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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 cuboctahedron 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 cuboctahedron (referred to as the descriptive). Then by using the operators of the isometry group on this cuboctahedron, a compound of cuboctahedra belonging to that group is obtained.
For example one can obtain a compound of cuboctahedra 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 cuboctahedra. Since the central inversion leaves all these cuboctahedra invariant, the compound of these cuboctahedra will belong to the symmetry group S_{4} x I. It is however possible that some of the cuboctahedra end up in the same position, leading to a compound of less than 24 cuboctahedra. For instance if the descriptive would have been positioned in such a way that the order 4 symmetry axes of the cuboctahedron would have been formed by the axes of the coordinate system, then all the isometries leave the cuboctahedron invariant, thus leading to the trivial compound of 1 cuboctahedron.
The set symmetry operations of the final symmetry that map the descriptive on itself form a group, and it is a subgroup of the final symmetery. This subgroup 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.
Using the above method for the finite groups of isometries one can get a first list of possible compounds of cuboctahedra, as shown in the following table.
In the interactive scenes in the table below the descriptive is rotated around 1 axis where in fact any combination of axes could be used.
Compound^{1}  Static Model  Interactive Model 
n  C_{n} x I / E x I
n ≥ 2 

2n  D_{n} x I / E x I
n ≥ 1 

12  A_{4} x I / E x I  
24  S_{4} x I / E x I  
60  A_{5} x I / E x I 
This basic list of cuboctahedron 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.
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.
Compound^{1}  3D model  Description 
nA  D_{n} x I / C_{2} x I  μ
n ≥ 3 μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[ 
Static Examples:
Interactive models: 
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 
n  D_{4n} x I / D_{4} x I  
with μ = μ_{0}
and n=2m, m is odd 
m  D_{4m} x I / D_{4} x I  
with μ = μ_{0}
and n=4m 
m  D_{4m} x I / D_{4} x I  
with μ = μ_{1}^{2}
and n=3 
  
with μ = μ_{1}
and n is coprime with 3 
n  D_{3n} x I / D_{3} x I  
with μ = μ_{1}
and n=3m (m>1) 
m  D_{3m} x I / D_{3} x I  
with μ = μ_{2}
and n is odd 
n  D_{2n} x I / D_{2} x I  
with μ = μ_{2}
and n=2m 

m  D_{2m} x I / D_{2} x I 
with μ = μ_{3}
and n=2m 
() 
m x 2  D_{4} x I / D_{2} x I 
with μ = μ_{4}^{2}
and n=2m 

m x 2  D_{6} x I / D_{3} x I 
with μ = μ_{5}^{2}
and n=4m 
() 
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}[ 
Static Examples:
Interactive models: 
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}) μ_{4} = α, with cos(α)  sin(α) = ^{cos(2π/5)}/_{sin(2π/5)} 
with μ = μ_{0}
and n is odd 
n  D_{4n} x I / D_{4} x I  
with μ = μ_{0}
and n=2m, m is odd 
m  D_{4m} x I / D_{4} x I  
with μ = μ_{0}
and n=4m 
m  D_{4m} x I / D_{4} x I  
with μ = μ_{1}
and n is odd 
n  D_{2n} x I / D_{2} x I  
with μ = μ_{1}^{2}
and n=2 
  
with μ = μ_{1}
and n=2m (m>1) 

m  D_{2m} x I / D_{2} x I 
with μ = μ_{2}
and n=2m 

m x 2  D_{8} x I / D_{4} x I 
with μ = μ_{3}^{2}
and n=3m 
() 
m x 3  S_{4} x I / D_{4} x I 
with μ = μ_{4}^{2}
and n=5m 

m x 5  A_{5} x I / A_{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_{2}[, for n ≥ 2 
Static Examples:
Interactive models: 
2n  D_{n} x I / E x I, for which the descriptive shares an order 2 symmetry axis with the nfold axis of D_{n}. The compound consists of pairs of n  D_{2n} x I / D_{2} x I, which rotate in opposite directions.
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 
  
with μ = μ_{0}
and n ≥ 2 
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  
with μ = μ_{2}
and m = 2n 

m  D_{2m} x I / D_{2} x I 
with μ = μ_{3}^{2}
and n ≥ 2 

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}[ 
Static Examples:
Interactive models: 
2n  D_{n} x I / E x I, for which the descriptive shares an order 3 symmetry axis with the nfold axis of D_{n}. The compound consists of pairs of n  D_{3n} x I / D_{3} x I, which rotate in opposite directions.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{6n} 
with μ = μ_{0}
and n = 1 
  
with μ = μ_{0}
and n ≥ 2 
n  D_{3n} x I / D_{3} x I  
with μ = μ_{1}
and m = 2n 
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}[ 
Static Examples:
Interactive models: 
2n  D_{n} x I / E x I, for which the descriptive shares an order 4 symmetry axis with the nfold axis of D_{n}. The compound consists of pairs of n  D_{4n} x I / D_{4} x I, which rotate in opposite directions.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{8n} 
with μ = μ_{0}
and n = 1 
  
with μ = μ_{0}
and n ≥ 2 
n  D_{4n} x I / D_{4} x I  
with μ = μ_{1}
and m = 2n 
m  D_{4m} x I / D_{4} x I  
6  A_{4} x I / C_{2} x I  μ
μ ∈ ]μ_{0}, μ_{1}[ 

