to as the descriptive). Then by using the operators of the isometry group on this cube, a compound of cubes belonging to that group is obtained.
For example one can obtain a compound of cubes belonging to the symmetry group S_{4} x I (the symmetry group of the cube and the octahedron). Choose the orientation of the S_{4} in such a way that the order 4 symmetry axes are formed by the Euclidean coordinate axes. Now position the descriptive in a random position but with its centre in the origin of the Euclidean space. Finally use all the symmetries in S_{4}, i.e.
to obtain 1 + 3*3 + 2*4 + 6 = 24 cubes. Since the central inversion leaves all these cubes invariant, the compound of these cubes will belong to the symmetry group S_{4} x I. It is however possible that some of the cubes end up in the same position, leading to a compound of less than 24 cubes. For instance if the descriptive would have been positioned in such a way that the order 4 symmetry axes of the cube would have been formed by the axes of the coordinate system, then all the isometries leave the cube invariant, thus leading to the trivial compound of 1 cube.
To be able to view and investigate the 3D models you need a VRML player. See note^{2}.
Using the above method for the finite groups of isometries one can get a first list of possible compounds of cubes, as shown in the following table.
This basic list of cube compounds is the list of compounds with central freedom; i.e. after having specified the centre of the descriptive, its position can be chosen freely, except for some special cases. In the animations in the table above the descriptive is rotated around 2 distinct axes in a linear way.
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 one or more symmetry axes of the descriptive are shared with one or more symmetry axes of the whole compound. If one axis is shared then the descriptive can be rotated freely around this axis, without changing that property. This leads to a group of compounds with rotational freedom.
In the table below these are summarised. The animations stop at the special angles as mentioned in Symmetry Orbits.
Compound^{1} | VRML^{2} model | Description |
nA | D_{n} x I / C_{2} x I | μ
n ≥ 3 μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[ |
Images:
for n=3 for n=4 for n=6 Animations: for n=3 for n=4 for n=6 |
n | C_{n} x I / E x I, for which the descriptive shares a mirror with a mirror in D_{n}. The mirror contains 2 opposite edges of the descriptive.
The special angles are: μ_{0} = 0 μ_{1} = acos(^{1}/_{√3}) μ_{2} = ^{π}/_{2} μ_{3} = ^{π}/_{4} μ_{4} = asin(^{1}/_{√3}) μ_{5} = asin(^{2√2}/_{3}) |
with μ = μ_{0}
and n is odd |
3 | D_{12} x I / D_{4} x I | n | D_{4n} x I / D_{4} x I |
with μ = μ_{0}
and n=2m, m is odd |
3 | D_{12} x I / D_{4} x I | m | D_{4m} x I / D_{4} x I |
with μ = μ_{0}
and n=4m |
3 | D_{12} x I / D_{4} x I | m | D_{4m} x I / D_{4} x I |
