With μ = degrees
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In the book [HVER01] Mr. Verheyen derives all possible groups of cube compounds. This page is based on his book. Later he wrote [HVER02] which holds the classication of compounds of all polyhedra. Unfortunately this book was never published. Nevertheless, this page is based on that book and it tries to summarise the compounds of tetrahedra with symmetries that aren't dihedral or cyclic based, which I refer to as Platonic symmetries. The special angles of the the compounds with rotational freedom aren't part of [HVER02] and are calculated by me.
A rough description of a method to obtain a certain tetrahedron 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 tetrahedron (referred to as the descriptive). Then by using the operators of the isometry group on this tetrahedron, a compound of tetrahedra belonging to that group is obtained. When the tetrahedron shares symmetries with the final compound, then some of them can be skipped. The former is referred to as the stabiliser symmetry group and it is an algebraic subgroup of the final symmetry, hence the naming n | G / F, where n refers to the amount of tetrahedra, G is the symmetry group of the compound and F is the stabiliser symmetry.
This page and the models are generated by a Python program [MTUN01]. They are displayed with help of a JavaScript bundle [MTUN02].
Using the above method for the finite groups of isometries one can get a first list of possible compounds of tetrahedra, as shown in the table below.
In the interactive scenes in the table below the descriptive is rotated around 1 specific axis where in fact any axis could be used. Therefore it would make more sense to use two slide-bars for two different axes.
| Compound | Static Model | Interactive Model |
| 12 | A4 / E | ||
| 24 | A4 x I / E | ||
| 24 | S4A4 / E | ||
| 24 | S4 / E | ||
| 48 | S4 x I / E | ||
| 60 | A5 / E | ||
| 120 | A5 x I / E |
This basic list of tetrahedron 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.
In [HVER01] the domain and other special angles were specified for the cube compounds with rotational freedom. No such angles are mentioned in [HVER02]. However I have tried to list them in the table below. I don't guarantee that all special angles are mentioned.
| Compound | 3D model | Description |
| 4 | A4 / C3 | μ
μ ∈ ]μ0, μ1[ |
|
12 | A4 / E, for which the descriptive shares a 3-fold axis with a 3-fold axis of A4. The interactive image contains the double domain, where
the first half and the second half give in principle the same
polyhedra, except that one is the dextro and one is the laevo
version.
The special angles are: μ0 = 0 μ1 = 60° μ2 = -2atan(2√3 - √15) ≈ 44.48° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | This is a 5 | A5 / A4 with one removed; it is shown in static example | |
| 8 | A4 x I / C3 | μ
μ ∈ ]μ0, μ1[ |
|
24 | A4 x I / E, for which the descriptive shares a 3-fold axis with a 3-fold axis of A4 x I. The compound consists of four Stella Octangulae rotating around 3-fold
axes.
The special angles are: μ0 = 0 μ1 = 60° μ2 = -2atan(2√3 - √15) ≈ 44.48° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | The 2 x 4 | D3 x I / C3 can be recognised in different ways, the 3D model shows an example. The compound is also a 10 | A5 x I / A4 with one Stella Octangula removed | |
| 12 | A4 x I / C2C1 | μ
μ ∈ ]μ0, μ1[ |
|
24 | A4 x I / E, for which the descriptive shares a reflection plane with a reflection plane of A4 x I. The compound consists of three pairs of Stella Octangulae rotating around 2-fold
axes. The pair components rotate in oppostie directions.
