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 | S_{4}A_{4} / C_{4}C_{2} is obtained from 24 | S_{4}A_{4} / 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:
Compound^{2} | VRML^{1} model | Description |
n | D_{n}C_{n} / C_{2}C_{1} | μ
for n>1 μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[ |
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 | D_{n}C_{n} / E, for which the descriptive shares a mirror with a mirror in D_{n}C_{n}.
The special angles are: μ_{0} = 0 μ_{1} = atan(√2) μ_{2} = ^{π}/_{2} μ_{3} = atan(^{1}/_{√2}) |
with μ = μ_{0}
and n is odd |
3 | D_{12}D_{6} / D_{4}D_{2} | n | D_{4n}D_{2n} / D_{4}D_{2} |
with μ = μ_{0}
and n = 2 |
1 | S_{4}A_{4} / S_{4}A_{4} | - |
with μ = μ_{0}
and n = 4 |
2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{0}
and n = 2m, m > 1, and m odd |
3 | D_{12}D_{6} / D_{4}D_{2} | m | D_{4m}D_{2m} / D_{4}D_{2} |
with μ = μ_{0}
and n = 4m, m > 1 |
4 | D_{8} x I / D_{4}D_{2} | 2m | D_{4m} x I / D_{4}D_{2} |
with μ = μ_{1}
and n is coprime with 3 |
2 | D_{6}C_{6} / D_{3}C_{3} | n | D_{3n}C_{3n} / D_{3}C_{3} |
with μ = μ_{1}
and n = 3 |
1 | S_{4}A_{4} / S_{4}A_{4} | - |
with μ = μ_{1}
and n = 3m, where m > 1 |
2 | D_{6}C_{6} / D_{3}C_{3} | m | D_{3m}C_{3m} / D_{3}C_{3} |
with μ = μ_{2}
and n = 2 |
2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{2}
and n is odd |
3 | D_{6}D_{3} / D_{2}C_{2} | n | D_{2n}D_{n} / D_{2}C_{2} |
with μ = μ_{2}
and n = 2m and m > 1 |
4 | D_{4} x I / D_{2}C_{2} | 2m | D_{2m} x I / D_{2}C_{2} |
with μ = μ_{3}
and n = 2 |
2 | D_{6}D_{3} / D_{3}C_{3} | 2 | D_{6}D_{3} / D_{3}C_{3} |
2n | D_{3n}C_{3n} / C_{3} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
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 | D_{m}C_{m} / E, for which the descriptive shares a 3-fold axis with an m-fold axis in D_{m}C_{m}, where m = 3*n.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{6n} |
with μ = μ_{0}
and n = 1 |
1 | S_{4}A_{4} / S_{4}A_{4} | - |
with μ = μ_{0}
and n > 1 |
2 | D_{6}C_{6} / D_{3}C_{3} | n | D_{3n}C_{3n} / D_{3}C_{3} |
with μ = μ_{1}
and m = 2n |
2 | D_{6}C_{6} / D_{3}C_{3} | m | D_{3m}C_{3m} / D_{3}C_{3} |
n | D_{n} / C_{2} | μ
for n>2 μ ∈ ]μ_{0}, μ_{1}[ |
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 | D_{n} / E, for which the descriptive shares a 2-fold axis with a half turn in D_{n} (i.e. not with the principal axis).
If n = 2 then 2 | D_{2} / C_{2} becomes a 2k | D_{4k}D_{2k} / C_{4}C_{2}(with k=1) The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} |
with μ = μ_{0}
and n = 4 |
2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{0}
and n is odd |
3 | D_{12}D_{6} / D_{4}D_{2} | n | D_{4n}D_{2n} / D_{4}D_{2} |
with μ = μ_{0}
and n = 2m, m > 1, and m odd |
3 | D_{12}D_{6} / D_{4}D_{2} | m | D_{4m}D_{2m} / D_{4}D_{2} |
with μ = μ_{0}
and n = 4m, m > 1 |
4 | D_{8} x I / D_{4}D_{2} | 2m | D_{4m} x I / D_{4}D_{2} |
with μ = μ_{1}
and n is odd |
3 | D_{6}D_{3} / D_{2}C_{2} | n | D_{2n}D_{n} / D_{2}C_{2} |
with μ = μ_{1}
and n = 2m |
6 | D_{6} x I / D_{2}C_{2} | 2m | D_{2m} x I / D_{2}C_{2} |
2n | D_{3n} / C_{3} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
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 | D_{m} / E, for which the descriptive shares a 3-fold axis with an m-fold axis in D_{m}, where m = 3*n.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{6n} |
with μ = μ_{0}
and n is odd |
2 | D_{6}D_{3} / D_{3}C_{3} | 2n | D_{6n}D_{3n} / D_{3}C_{3} |
with μ = μ_{0}
and n = 2m |
4 | D_{6} x I / D_{3}C_{3} | 2n | D_{3n} x I / D_{3}C_{3} |
with μ = μ_{1}
and n = 1 |
2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{0}
and n is odd and n > 1 |
6 | D_{9} x I / D_{3}C_{3} | 2n | D_{3n} x I / D_{3}C_{3} |
with μ = μ_{0}
and n = 2m |
4 | D_{12}D_{6} / D_{3}C_{3} | 2n | D_{6n}D_{3n} / D_{3}C_{3} |
2n | D_{2n}D_{n} / C_{2} | μ
for n>1 μ ∈ ]μ_{0}, μ_{1}[ |
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 | D_{2n}D_{n} / E, for which the descriptive shares a 2-fold axis with a half turn in D_{2n}D_{n} (i.e. not with the principal axis).
