Compounds of Octahedra

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Contents

Introduction

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 octahedron 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 octahedron (referred to as the descriptive). Then by using the operators of the isometry group on this octahedron, a compound of octahedra belonging to that group is obtained.

For example one can obtain a compound of octahedra belonging to the symmetry group S4 x I (the symmetry group of the cube and the octahedron). Choose the orientation of the S4 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 S4, i.e.

to obtain 1 + 3*3 + 2*4 + 6 = 24 octahedra. Since the central inversion leaves all these octahedra invariant, the compound of these octahedra will belong to the symmetry group S4 x I. It is however possible that some of the octahedra end up in the same position, leading to a compound of less than 24 octahedra. For instance if the descriptive would have been positioned in such a way that the order 4 symmetry axes of the octahedron would have been formed by the axes of the coordinate system, then all the isometries leave the octahedron invariant, thus leading to the trivial compound of 1 octahedron.

The set symmetry operations of the final symmetry that map the descriptive on itself form a group, and it is a sub-group of the final symmetery. This sub-group is part of the right part of the compound notation, the one starting with a '/'. Algebra can be used to obtain which operations are needed to map the descriptive onto a cube that isn't the descriptive.

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Compounds of Octahedra with Central Freedom

Using the above method for the finite groups of isometries one can get a first list of possible compounds of octahedra, as shown in the following table.

Compound1 VRML image VRML animation
n | Cn x I / E x I
n ≥ 2 		for n=3 for n=3
2n | Dn x I / E x I
n ≥ 1 		for n=3 for n=2
12 | A4 x I / E x I Image Animation
24 | S4 x I / E x I Image Animation
60 | A5 x I / E x I Image Animation

This basic list of octahedron 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.

Compounds of Octahedra 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 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.