The special angles are: μ0 = 0 μ1 = 45° μ2 = asin(1/√3) ≈ 35.26° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | - | |
| 6 | S4A4 / C4C2 | μ
μ ∈ ]μ0, μ1[ |
|
24 | S4A4 / E, for which the descriptive shares a 2-fold axis with a 2-fold axis of S4A4. The compound consists of 3 pairs of tetrahedra of which the tetrahedra
rotate in opposite directions around a 2-fold axis
The special angles are: μ0 = 0 μ1 = 45° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| 8 | S4A4 / C3 | μ
μ ∈ ]μ0, μ1[ |
|
24 | S4A4 / E, for which the descriptive shares a 3-fold axis with a 3-fold axis of S4A4. The compound consists of 4 pairs of tetrahedra of which the tetrahedra
rotate in opposite directions around a 3-fold axis
The special angles are: μ0 = 0 μ1 = 60° μ2 = 30° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | The sub-compound of 2 | D3C3 / C3 gets a higher stability 2 | D6C6 / D3C3 | |
| 12 | S4A4 / C2C1 | μ
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ |
|
24 | S4A4 / E, for which the descriptive shares a reflection plane with a reflection plane of S4A4. The compound consists of 6 pairs of tetrahedra of which the tetrahedra
rotate in opposite directions around a normal of a reflection plane
The special angles are: μ0 = 0 μ1 = 2asin(1/√3) ≈ 70.53° μ2 = 90° μ3 = acos(1/√3) ≈ 54.73° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | - | |
| with μ = μ3 | A sub-compound 4 | D4D2 / C2C1 obtains a higher stability 4 | D12D6 / D3C3. | |
| 8 | S4 / C3 | μ
μ ∈ ]μ0, μ1[ |
|
24 | S4 / E, for which the descriptive shares a 3-fold axis with a 3-fold axis of S4. The compound consists of 4 antipairs of tetrahedra of which the tetrahedra
rotate in opposite directions around a 3-fold axis The interactive image contains the double domain, where
the first half and the second half give in principle the same
polyhedra, except that one is the dextro and one is the laevo
version.
The special angles are: μ0 = 0 μ1 = 60° μ2 = 30° μ3 = -2atan(2√3 - √15) ≈ 44.48° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | The sub-compound 2 | D3 / C3 gets a higher stability 2 | D6D3 / D3C3 | |
| with μ = μ3 | For the each angle holds already that it consists of 4 x 2 | D3 / C3, where order 3 axes are shared as indicated by the colouring of the standard model, but for this angle there are also 4 x 2 | D3 / C3 for which the order 3 axes are shared (in different ways) as shown in the example. All vertices are organised in regular pentagons; more specifically the vertices of this compound are the same as the vertices of a classic compound of two dodecahedra 2 | S4 x I / A4 x I, where one dodecahedron is rotated 90 degrees around a 2-fold axis. | |
| 12 | S4 / C2 | μ
μ ∈ ]μ0, μ1[ |
|
24 | S4 / E, for which the descriptive shares a 2-fold axis with a 2-fold axis of S4. The compound consists of 6 antipairs of tetrahedra of which the tetrahedra
rotate in opposite directions around a 2-fold axis The interactive image contains the double domain, where
the first half and the second half give in principle the same
polyhedra, except that one is the dextro and one is the laevo
version.
The special angles are: μ0 = 0 μ1 = 45° μ2 = atan(½(√2 - 1)(√5 - 1)) ≈ 14.36° μ3 = ½atan(√2/4) ≈ 9.74° μ4 = asin(1/√3) ≈ 35.26° μ5 = atan(√2/4) ≈ 19.47° μ6 = atan(½(√2 - 1)(√5 + 1)) ≈ 33.83° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | A reflection of a tetrahedron is shared with a compound reflection plane through a halfturn axis. | |
| with μ = μ3 | A halfturn axis of a tetrahedron is shared with an order 3 symmetry axis of the compound. | |
| with μ = μ4 | A halfturn of the final compound ends up in a reflection plane of the descriptor. For each of the 4 non-fundamental compounds 3 | D3 / C2 one can choose any pair which will be a 2 | D2C2 / C2C1. I.e. this sub-compound contains a reflection whereas the final symmetry lacks any opposite symmetries.It isn't possible to find 6 separate pairs and the static model on the left shows 4 pairs emphasised with colours that aren't gray. | |
| with μ = μ5 | The 3 fold axes of two elements coincide. For each of the 4 non-fundamental compounds 3 | D3 / C2 one can choose any pair which will be a 2 | D3 / C3. It isn't possible to find 6 separate pairs and the static model on the left shows 4 pairs emphasised with colours that aren't gray. | |
| with μ = μ6 | The 3 fold axes of two elements coincide. For each of the 4 non-fundamental compounds 3 | D3 / C2 one can choose any pair which will be a 2 | D3 / C3.It isn't possible to find 6 separate pairs and the static model on the left shows 4 pairs emphasised with colours that aren't gray. | |
| 12 | S4 x I / C4C2 | μ
μ ∈ ]μ0, μ1[ |
|
48 | S4 x I / E, for which the descriptive shares a 2-fold axis with a 4-fold axis of S4 x I. The compound consists of three pairs of Stella Octangulae rotating around 4-fold
axes. The pair components rotate in oppostie directions.