If n = 1 then 2 | D_{2}D_{1} / C_{2} becomes a 2k | D_{4k}D_{2k} / C_{4}C_{2}(with k=1) The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} |
with μ = μ_{0}
and n = 2 |
1 | S_{4}A_{4} / S_{4}A_{4} | - |
with μ = μ_{0}
and n is odd |
6 | D_{12} x I / D_{4}D_{2} | 2n | D_{4n} x I / D_{4}D_{2} |
with μ = μ_{0}
and n = 2m and m > 1 |
4 | D_{8} x I / D_{4}D_{2} | 2m | D_{4m} x I / D_{4}D_{2} |
with μ = μ_{1}
and n is odd |
3 | D_{6}D_{3} / D_{2}C_{2} | n | D_{2n}D_{n} / D_{2}C_{2} |
with μ = μ_{1}
and n = 2m |
4 | D_{4} x I / D_{2}C_{2} | 2n | D_{2n} x I / D_{2}C_{2} |
2n | D_{2n}D_{n} / C_{2}C_{1} | μ
for n>1 μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[ |
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 | D_{2n}D_{n} / E, for which the descriptive shares a mirror with a mirror (through the pricipal axis) in D_{2n}D_{n}.
If n = 1 then 2 | D_{2}D_{1} / C_{2}C_{1} equals to a k | D_{k}C_{k} / C_{2}C_{1}(with k=2) The special angles are: μ_{0} = 0 μ_{1} = atan(√2) μ_{2} = ^{π}/_{2} |
with μ = μ_{0}
and n = 2 |
1 | S_{4}A_{4} / S_{4}A_{4} | - |
with μ = μ_{0}
and n is odd |
6 | D_{12} x I / D_{4}D_{2} | 2n | D_{4n} x I / D_{4}D_{2} |
with μ = μ_{0}
and n = 2m and m > 1 |
4 | D_{8} x I / D_{4}D_{2} | 2m | D_{4m} x I / D_{4}D_{2} |
with μ = μ_{1}
and n is coprime with 3 |
4 | D_{12}D_{6} / D_{3}C_{3} | 2n | D_{6n}D_{3n} / D_{3}C_{3} |
with μ = μ_{1}
and n = 3m |
2 | D_{6}D_{3} / D_{3}C_{3} | 2m | D_{6m}D_{3m} / D_{3}C_{3} |
with μ = μ_{2}
and n is odd |
3 | D_{6}D_{3} / D_{2}C_{2} | n | D_{2n}D_{n} / D_{2}C_{2} |
with μ = μ_{2}
and n = 2m |
4 | D_{4} x I / D_{2}C_{2} | 2n | D_{2n} x I / D_{2}C_{2} |
2n | D_{2n}D_{n} / D_{1}C_{1} | μ
for n>1 and n is odd μ ∈ ]μ_{0}, μ_{1}[ |
Examples:
for n=3 for n=5 Animations: for n=3 for n=5 Interactive Models: for n=3 for n=5 |
4n | D_{2n}D_{n} / E, for which the descriptive shares a mirror with a mirror that is perpendicular to the pricipal axis in D_{2n}D_{n} (thus requiring that n is odd).