Compound1 VRML model Description
nA | Dn x I / C2 x I | μ
n ≥ 3
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ 		Examples:
for n=3
for n=4
for n=6
Animations:
for n=3
for n=4
for n=6 		n | Cn x I / E x I, for which the descriptive shares a mirror with a mirror in Dn. The mirror contains 2 opposite edges of the descriptive.
The special angles are:
μ0 = 0
μ1 = acos(1/√3)
μ2 = π/2
μ3 = π/4
μ4 = asin(1/√3)
μ5 = asin(2√2/3)
with μ = μ0
and n is odd 		3 | D12 x I / D4 x I (n=3) n | D4n x I / D4 x I
with μ = μ0
and n=2m, m is odd 		3 | D12 x I / D4 x I (n=6) m | D4m x I / D4 x I
with μ = μ0
and n=4m 		3 | D12 x I / D4 x I (n=12) m | D4m x I / D4 x I
with μ = μ12
and n=3 		1 | S4 x I / S4 x I -
with μ = μ1
and n is coprime with 3 		2 | D6 x I / D3 x I (n=2) n | D3n x I / D3 x I
with μ = μ1
and n=3m (m>1) 		3 | D9 x I / D3 x I (n=9) m | D3m x I / D3 x I
with μ = μ2
and n is odd 		3 | D6 x I / D2 x I (n=3) n | D2n x I / D2 x I
with μ = μ2
and n=2m 		2 | D4 x I / D2 x I (n=4)
3 | D6 x I / D2 x I (n=6) 		m | D2m x I / D2 x I
with μ = μ3
and n=2m 		for n = 4
for n = 6 (only one emphasised) 		m x 2 | D4 x I / D2 x I
with μ = μ42
and n=2m 		for n = 4
for n = 6 		m x 2 | D6 x I / D3 x I
with μ = μ52
and n=4m 		for n = 4
for n = 8
for n = 12 (only one emphasised) 		m x 4 | S4 x I / D3 x I
nB | Dn x I / C2 x I | μ
n ≥ 3
μ ∈ ]μ0, μ1[ 		Examples:
for n=3
for n=4
for n=6
Animations:
for n=3
for n=4
for n=6 		n | Cn x I / E x I, for which the descriptive shares a mirror with a mirror in Dn. The mirror contains a ring of four face centres of the descriptive.
The special angles are:
μ0 = 0
μ1 = π/4
μ2 = π/8
μ3 = asin(1/√3)
μ4 = α, with
cos(α) - sin(α) = cos(2π/5)/sin(2π/5)
with μ = μ0
and n is odd 		3 | D12 x I / D4 x I (n=3) n | D4n x I / D4 x I
with μ = μ0
and n=2m, m is odd 		3 | D12 x I / D4 x I (n=6) m | D4m x I / D4 x I
with μ = μ0
and n=4m 		3 | D12 x I / D4 x I (n=12) m | D4m x I / D4 x I
with μ = μ1
and n is odd 		3 | D6 x I / D2 x I (n=3) n | D2n x I / D2 x I
with μ = μ12
and n=2 		1 | S4 x I / S4 x I -
with μ = μ1
and n=2m (m>1) 		2 | D4 x I / D2 x I (n=4)
3 | D6 x I / D2 x I (n=6) 		m | D2m x I / D2 x I
with μ = μ2
and n=2m 		for n = 4
for n = 6 		m x 2 | D8 x I / D4 x I
with μ = μ32
and n=3m 		for n = 3
for n = 6
for n = 9 (only one emphasised) 		m x 3 | S4 x I / D4 x I
with μ = μ42
and n=5m 		for n = 5
for n = 10 		m x 5 | A5 x I / A4 x I
2n | D2n x I / D1 x I | μ
n ≥ 1
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[, for n = 1
μ ∈ ]μ0, μ1[, for n ≥ 2 		Examples:
for n=2
for n=3
Animations:
for n=2
for n=3 		2n | Dn x I / E x I, for which the descriptive shares an order 2 symmetry axis with the n-fold axis of Dn. The compound consists of pairs of n | D2n x I / D2 x I, which rotate in opposite directions. See for instance for n=2 or for n=3s
The special angles are:
μ0 = 0
μ1 = asin(1/√3)
μ2 = π/4n
μ3 = |asin(1/√3) - kπ/2n| for k ∈ ℕ
with μ = μ0
and n = 1 		1 | S4 x I / S4 x I -
with μ = μ0
and n ≥ 2 		2 | D4 x I / D2 x I (n=2) n | D2n x I / D2 x I
with μ = μ1
and n = 1 		2 | D6 x I / D3 x I 2 | D6 x I / D3 x I
with μ = μ2
and m = 2n 		2 | D4 x I / D2 x I (n=1)
4 | D8 x I / D2 x I (n=2) 		m | D2m x I / D2 x I
with μ = μ32
and n ≥ 2 		for n = 2
for n = 3 		n x 2 | D6 x I / D3 x I
2n | D3n x I / C3 x I | μ
n ≥ 1
μ ∈ ]μ0, μ1[ 		Examples:
for n=1
for n=2
Animations:
for n=1
for n=2