The special angles are: μ0 = 0 μ1 = 45° μ2 = 22.5° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | The sub-compound 4 | D4 x I / C4C2 gets a higher stability 4 | D8 x I / D4D2 | |
| 16 | S4 x I / C3 | μ
μ ∈ ]μ0, μ1[ |
|
48 | S4 x I / E, for which the descriptive shares a 3-fold axis with a 3-fold axis of S4 x I. The compound consists of four pairs of Stella Octangulae rotating around 3-fold
axes. The pair components rotate in oppostie directions.
The special angles are: μ0 = 0 μ1 = 60° μ2 = 30° μ3 = -2atan(2√3 - √15) ≈ 44.48° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | Each sub-compound 4 | D3 x I / C3 reaches its special angle and becomes a 4 | D6 x I / D3C3. | |
| with μ = μ3 |
() |
This occurs when a 3-fold axis of the descriptive ends up in a reflection plane of S4 x I. Then it becomes a compound of two 8 | A4 x I / C3 | μ2, which is a classic compound of ten tetrahedra with on Stella Octangula removed. These are put together by rotating one 90 degrees around a 2-fold axis. It is also a compound of two 8 | S4 / C3 | μ3 where one is mapped on the other be a central inversion. Both options are shown by using two different colours. |
| 24 | S4 x I / C2 | μ
μ ∈ ]μ0, μ1[ |
|
48 | S4 x I / E, for which the descriptive shares a 2-fold axis with a 2-fold axis of S4 x I. The compound consists of six pairs of Stella Octangulae rotating around 2-fold
axes. The pair components rotate in oppostie directions.
The special angles are: μ0 = 0 μ1 = 45° μ2 = ½atan(√2/4) ≈ 9.74° μ3 = asin(1/√3) ≈ 35.26° μ4 = 22.5° μ5 = atan(√2/4) ≈ 19.47° μ6 = atan(½(√2 - 1)(√5 - 1)) ≈ 14.36° μ7 = atan(½(√2 - 1)(√5 + 1)) ≈ 33.83° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | The sub-compound 6 | D3C3 / E gets a higher stability 6 | D6 x I / D2C2 | |
| with μ = μ3 | The sub-compound 6 | D3 x I / C2 gets a higher stability 6 | D12 x I / D4D2 | |
| with μ = μ4 | The sub-compound 4 | D4 x I / C4C2 gets a higher stability 4 | D8 x I / D4D2 | |
| with μ = μ5 | The sub-compound 6 | D3C3 / E gets a higher stability 6 | S4 x I / D4D2 | |
| with μ = μ6 | The 3 fold axes of two elements coincide and as a consequence the sub-compound with rotational freedom 6 | D3 x I / C2, sharing all its symmetries with the final symmetry, reaches a special angle where its sub-compound gets a higher stability, 2 | D3C3 / C3 instead of 2 | D2C2 / C2C1 | |
| with μ = μ7 | Same a before with a different sub-compound. The 3 fold axes of two elements coincide and as a consequence the sub-compound with rotational freedom 6 | D3 x I / C2, sharing all its symmetries with the final symmetry, reaches a special angle where its sub-compound gets a higher stability, 2 | D3C3 / C3 instead of 2 | D2C2 / C2C1 | |
| 24 | S4 x I / C2C1 | μ
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ |
|
48 | S4 x I / E, for which the descriptive shares reflection plane with reflection plane (through a 2-fold axis) of S4 x I. The compound consists of six pairs of Stella Octangulae rotating around 2-fold
axes. The pair components rotate in oppostie directions.