If n = 1 then 2 | D_{2}D_{1} / D_{1}C_{1} equals to a k | D_{k}C_{k} / C_{2}C_{1}(with k=2) The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{2n} |
with μ = μ_{0} | 3 | D_{6}D_{3} / D_{2}C_{2} | n | D_{2n}D_{n} / D_{2}C_{2} |
with μ = μ_{1} | 6 | D_{6} x I / D_{2}C_{2} | 2n | D_{2n} x I / D_{2}C_{2} |
2n | D_{4n}D_{2n} / C_{4}C_{2} | μ
for n is odd μ ∈ ]μ_{0}, μ_{1}[ |
Examples:
for n=1 for n=3 Animations: for n=1 for n=3 Interactive Models: for n=1 for n=3 |
4m | D_{2m}D_{m} / E, for which the descriptive shares a half turn with an m-fold axis in D_{2m}D_{m}, where m = 2*n.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4n} |
with μ = μ_{0}
and n = 1 |
2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{0} | 6 | D_{12} x I / D_{4}D_{2} | 2n | D_{4n} x I / D_{4}D_{2} |
with μ = μ_{1}
and n = 1 |
1 | S_{4}A_{4} / S_{4}A_{4} | - |
with μ = μ_{1} | 3 | D_{12}D_{6} / D_{4}D_{2} | n | D_{4n}D_{2n} / D_{4}D_{2} |
4n | D_{6n}D_{3n} / C_{3} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
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 | D_{2m}D_{m} / E, for which the descriptive shares a 3-fold axis with an m-fold axis in D_{2m}D_{m}, where m = 3*n.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{6n} |
with μ = μ_{0} | 4 | D_{12}D_{6} / D_{3}C_{3} | 2n | D_{6n}D_{3n} / D_{3}C_{3} |
with μ = μ_{1}
and m = 2n |
8 | D_{12} x I / D_{3}C_{3} | 2m | D_{3m} x I / D_{3}C_{3} |
2n | D_{n} x I / C_{2} | μ
for n>2 μ ∈ ]μ_{0}, μ_{1}[ |
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 | D_{n} x I / E, for which the descriptive shares a 2-fold axis with a half turn in D_{n}xI (i.e. not with the principal axis).
If n = 1 then 2 | D_{1} x I / C_{2} becomes a 2 | S_{4} x I / S_{4}A_{4} If n = 2 then 4 | D_{2} x I / C_{2} becomes a 4k | D_{4k} x I / C_{4}C_{2}(with k=1) The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{2} |
with μ = μ_{0}
and n = 4 |
2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{0}
and n is odd |
6 | D_{12} x I / D_{4}D_{2} | 2n | D_{4n} x I / D_{4}D_{2} |
with μ = μ_{0}
and n = 2m and m odd |
6 | D_{12} x I / D_{4}D_{2} | 2m | D_{4m} x I / D_{4}D_{2} |
with μ = μ_{0}
and n = 4m and m > 1 |
4 | D_{8} x I / D_{4}D_{2} | 2m | D_{4m} x I / D_{4}D_{2} |
with μ = μ_{1}
and n is odd |
6 | D_{6} x I / D_{2}C_{2} | 2n | D_{2n} x I / D_{2}C_{2} |
with μ = μ_{1}
and n = 2m |
4 | D_{4} x I / D_{2}C_{2} | 2m | D_{2m} x I / D_{2}C_{2} |
2n | D_{n} x I / C_{2}C_{1} | μ
for n>1 μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[ |
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 | D_{n} x I / E, for which the descriptive shares a mirror with a mirror (through the pricipal axis) in D_{n}xI.
If n = 1 then 2 | D_{1} x I / C_{2} becomes a 2 | S_{4} x I / S_{4}A_{4} The special angles are: μ_{0} = 0 μ_{1} = atan(√2) μ_{2} = ^{π}/_{2} |
with μ = μ_{0}
and n = 2 or n = 4 |
2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{0}
and n is odd |
6 | D_{12} x I / D_{4}D_{2} | 2n | D_{4n} x I / D_{4}D_{2} |
with μ = μ_{0}
and n = 2m and m odd |
6 | D_{12} x I / D_{4}D_{2} | 2m | D_{4m} x I / D_{4}D_{2} |
with μ = μ_{0}
and n = 4m and m > 1 |
4 | D_{8} x I / D_{4}D_{2} | 2m | D_{4m} x I / D_{4}D_{2} |
with μ = μ_{1}
and n = 3 |
2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{1}
and n is coprime with 3 |
4 | D_{6} x I / D_{3}C_{3} | 2n | D_{3n} x I / D_{3}C_{3} |
with μ = μ_{1}
and n = 3m and m > 1 |
4 | D_{6} x I / D_{3}C_{3} | 2m | D_{3m} x I / D_{3}C_{3} |
with μ = μ_{2}
and n = 2 |
2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{2}
and n is odd |
6 | D_{6} x I / D_{2}C_{2} | 2n | D_{2n} x I / D_{2}C_{2} |
with μ = μ_{2}
and n = 2m |
4 | D_{4} x I / D_{2}C_{2} | 2m | D_{2m} x I / D_{2}C_{2} |
4n | D_{2n} x I / D_{1}C_{1} | μ
for n>1 μ ∈ ]μ_{0}, μ_{1}[ |
Examples:
for n=2 for n=3 Animations: for n=2 for n=3 Interactive Models: for n=2 for n=3 |
4m | D_{m} x I / E, for which the descriptive shares a mirror with a mirror that is perpendicular to the pricipal axis in D_{m}xI, where m = 2*n.