for n=3 		2n | Dn x I / E x I, for which the descriptive shares an order 3 symmetry axis with the n-fold axis of Dn. The compound consists of pairs of n | D3n x I / D3 x I, which rotate in opposite directions. See for instance for n=2 or for n=3s
The special angles are:
μ0 = 0
μ1 = π/6n
with μ = μ0
and n = 1 		1 | S4 x I / S4 x I -
with μ = μ0
and n ≥ 2 		3 | D9 x I / D3 x I n | D3n x I / D3 x I
with μ = μ1
and m = 2n 		2 | D6 x I / D3 x I m | D3m x I / D3 x I
2n | D4n x I / C4 x I | μ
n ≥ 1
μ ∈ ]μ0, μ1[ 		Examples:
for n=2
for n=3
Animations:
for n=2
for n=3 		2n | Dn x I / E x I, for which the descriptive shares an order 4 symmetry axis with the n-fold axis of Dn. The compound consists of pairs of n | D4n x I / D4 x I, which rotate in opposite directions. See for instance for n=2s
The special angles are:
μ0 = 0
μ1 = π/8n
with μ = μ0
and n = 1 		1 | S4 x I / S4 x I -
with μ = μ0
and n ≥ 2 		3 | D12 x I / D4 x I n | D4n x I / D4 x I
with μ = μ1
and m = 2n 		4 | D16 x I / D4 x I m | D4m x I / D4 x I
6 | A4 x I / C2 x I | μ
μ ∈ ]μ0, μ1[ 		Example
Animation 		12 | A4 x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 2-fold axis of A4.
The special angles are:
μ0 = 0
μ1 = π/4
μ2 = asin(1/√3)
with μ = μ0 3 | S4 x I / D4 x I -
with μ = μ1 6 | S4 x I / D2 x I -
with μ = μ2 3 x 2 | D6 x I / D3 x I -
4 | A4 x I / C3 x I | μ
μ ∈ ]μ0, μ1[ 		Example
Animation 		12 | A4 x I / E x I, for which the descriptive shares an order 3 symmetry axis with a 3-fold axis of A4.
The special angles are:
μ0 = 0
μ1 = π/3
μ2 = acos(-1 + 3√5/8)
with μ = μ0 1 | S4 x I / S4 x I -
with μ = μ1 4 | S4 x I / D3 x I -
with μ = μ2 2 x 2 | D3 x I / C3 x I | ½μ2 -
12A | S4 x I / C2 x I | μ
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ 		Example
Animation 		24 | S4 x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 2-fold axis of S4.
The special angles are:
μ0 = 0
μ1 = asin(2√2/3)
μ2 = π/2
μ3 = asin(1/√3)
μ4 = acos(1/√3)
μ5 = π/4
with μ = μ0 1 | S4 x I / S4 x I -
with μ = μ1 4 | S4 x I / D3 x I -
with μ = μ2 6 | S4 x I / D2 x I -
with μ = μ3 4 x 3 | D6 x I / D2 x I or
6 x 2 | D6 x I / D3 x I 		-
with μ = μ4 3 x 4 | D12 x I / D3 x I or
4 x 3 | D9 x I / D3 x I 		-
with μ = μ5 6 x 2 | D4 x I / D2 x I -
12B | S4 x I / C2 x I | μ
μ ∈ ]μ0, μ1[ 		Example
Animation 		24 | S4 x I / E x I, for which the descriptive shares an order 4 symmetry axis with a 2-fold axis of S4.
The special angles are:
μ0 = 0
μ1 = π/4
μ2 = acos(1/√3) - π/4
μ3 = asin(1/√3)
μ4 = π/8
μ5 = ½ atan(4√2/7)
with μ = μ0 3 | S4 x I / D4 x I -
with μ = μ1 6 | S4 x I / D2 x I -
with μ = μ2 4 x 3 | D6 x I / D2 x I -
with μ = μ3 4 x 3 | D12 x I / D4 x I -
with μ = μ4 6 x 2 | D8 x I / D4 x I -
with μ = μ52 4 x 3 | S4 x I / D4 x I
(only one emphasised) 		-
12 | S4 x I / D1 x I | μ
μ ∈ ]μ0, μ1[ 		Example
Animation 		24 | S4 x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 4-fold axis of S4. 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 | D2 x I / D1 x I | μ. There is an alternative way, for which the compound consists of pairs of 2 | D4 x I / D2 x I, which rotate in opposite directions. See for instance this animations
The special angles are:
μ0 = 0
μ1 = π/4
μ2 = acos(1/√3) - π/4
μ3 = asin(1/√3)
μ4 = π/8
μ5 = ½ atan(4√2/7)
with μ = μ0 3 | S4 x I / D4 x I -
with μ = μ1 6 | S4 x I / D2 x I -
with μ = μ2 6 x 2 | D6 x I / D3 x I -
with μ = μ3 3 x 4 | D12 x I / D3 x I -
with μ = μ4 3 x 4 | D8 x I / D2 x I -