with μ = μ_{1}
and n is coprime with 3 |
3 | D_{9} x I / D_{3} x I | n | D_{3n} x I / D_{3} x I |
with μ = μ_{1}
and n=3m |
3 | D_{9} x I / D_{3} x I | m | D_{3m} x I / D_{3} x I |
with μ = μ_{2}
and n is odd |
2 | D_{4} x I / D_{2} x I
3 | D_{6} x I / D_{2} x I |
n | D_{2n} x I / D_{2} x I |
with μ = μ_{2}
and n=2m |
2 | D_{4} x I / D_{2} x I
3 | D_{6} x I / D_{2} x I |
m | D_{2m} x I / D_{2} x I |
with μ = μ_{3}
and n=2m |
for n = 4
for n = 6 (only one emphasised) |
m x 2 | D_{4} x I / D_{2} x I |
with μ = μ_{4}^{3}
and n=2m |
for n = 4
for n = 6 |
m x 2 | D_{6} x I / D_{3} x I |
with μ = μ_{5}^{3}
and n=4m |
for n = 4
for n = 8 for n = 12 (only one emphasised) |
m x 4 | S_{4} x I / D_{3} x I |
nB | D_{n} x I / C_{2} x I | μ
n ≥ 3 μ ∈ ]μ_{0}, μ_{1}[ |
Images:
for n=3 for n=4 for n=6 Animations: for n=3 for n=4 for n=6 |
n | C_{n} x I / E x I, for which the descriptive shares a mirror with a mirror in D_{n}. The mirror contains a ring of four face centres of the descriptive.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} μ_{2} = ^{π}/_{8} μ_{3} = asin(^{1}/_{√3}) |
with μ = μ_{0}
and n is odd |
3 | D_{12} x I / D_{4} x I | n | D_{4n} x I / D_{4} x I |
with μ = μ_{0}
and n=2m, m is odd |
3 | D_{12} x I / D_{4} x I | m | D_{4m} x I / D_{4} x I |
with μ = μ_{0}
and n=4m |
3 | D_{12} x I / D_{4} x I | m | D_{4m} x I / D_{4} x I |
with μ = μ_{1}
and n is odd |
2 | D_{4} x I / D_{2} x I
3 | D_{6} x I / D_{2} x I |
n | D_{2n} x I / D_{2} x I |
with μ = μ_{1}
and n=2m |
2 | D_{4} x I / D_{2} x I
3 | D_{6} x I / D_{2} x I |
m | D_{2m} x I / D_{2} x I |
with μ = μ_{2}
and n=2m |
for n = 4
for n = 6 |
m x 2 | D_{8} x I / D_{4} x I |
with μ = μ_{3}^{3}
and n=3m |
for n = 3
for n = 6 for n = 9 (only one emphasised) |
m x 3 | S_{4} x I / D_{4} x I |
2n | D_{2n} x I / D_{1} x I | μ
n ≥ 1 μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[, for n = 1 μ ∈ ]μ_{0}, μ_{1}[, for n ≥ 2 |
Images:
for n=2 for n=3 Animations: for n=2 for n=3 |
2n | D_{n} x I / E x I, for which the descriptive shares an order 2 symmetry axis with the n-fold axis of D_{n}. The compound consists of pairs of n | D_{2n} x I / D_{2} x I, which rotate in opposite directions. See for instance for n=2 or for n=3
The special angles are: μ_{0} = 0 μ_{1} = asin(^{1}/_{√3}) μ_{2} = ^{π}/_{4n} μ_{3} = |asin(^{1}/_{√3}) - ^{kπ}/_{2n}| for k ∈ ℕ |
with μ = μ_{0}
and n = 1 |
1 | S_{4} x I / S_{4} x I | - |
with μ = μ_{0}
and n ≥ 2 |
2 | D_{4} x I / D_{2} x I | n | D_{2n} x I / D_{2} x I |
with μ = μ_{1}
and n = 1 |
2 | D_{6} x I / D_{3} x I | 2 | D_{6} x I / D_{3} x I |
with μ = μ_{2}
and m = 2n |
2 | D_{4} x I / D_{2} x I
4 | D_{8} x I / D_{2} x I |
m | D_{2m} x I / D_{2} x I |
with μ = μ_{3}^{3}
and n ≥ 2 |
for n = 2
for n = 3 |
n x 2 | D_{6} x I / D_{3} x I |
2n | D_{3n} x I / C_{3} x I | μ
n ≥ 1 μ ∈ ]μ_{0}, μ_{1}[ |
Images:
for n=1 for n=2 Animations: for n=1 for n=2 for n=3 |