The special angles are: μ0 = 0 μ1 = 2asin(1/√3) ≈ 70.53° μ2 = 90° μ3 = asin(1/√3) ≈ 35.26° μ4 = acos(1/√3) ≈ 54.73° μ5 = 45° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | - | |
| with μ = μ3 | The sub-compound 4 | D2 x I / C2C1 gets a higher stability 4 | D6 x I / D3C3 | |
| with μ = μ4 | The sub-compound 8 | D4 x I / C2C1 gets a higher stability 8 | D12 x I / D3C3 | |
| with μ = μ5 | The sub-compound 4 | D2 x I / C2C1 gets a higher stability 4 | D4 x I / D2C2. The 4-fold axis of this sub-compound is shared with the 2-fold axis of the final compound. | |
| 24 | S4 x I / D1C1 | μ
μ ∈ ]μ0, μ1[ |
|
48 | S4 x I / E, for which the descriptive shares reflection plane with reflection plane (through 4-fold axes only) of S4 x I. The compound consists of six pairs of Stella Octangulae rotating around 2-fold
axes. The pair components rotate in oppostie directions.
The special angles are: μ0 = 0 μ1 = 45° μ2 = ½atan(√2/4) ≈ 9.74° μ3 = atan(√2/4) ≈ 19.47° μ4 = 22.5° μ5 = asin(1/√3) ≈ 35.26° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | The sub-compound 2 | D2C2 / C2C1 gets a higher stability 2 | D6C6 / D3C3 | |
| with μ = μ3 | The sub-compound 8 | D4 x I / D1C1 gets a higher stability 8 | S4 x I / D3C3 | |
| with μ = μ4 | The sub-compound 8 | D4 x I / D1C1 gets a higher stability 8 | D8 x I / D2C2 | |
| with μ = μ5 | The sub-compound 8 | D4 x I / C2C1 gets a higher stability 8 | D12 x I / D3C3 | |
| 20 | A5 / C3 | μ
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ |
|
60 | A5 / E, for which the descriptive shares a 3-fold axis with a 3-fold axis of A5. The compound consists of 10 antipairs of tetrahedra of which the tetrahedra
rotate in opposite directions around a 3-fold axis
The special angles are: μ0 = 0 μ1 = acos(√10/4) ≈ 37.76° μ2 = -acos(√2(3+√5)/8) ≈ -22.24° μ3 = ½acos(1+3√5/8) ≈ 7.76° μ4 = acos(1+3√5/8) ≈ 15.52° μ5 = acos(-7 + 3√5 + 3√(2+2√5)/8) ≈ 23.43 ° μ6 = acos(7+3√5/16) ≈ 31.04° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | - | |
| with μ = μ3 | The sub-compound 2 | D3C3 / C3 gets a higher stability 2 | D6D3 / D3C3 | |
| with μ = μ4 | The sub-compound 4 | A4 / C3 gets a higher stability 4 | S4A4 / D3C3. | |
| with μ = μ5 | The sub-compound 2 | C2 / E gets a higher stability 2 | D2C2 / C2C1.There are many such pairs. The way they are coloured in the example is by surrounding one 5-fold dihedral symmetry axis. | |
| with μ = μ6 | The sub-compound 2 | C2 / E gets a higher stability 2 | D3 / C3. | |
| 30 | A5 / C2 | μ
μ ∈ ]μ0, μ1[ |
|
60 | A5 / E, for which the descriptive shares a 2-fold axis with a 2-fold axis of A5. The compound consists of 15 pairs of tetrahedra of which the tetrahedra
rotate in opposite directions around a 2-fold axis
The special angles are: μ0 = 0 μ1 = 45° μ2 = atan(√5-2) ≈ 13.28° μ3 = acos(√5 + 1/2√3) ≈ 20.91° μ4 = atan(1/√5) ≈ 24.09° μ5 = ~26.57° (calculate algebraic expression) μ6 = acos(√(5+√5)/√10) ≈ 31.72° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | The sub-compound of 5 | D5 / C2 gets a higher stability 5 | D10D5 / D2C2 | |
| with μ = μ3 | A halfturn axis of a tetrahedron is shared with an order 3 symmetry axis of the compound and the sub-compound of 3 | D3 / C2 gets a higher stability 3 | D12D6 / D4D2 | |
| with μ = μ4 | The sub-compound of 3 | D3 / C2 gets a higher stability 3 | D6D3 / D2C2 | |
| with μ = μ5 | The sub-compound of 5 | D5 / C2 gets a higher stability 5 | A5 / A4 | |
| with μ = μ6 | A halfturn axis of a tetrahedron is shared with an order 5 symmetry axis of the compound and the sub-compound of 5 | D5 / C2 gets a higher stability 5 | D20D10 / D4D2 | |
| 40 | A5 x I / C3 | μ
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ |
|
120 | A5 x I / E, for which the descriptive shares a 3-fold axis with a 3-fold axis of A5. The compound consists of 10 pairs of Stella Octangulae rotating around 3-fold
axes. The pair components rotate in oppostie directions.