with μ = μ52 3 x 4 | S4 x I / D3 x I
(only one emphasised) 		-
8 | S4 x I / C3 x I | μ
μ ∈ ]μ0, μ1[ 		Example
Animation 		24 | S4 x I / E x I, for which the descriptive shares an order 3 symmetry axis with a 3-fold axis of S4.
The special angles are:
μ0 = 0
μ1 = π/3
μ2 = acos(-1 + 3√5/8)
μ3 = π/6
with μ = μ0 1 | S4 x I / S4 x I -
with μ = μ0 4 | S4 x I / D3 x I -
with μ = μ2 2 x 4 | A4 x I / C3 x I | μ2
(alternative colour arrangement,
alternative colour arrangement,
alternative colour arrangement) 		-
with μ = μ3 4 x 2 | D6 x I / D3 x I -
6 | S4 x I / C4 x I | μ
μ ∈ ]μ0, μ1[ 		Example
Animation 		24 | S4 x I / E x I, for which the descriptive shares an order 4 symmetry axis with a 4-fold axis of S4.
The special angles are:
μ0 = 0
μ1 = π/4
μ2 = π/8
with μ = μ0 1 | S4 x I / S4 x I -
with μ = μ1 3 | S4 x I / D4 x I -
with μ = μ2 3 x 2 | D8 x I / D4 x I -
30A | A5 x I / C2 x I | μ
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ ∪ ]μ2, μ3[ 		Example
Animation 		60 | A5 x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 2-fold axis of A5.
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
μ4 = asin(1/√3)
μ5 = acos(1/√3)
μ6 = acos(√5+1/2√3)
μ7 = acos(√(5+√5)/√10)
μ8 = acos(√(5-√5)/√10)
μ9 = acos(√5-1/2√3)
μ10 = π/4
with μ = μ0 15 | A5 x I / D2 x I -
with μ = μ1 10A | A5 x I / D3 x I -
with μ = μ2 10B | A5 x I / D3 x I -
with μ = μ3 15 | A5 x I / D2 x I -
with μ = μ4 15 x 2 | D6 x I / D3 x I
(only one emphasised) 		-
with μ = μ5 15 x 2 | D6 x I / D3 x I
(only one emphasised) 		-
with μ = μ6 10 x 3 | D6 x I / D2 x I
(only one emphasised) 		-
with μ = μ7 6 x 5 | D20 x I / D4 x I
(only one emphasised) 		-
with μ = μ8 6 x 5 | D10 x I / D2 x I
(only one emphasised) 		-
with μ = μ9 10 x 3 | D12 x I / D4 x I
(only one emphasised) 		-
with μ = μ10 15 x 2 | D4 x I / D2 x I
(only one emphasised) 		-
30B | A5 x I / C2 x I | μ
μ ∈ ]μ0, μ1[ 		Example
Animation 		60 | A5 x I / E x I, for which the descriptive shares an order 4 symmetry axis with a 2-fold axis of A5.
The special angles are:
μ0 = 0
μ1 = π/4
μ2 = acos(√(5+2√5)/√10)
μ3 = acos(√5+1/2√3)
μ4 = acos(√5/√6)
μ5 = acos(√(5+√5)/√10)
μ6 = π/8
with μ = μ0 5 | A5 x I / A4 x I -
with μ = μ1 15 | A5 x I / D2 x I -
with μ = μ2 6 x 5 | D10 x I / D2 x I
(only one emphasised) 		-
with μ = μ3 10 x 3 | D12 x I / D4 x I
(only one emphasised) 		-
with μ = μ4 10 x 3 | D6 x I / D2 x I
(only one emphasised) 		-
with μ = μ5 6 x 5 | D20 x I / D4 x I
(only one emphasised) 		-
with μ = μ6 15 x 2 | D8 x I / D4 x I
(only one emphasised) 		-
20 | A5 x I / C3 x I | μ
μ ∈ ]μ2, μ0[ ∪ ]μ0, μ1[ 		Example
Animation 		60 | A5 x I / E x I, for which the descriptive shares an order 3 symmetry axis with a 3-fold axis of A5.
The special angles are:
μ0 = 0
μ1 = acos(√10/4)
μ2 = -acos(√2(3+√5)/8)
μ3 = acos(7+3√5/16)
μ4 = acos(-7 + 3√5 + 3√(2+2√5)/8)
μ5 = acos(1+3√5/8)
μ6 = ½acos(1+3√5/8)
with μ = μ0 5 | A5 x I / A4 x I -
with μ = μ1 10A | A5 x I / D3 x I -
with μ = μ2 10B | A5 x I / D3 x I -
with μ = μ3 5 x 4 | A4 x I / C3 x I | μ3
(only one emphasised) or
5 x 4 | A4 x I / C3 x I | 2μ1 - μ3
(only one emphasised) 		-
with μ = μ4 5 x 4 | A4 x I / C3 x I | μ4
(only one emphasised) or
5 x 4 | A4 x I / C3 x I | 2μ1 - μ4
(only one emphasised) 		-
with μ = μ5 5 x 4 | S4 x I / D3 x I
(only one emphasised) or
5 x 4 | A4 x I / C3 x I | μ5
(only one emphasised) 		-
with μ = μ6 10 x 2 | D6 x I / D3 x I
(only one emphasised) 		-