2n | D_{n} x I / E x I, for which the descriptive shares an order 3 symmetry axis with the n-fold axis of D_{n}. The compound consists of pairs of n | D_{3n} x I / D_{3} x I, which rotate in opposite directions. See for instance for n=2 or for n=3
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{6n} |
with μ = μ_{0}
and n = 1 |
1 | S_{4} x I / S_{4} x I | - |
with μ = μ_{0}
and n ≥ 2 |
3 | D_{9} x I / D_{3} x I | n | D_{3n} x I / D_{3} x I |
with μ = μ_{1}
and m = 2n |
2 | D_{6} x I / D_{3} x I | m | D_{3m} x I / D_{3} x I |
2n | D_{4n} x I / C_{4} x I | μ
n ≥ 1 μ ∈ ]μ_{0}, μ_{1}[ |
Images:
for n=2 for n=3 Animations: for n=2 for n=3 |
2n | D_{n} x I / E x I, for which the descriptive shares an order 4 symmetry axis with the n-fold axis of D_{n}. The compound consists of pairs of n | D_{4n} x I / D_{4} x I, which rotate in opposite directions. See for instance for n=2
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{8n} |
with μ = μ_{0}
and n = 1 |
1 | S_{4} x I / S_{4} x I | - |
with μ = μ_{0}
and n ≥ 2 |
3 | D_{12} x I / D_{4} x I | n | D_{4n} x I / D_{4} x I |
with μ = μ_{1}
and m = 2n |
4 | D_{16} x I / D_{4} x I | m | D_{4m} x I / D_{4} x I |
6 | A_{4} x I / C_{2} x I | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Image
Animation |
12 | A_{4} x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 2-fold axis of A_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} μ_{2} = asin(^{1}/_{√3}) |
with μ = μ_{0} | 3 | S_{4} x I / D_{4} x I | - |
with μ = μ_{1} | 6 | S_{4} x I / D_{2} x I | - |
with μ = μ_{2} | 3 x 2 | D_{6} x I / D_{3} x I | - |
4 | A_{4} x I / C_{3} x I | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Image
Animation |
12 | A_{4} x I / E x I, for which the descriptive shares an order 3 symmetry axis with a 3-fold axis of A_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{3} μ_{2} = acos(^{-1 + 3√5}/_{8}) |
with μ = μ_{0} | 1 | S_{4} x I / S_{4} x I | - |
with μ = μ_{1} | 4 | S_{4} x I / D_{3} x I | - |
with μ = μ_{2} | 2 x 2 | D_{3} x I / C_{3} x I | ½μ_{2} | - |
12A | S_{4} x I / C_{2} x I | μ
μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[ |
Image
Animation |
24 | S_{4} x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 2-fold axis of S_{4}.
The special angles are: μ_{0} = 0 μ_{1} = asin(^{2√2}/_{3}) μ_{2} = ^{π}/_{2} μ_{3} = asin(^{1}/_{√3}) μ_{4} = acos(^{1}/_{√3}) μ_{5} = ^{π}/_{4} |
with μ = μ_{0} | 1 | S_{4} x I / S_{4} x I | - |
with μ = μ_{1} | 4 | S_{4} x I / D_{3} x I | - |
with μ = μ_{2} | 6 | S_{4} x I / D_{2} x I | - |
with μ = μ_{3} | 4 x 3 | D_{6} x I / D_{2} x I or
6 x 2 | D_{6} x I / D_{3} x I |
- |
with μ = μ_{4} | 3 x 4 | D_{12} x I / D_{3} x I or
4 x 3 | D_{12} x I / D_{4} x I |
- |
with μ = μ_{5} | 6 x 2 | D_{4} x I / D_{2} x I | - |
12B | S_{4} x I / C_{2} x I | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Image
Animation |