The special angles are: μ0 = 0 μ1 = acos(√10/4) ≈ 37.76° μ2 = -acos(√2(3+√5)/8) ≈ -22.24° μ3 = ½acos(1+3√5/8) ≈ 7.76° μ4 = acos(1+3√5/8) ≈ 15.52° μ5 = acos(-7 + 3√5 + 3√(2+2√5)/8) ≈ 23.43 ° μ6 = acos(7+3√5/16) ≈ 31.04° |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | - | |
| with μ = μ3 | The sub-compound 4 | D3 x I / C3 gets a higher stability 4 | D6 x I / D3C3 | |
| with μ = μ4 | The sub-compound 8 | S4A4 / C3 gets a higher stability 8 | S4 x I / D3C3 | |
| with μ = μ5 | The non-fundamental sub-compound 2 x 2 | C2 / E gets a higher stability 4 | C2 x I / E.There are many such pairs. The way they are coloured in the example is by surrounding one 5-fold dihedral symmetry axis. (the one from top-back to bottom-front). | |
| with μ = μ6 | The sub-compound 8 | A4 x I / C3 reaches a special angle and becomes 8 | A4 x I / C3 | μ2 | |
| 60 | A5 x I / C2 | μ
μ ∈ ]μ0, μ1[ |
|
120 | A5 x I / E, for which the descriptive shares a 2-fold axis with a 2-fold axis of A5 x I. The compound consists of 15 pairs of Stella Octangulae rotating around 2-fold
axes. The pair components rotate in oppostie directions.
The special angles are: μ0 = 0 μ1 = 45° μ2 = atan(√5-2) ≈ 13.28° μ3 = acos(√5 + 1/2√3) ≈ 20.91° μ4 = atan(1/√5) ≈ 24.09° μ5 = ~26.57° (calculate algebraic expression) μ6 = acos(√(5+√5)/√10) ≈ 31.72° μ7 = atan(½(√2 - 1)(√5 + 1)) ≈ 33.83° μ8 = ~41.81° (calculate algebraic expression) |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | The sub-compound of 10 | D5 x I / C2 gets a higher stability 10 | D10 x I / D2C2 | |
| with μ = μ3 | A halfturn axis of a tetrahedron is shared with an order 3 symmetry axis of the compound and the sub-compound of 6 | D3 x I / C2 gets a higher stability 6 | D12 x I / D4D2 | |
| with μ = μ4 | The sub-compound of 6 | D3 x I / C2 gets a higher stability 6 | D6 x I / D2C2 | |
| with μ = μ5 | The sub-compound of 10 | D5 x I / C2 gets a higher stability 10 | A5 x I / A4 | |
| with μ = μ6 | A halfturn axis of a tetrahedron is shared with an order 5 symmetry axis of the compound and the sub-compound of 10 | D5 x I / C2 gets a higher stability 10 | D20 x I / D4D2 | |
| with μ = μ7 | the sub-compound of 6 | D3 x I / C2 gets a higher stability 6 | S4 x I / D4D2 | |
| with μ = μ8 | the sub-compound of 2 | D2C2 / C2C1 gets a higher stability 4 | D6C6 / C3. The model uses ten colours, i.e. three of the sub-compounds use the same colour. | |
| 60 | A5 x I / C2C1 | μ
μ ∈ ]μ0, μ1[ ∪ ]μ1, μ2[ |
|
120 | A5 x I / E, for which the descriptive shares reflection plane with a reflection plane of A5 x I. The compound consists of 15 pairs of Stella Octangulae rotating around 2-fold
axes. The pair components rotate in oppostie directions.
The special angles are: μ0 = 0 μ1 = 33.83 degrees (TODO calculate) μ2 = 75.64 degrees (TODO calculate) |
| with μ = μ0 | - | |
| with μ = μ1 | - | |
| with μ = μ2 | - |
For some angles the cube compounds with rotational freedom obtain a higher order symmetry. These are referred to as rigid compounds of tetrahedra. The table below summarises these and specifies from which compounds with rotational freedom they can be obtained.
2024-04-16, 07:38 CET