Rigid Compounds of Octahedra

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.

Compound1 Special Case of
n | D4n x I / D4 x I, e.g.
3 | D12 x I / D4 x I 		nA | Dn x I / C2 x I | μ0
nB | Dn x I / C2 x I | μ0
2n | D4n x I / C4 x I | μ0
2n | D4n x I / C4 x I | μ1
n | D3n x I / D3 x I, e.g.
2 | D6 x I / D3 x I
3 | D9 x I / D3 x I 		nA | Dn x I / C2 x I | μ1
2 | D2 x I / D1 x I | μ1
2n | D3n x I / C3 x I | μ0
2n | D3n x I / C3 x I | μ1
n | D2n x I / D2 x I, e.g.
2 | D4 x I / D2 x I
3 | D6 x I / D2 x I 		nA | Dn x I / C2 x I | μ2
nB | Dn x I / C2 x I | μ1
2n | D2n x I / D1 x I | μ0
2n | D2n x I / D1 x I | μ2
1 | S4 x I / S4 x I 2 | D2 x I / D1 x I | μ0
4 | A4 x I / C3 x I | μ0
12A | S4 x I / C2 x I | μ0
8 | S4 x I / C3 x I | μ0
6 | S4 x I / C4 x I | μ0
3 | S4 x I / D4 x I 6 | A4 x I / C2 x I | μ0
12B | S4 x I / C2 x I | μ0
12 | S4 x I / D1 x I | μ0
6 | S4 x I / C4 x I | μ1
3B | D3 x I / C2 x I | μ3
4 | S4 x I / D3 x I 4 | A4 x I / C3 x I | μ1
12A | S4 x I / C2 x I | μ1
8 | S4 x I / C3 x I | μ1
4A | D4 x I / C2 x I | μ5
6 | S4 x I / D2 x I 6 | A4 x I / C2 x I | μ1
12A | S4 x I / C2 x I | μ2
12B | S4 x I / C2 x I | μ1
12 | S4 x I / D1 x I | μ1
5 | A5 x I / A4 x I 30B | A5 x I / C2 x I | μ0
20 | A5 x I / C3 x I | μ0
10A | A5 x I / D3 x I 30A | A5 x I / C2 x I | μ1
20 | A5 x I / C3 x I | μ1
10B | A5 x I / D3 x I 30A | A5 x I / C2 x I | μ2
20 | A5 x I / C3 x I | μ2
15 | A5 x I / D2 x I 30A | A5 x I / C2 x I | μ0
30A | A5 x I / C2 x I | μ3
30B | A5 x I / C2 x I | μ1

Notes

1The table uses the notation as used in Symmetry Orbits, in which the first part states the amount of cubes the the middle part to which symmetry cube the compound belongs.
2This special case I found myself and is not mentioned in Symmetry Orbits. These special cases are:

References

Verheyen, Hugo F: Symmetry Orbits, Birkhauser; 1 edition (January 26, 1996)

Links

Last Updated

2019-02-10, 12:25 CET