24 | S_{4} x I / E x I, for which the descriptive shares an order 4 symmetry axis with a 2-fold axis of S_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} μ_{2} = acos(^{1}/_{√3}) - ^{π}/_{4} μ_{3} = asin(^{1}/_{√3}) μ_{4} = ^{π}/_{8} μ_{5} = ½ atan(^{4√2}/_{7}) |
with μ = μ_{0} | 3 | S_{4} x I / D_{4} x I | - |
with μ = μ_{1} | 6 | S_{4} x I / D_{2} x I | - |
with μ = μ_{2} | 4 x 3 | D_{6} x I / D_{2} x I | - |
with μ = μ_{3} | 4 x 3 | D_{12} x I / D_{4} x I | - |
with μ = μ_{4} | 6 x 2 | D_{8} x I / D_{4} x I | - |
with μ = μ_{5}^{3} | 4 x 3 | S_{4} x I / D_{4} x I
(only one emphasised) |
- |
12 | S_{4} x I / D_{1} x I | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Image
Animation |
24 | S_{4} x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 4-fold axis of S_{4}. One order 4 axis of the compound is shared with the 2-fold axis of 4 elements. Hence there are 2 different ways to make pairs of these. By its colour arrangement the animation emphasises the case where the 4 elements consist of a pair of 2 | D_{2} x I / D_{1} x I | μ. There is an alternative way, for which the compound consists of pairs of 2 | D_{4} x I / D_{2} x I, which rotate in opposite directions. See for instance this animation
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} μ_{2} = acos(^{1}/_{√3}) - ^{π}/_{4} μ_{3} = asin(^{1}/_{√3}) μ_{4} = ^{π}/_{8} μ_{5} = ½ atan(^{4√2}/_{7}) |
with μ = μ_{0} | 3 | S_{4} x I / D_{4} x I | - |
with μ = μ_{1} | 6 | S_{4} x I / D_{2} x I | - |
with μ = μ_{2} | 6 x 2 | D_{6} x I / D_{3} x I | - |
with μ = μ_{3} | 3 x 4 | D_{12} x I / D_{3} x I | - |
with μ = μ_{4} | 3 x 4 | D_{8} x I / D_{2} x I | - |
with μ = μ_{5}^{3} | 3 x 4 | S_{4} x I / D_{3} x I
(only one emphasised) |
- |
8 | S_{4} x I / C_{3} x I | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Image
Animation |
24 | S_{4} x I / E x I, for which the descriptive shares an order 3 symmetry axis with a 3-fold axis of S_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{3} μ_{2} = acos(^{-1 + 3√5}/_{8}) μ_{3} = ^{π}/_{6} |
with μ = μ_{0} | 1 | S_{4} x I / S_{4} x I | - |
with μ = μ_{0} | 4 | S_{4} x I / D_{3} x I | - |
with μ = μ_{2} | 2 x 4 | A_{4} x I / C_{3} x I | μ_{2}
(alternative colour arrangement, alternative colour arrangement, alternative colour arrangement) |
- |
with μ = μ_{3} | 4 x 2 | D_{6} x I / D_{3} x I | - |
6 | S_{4} x I / C_{4} x I | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Image
Animation |
24 | S_{4} x I / E x I, for which the descriptive shares an order 4 symmetry axis with a 4-fold axis of S_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} μ_{2} = ^{π}/_{8} |
with μ = μ_{0} | 1 | S_{4} x I / S_{4} x I | - |
with μ = μ_{1} | 3 | S_{4} x I / D_{4} x I | - |
with μ = μ_{2} | 3 x 2 | D_{8} x I / D_{4} x I | - |
30A | A_{5} x I / C_{2} x I | μ
μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[ ∪ ]μ_{2}, μ_{3}[ |
Image
Animation |
60 | A_{5} x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 2-fold axis of A_{5}.