12  A_{4} x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 2fold axis of A_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} μ_{2} = asin(^{1}/_{√3}) 
with μ = μ_{0}    
with μ = μ_{1}    
with μ = μ_{2}    
4  A_{4} x I / C_{3} x I  μ
μ ∈ ]μ_{0}, μ_{1}[ 

12  A_{4} x I / E x I, for which the descriptive shares an order 3 symmetry axis with a 3fold 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}    
with μ = μ_{1}    
with μ = μ_{2}    
12A  S_{4} x I / C_{2} x I  μ
μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[ 

24  S_{4} x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 2fold 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}    
with μ = μ_{1}    
with μ = μ_{2}    
with μ = μ_{3}  or

 
with μ = μ_{4}  or

 
with μ = μ_{5}    
12B  S_{4} x I / C_{2} x I  μ
μ ∈ ]μ_{0}, μ_{1}[ 

24  S_{4} x I / E x I, for which the descriptive shares an order 4 symmetry axis with a 2fold 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}    
with μ = μ_{1}    
with μ = μ_{2}    
with μ = μ_{3}    
with μ = μ_{4}    
with μ = μ_{5}^{2} 
() 
 
12  S_{4} x I / D_{1} x I  μ
μ ∈ ]μ_{0}, μ_{1}[ 

24  S_{4} x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 4fold axis of S_{4}. One order 4 axis of the compound is shared with the 2fold axis of 4 elements. Hence there are 2 different ways to make pairs of these.
One way is where the 4 elements consist of a pair if 2  D_{2} x I / D_{1} x I  μ. For the other way
the compound consists of pairs of 2  D_{4} x I / D_{2} x I, which rotate in opposite directions.
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}    
with μ = μ_{1}    
with μ = μ_{2}    
with μ = μ_{3}    
with μ = μ_{4}    
with μ = μ_{5}^{2} 
() 
 
8  S_{4} x I / C_{3} x I  μ
μ ∈ ]μ_{0}, μ_{1}[ 

24  S_{4} x I / E x I, for which the descriptive shares an order 3 symmetry axis with a 3fold 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}    
with μ = μ_{1}    
with μ = μ_{2} 
(, , ) 
 
with μ = μ_{3}    
6  S_{4} x I / C_{4} x I  μ
μ ∈ ]μ_{0}, μ_{1}[ 

24  S_{4} x I / E x I, for which the descriptive shares an order 4 symmetry axis with a 4fold axis of S_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} μ_{2} = ^{π}/_{8} 
with μ = μ_{0}    
with μ = μ_{1}    
with μ = μ_{2}    
30A  A_{5} x I / C_{2} x I  μ
μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[ ∪ ]μ_{2}, μ_{3}[ 

60  A_{5} x I / E x I, for which the descriptive shares an order 2 symmetry axis with a 2fold axis of A_{5}.
The special angles are: μ_{0} = 0 μ_{1} = acos(^{(√2+1)√5 + √21}/_{6}) μ_{2} = acos(^{(√21)√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(^{√51}/_{2√3}) μ_{10} = ^{π}/_{4} 
with μ = μ_{0}    
with μ = μ_{1}    
with μ = μ_{2}    
with μ = μ_{3}    
with μ = μ_{4} 
() 
 
with μ = μ_{5} 
() 
 
with μ = μ_{6} 
() 
 
with μ = μ_{7} 
() 
 
with μ = μ_{8} 
() 
 
with μ = μ_{9} 
() 
 
with μ = μ_{10} 
() 
 
30B  A_{5} x I / C_{2} x I  μ
μ ∈ ]μ_{0}, μ_{1}[ 

60  A_{5} x I / E x I, for which the descriptive shares an order 4 symmetry axis with a 2fold 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}    
with μ = μ_{1}    
with μ = μ_{2} 
() 
 
with μ = μ_{3} 
() 
 
with μ = μ_{4} 
() 
 
with μ = μ_{5} 
() 
 
with μ = μ_{6} 
() 
 
20  A_{5} x I / C_{3} x I  μ
μ ∈ ]μ_{2}, μ_{0}[ ∪ ]μ_{0}, μ_{1}[ 

60  A_{5} x I / E x I, for which the descriptive shares an order 3 symmetry axis with a 3fold 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}    
with μ = μ_{1}    
with μ = μ_{2}    
with μ = μ_{3} 
() or () 
 
with μ = μ_{4} 
() or () 
 
with μ = μ_{5} 
() or () 
 
with μ = μ_{6} 
() 
 
For some angles the cube compounds with rotational freedom obtain a higher order symmetry. These are referred to as rigid compounds of cuboctahedra. 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.

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.

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.

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} 
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} 

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  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  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} 

30B  A_{5} x I / C_{2} x I  μ_{0}
20  A_{5} x I / C_{3} x I  μ_{0} 

30A  A_{5} x I / C_{2} x I  μ_{1}
20  A_{5} x I / C_{3} x I  μ_{1} 

30A  A_{5} x I / C_{2} x I  μ_{2}
20  A_{5} x I / C_{3} x I  μ_{2} 

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} 
20240416, 17:29 CET