The special angles are: μ_{0} = 0 μ_{1} = acos(^{(√2+1)√5 + √2-1}/_{6}) μ_{2} = acos(^{(√2-1)√5 + √2+1}/_{6}) μ_{3} = ^{π}/_{2} μ_{4} = asin(^{1}/_{√3}) μ_{5} = acos(^{1}/_{√3}) μ_{6} = acos(^{√5+1}/_{2√3}) μ_{7} = acos(^{√(5+√5)}/_{√10}) μ_{8} = acos(^{√(5-√5)}/_{√10}) μ_{9} = acos(^{√5-1}/_{2√3}) μ_{10} = ^{π}/_{4} |
with μ = μ_{0} | 15 | A_{5} x I / D_{2} x I | - |
with μ = μ_{1} | 10A | A_{5} x I / D_{3} x I | - |
with μ = μ_{2} | 10B | A_{5} x I / D_{3} x I | - |
with μ = μ_{3} | 15 | A_{5} x I / D_{2} x I | - |
with μ = μ_{4} | 15 x 2 | D_{6} x I / D_{3} x I
(only one emphasised) |
- |
with μ = μ_{5} | 15 x 2 | D_{6} x I / D_{3} x I
(only one emphasised) |
- |
with μ = μ_{6} | 10 x 3 | D_{6} x I / D_{2} x I
(only one emphasised) |
- |
with μ = μ_{7} | 6 x 5 | D_{20} x I / D_{4} x I
(only one emphasised) |
- |
with μ = μ_{8} | 6 x 5 | D_{10} x I / D_{2} x I
(only one emphasised) |
- |
with μ = μ_{9} | 10 x 3 | D_{12} x I / D_{4} x I
(only one emphasised) |
- |
with μ = μ_{10} | 15 x 2 | D_{4} x I / D_{2} x I
(only one emphasised) |
- |
30B | A_{5} x I / C_{2} x I | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Image
Animation |
60 | A_{5} x I / E x I, for which the descriptive shares an order 4 symmetry axis with a 2-fold axis of A_{5}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} μ_{2} = acos(^{√(5+2√5)}/_{√10}) μ_{3} = acos(^{√5+1}/_{2√3}) μ_{4} = acos(^{√5}/_{√6}) μ_{5} = acos(^{√(5+√5)}/_{√10}) μ_{6} = ^{π}/_{8} |
with μ = μ_{0} | 5 | A_{5} x I / A_{4} x I | - |
with μ = μ_{1} | 15 | A_{5} x I / D_{2} x I | - |
with μ = μ_{2} | 6 x 5 | D_{10} x I / D_{2} x I
(only one emphasised) |
- |
with μ = μ_{3} | 10 x 3 | D_{12} x I / D_{4} x I
(only one emphasised) |
- |
with μ = μ_{4} | 10 x 3 | D_{6} x I / D_{2} x I
(only one emphasised) |
- |
with μ = μ_{5} | 6 x 5 | D_{20} x I / D_{4} x I
(only one emphasised) |
- |
with μ = μ_{6} | 15 x 2 | D_{8} x I / D_{4} x I
(only one emphasised) |
- |
20 | A_{5} x I / C_{3} x I | μ
μ ∈ ]μ_{2}, μ_{0}[ ∪ ]μ_{0}, μ_{1}[ |
Image
Animation |
60 | A_{5} x I / E x I, for which the descriptive shares an order 3 symmetry axis with a 3-fold axis of A_{5}.
The special angles are: μ_{0} = 0 μ_{1} = acos(^{√10}/_{4}) μ_{2} = -acos(^{√2(3+√5)}/_{8}) μ_{3} = acos(^{7+3√5}/_{16}) μ_{4} = acos(^{-7 + 3√5 + 3√(2+2√5)}/_{8}) μ_{5} = acos(^{1+3√5}/_{8}) μ_{6} = ½acos(^{1+3√5}/_{8}) |
with μ = μ_{0} | 5 | A_{5} x I / A_{4} x I | - |
with μ = μ_{1} | 10A | A_{5} x I / D_{3} x I | - |
with μ = μ_{2} | 10B | A_{5} x I / D_{3} x I | - |
with μ = μ_{3} | 5 x 4 | A_{4} x I / C_{3} x I | μ_{3}
(only one emphasised) or 5 x 4 | A_{4} x I / C_{3} x I | 2μ_{1} - μ_{3} (only one emphasised) |
- |
with μ = μ_{4} | 5 x 4 | A_{4} x I / C_{3} x I | μ_{4}
(only one emphasised) or 5 x 4 | A_{4} x I / C_{3} x I | 2μ_{1} - μ_{4} (only one emphasised) |
- |
with μ = μ_{5} | 5 x 4 | S_{4} x I / D_{3} x I
(only one emphasised) or 5 x 4 | A_{4} x I / C_{3} x I | μ_{5} (only one emphasised) |
- |
with μ = μ_{6} | 10 x 2 | D_{6} x I / D_{3} x I
(only one emphasised) |
- |
For some angles the cube compounds with rotational freedom obtain a higher order symmetry. These are refered to as rigid compounds of cubes. The table below summarises these and specifies from which compounds with rotational freedom they can be obtained.
Compound^{1} | Special Case of |
n | D_{4n} x I / D_{4} x I, e.g.
3 | D_{12} x I / D_{4} x I |
nA | D_{n} x I / C_{2} x I | μ_{0}
nB | D_{n} x I / C_{2} x I | μ_{0} 2n | D_{4n} x I / C_{4} x I | μ_{0} 2n | D_{4n} x I / C_{4} x I | μ_{1} |
n | D_{3n} x I / D_{3} x I, e.g.
2 | D_{6} x I / D_{3} x I 3 | D_{9} x I / D_{3} x I |
nA | D_{n} x I / C_{2} x I | μ_{1}
2 | D_{2} x I / D_{1} x I | μ_{1} 2n | D_{3n} x I / C_{3} x I | μ_{0} 2n | D_{3n} x I / C_{3} x I | μ_{1} |
n | D_{2n} x I / D_{2} x I, e.g.
2 | D_{4} x I / D_{2} x I 3 | D_{6} x I / D_{2} x I |
nA | D_{n} x I / C_{2} x I | μ_{2}
nB | D_{n} x I / C_{2} x I | μ_{1} 2n | D_{2n} x I / D_{1} x I | μ_{0} 2n | D_{2n} x I / D_{1} x I | μ_{2} |
1 | S_{4} x I / S_{4} x I | 2 | D_{2} x I / D_{1} x I | μ_{0}
4 | A_{4} x I / C_{3} x I | μ_{0} 12A | S_{4} x I / C_{2} x I | μ_{0} 8 | S_{4} x I / C_{3} x I | μ_{0} 6 | S_{4} x I / C_{4} x I | μ_{0} |
3 | S_{4} x I / D_{4} x I
Classical Compound of 3 Cubes |
6 | A_{4} x I / C_{2} x I | μ_{0}
12B | S_{4} x I / C_{2} x I | μ_{0} 12 | S_{4} x I / D_{1} x I | μ_{0} 6 | S_{4} x I / C_{4} x I | μ_{1} 3B | D_{3} x I / C_{2} x I | μ_{3} |
4 | S_{4} x I / D_{3} x I
Bakos' Compound of 4 Cubes |
4 | A_{4} x I / C_{3} x I | μ_{1}
12A | S_{4} x I / C_{2} x I | μ_{1} 8 | S_{4} x I / C_{3} x I | μ_{1} 4A | D_{4} x I / C_{2} x I | μ_{5} |
6 | S_{4} x I / D_{2} x I | 6 | A_{4} x I / C_{2} x I | μ_{1}
12A | S_{4} x I / C_{2} x I | μ_{2} 12B | S_{4} x I / C_{2} x I | μ_{1} 12 | S_{4} x I / D_{1} x I | μ_{1} |
5 | A_{5} x I / A_{4} x I
Classical Compound of 5 Cubes |
30B | A_{5} x I / C_{2} x I | μ_{0}
20 | A_{5} x I / C_{3} x I | μ_{0} |
10A | A_{5} x I / D_{3} x I | 30A | A_{5} x I / C_{2} x I | μ_{1}
20 | A_{5} x I / C_{3} x I | μ_{1} |
10B | A_{5} x I / D_{3} x I | 30A | A_{5} x I / C_{2} x I | μ_{2}
20 | A_{5} x I / C_{3} x I | μ_{2} |
15 | A_{5} x I / D_{2} x I | 30A | A_{5} x I / C_{2} x I | μ_{0}
30A | A_{5} x I / C_{2} x I | μ_{3} 30B | A_{5} x I / C_{2} x I | μ_{1} |
In all these compounds the cubes can be replaced by any other polyhedron with cube symmetry. This does not influence the properties for the whole compound. For instance one can use the same constructs to summarise all compounds of Octahedra or the compounds of Cuboctahedra.
2008-02-27, 07:45