If n = 1 then 4 | D_{2} x I / D_{1}C_{1} equals to a 2k | D_{k} x I / C_{2}C_{1}(with k=2) The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{2n} |
with μ = μ_{0} | 4 | D_{4} x I / D_{2}C_{2} | 2n | D_{2n} x I / D_{2}C_{2} |
with μ = μ_{1}
and m = 2n |
8 | D_{8} x I / D_{2}C_{2} | 2m | D_{2m} x I / D_{2}C_{2} |
4n | D_{3n} x I / C_{3} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Examples:
for n=1 for n=2 Animations: for n=1 for n=2 Interactive Models: for n=1 for n=2 |
4m | D_{m} x I / E, for which the descriptive shares a 3-fold axis with an m-fold axis in D_{m}xI, where m = 3*n.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{6n} |
with μ = μ_{0}
and n = 1 |
2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{0}
and n > 1 |
4 | D_{6} x I / D_{3}C_{3} | 2n | D_{3n} x I / D_{3}C_{3} |
with μ = μ_{1}
and m = 2n |
8 | D_{12} x I / D_{3}C_{3} | 2m | D_{3m} x I / D_{3}C_{3} |
4n | D_{4n} x I / C_{4}C_{2} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
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 | D_{m} x I / E, for which the descriptive shares a half turn with an m-fold axis in D_{m}xI, where m = 4*n.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{8n} |
with μ = μ_{0}
and n = 1 |
2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{0}
and n > 1 |
6 | D_{12} x I / D_{4}D_{2} | 2n | D_{4n} x I / D_{4}D_{2} |
with μ = μ_{1}
and m = 2n |
4 | D_{8} x I / D_{4}D_{2} | 2m | D_{4m} x I / D_{4}D_{2} |
The following table contains all compounds (with roational freedom) that do not have a dihedral and cyclic symmetry:
Compound^{2} | VRML^{1} model | Description |
4 | A_{4} / C_{3} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
12 | A_{4} / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in A_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{3} |
with μ = μ_{0} | 1 | S_{4}A_{4} / S_{4}A_{4} | - |
with μ = μ_{1} | 4 | S_{4}A_{4} / D_{3}C_{3} | - |
12 | A_{4} x I / C_{2}C_{1} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
24 | A_{4} x I / E, for which the descriptive shares a mirror with a mirror in A_{4}xI.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} |
with μ = μ_{0} | 6 | S_{4} x I / D_{4}D_{2} | - |
with μ = μ_{1} | 12 | S_{4} x I / D_{2}C_{2} | - |
8 | A_{4} x I / C_{3} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
24 | A_{4} x I / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in A_{4}xI.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{3} |
with μ = μ_{0} | 2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{1} | 8 | S_{4} x I / D_{3}C_{3} | - |
12 | S_{4}A_{4} / C_{2}C_{1} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
24 | S_{4}A_{4} / E, for which the descriptive shares a mirror with a mirror in S_{4}A_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{2} |
with μ = μ_{0} | 1 | S_{4}A_{4} / S_{4}A_{4} | - |
with μ = μ_{1} | 12 | S_{4} x I / D_{2}C_{2} | - |
8 | S_{4}A_{4} / C_{3} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
24 | S_{4}A_{4} / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in S_{4}A_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{3} |
with μ = μ_{0} | 1 | S_{4}A_{4} / S_{4}A_{4} | - |
with μ = μ_{1} | 4 | S_{4}A_{4} / D_{3}C_{3} | - |
6 | S_{4}A_{4} / C_{4}C_{2} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
24 | S_{4}A_{4} / E, for which the descriptive shares a half-turn with a half-turn in S_{4}A_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} |
with μ = μ_{0} | 1 | S_{4}A_{4} / S_{4}A_{4} | - |
with μ = μ_{1} | 6 | S_{4} x I / D_{4}D_{2} | - |
12 | S_{4} / C_{2} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
24 | S_{4} / E, for which the descriptive shares a half-turn with a half-turn in S_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} |
with μ = μ_{0} | 12 | S_{4} x I / D_{2}C_{2} | - |
with μ = μ_{1} | 6 | S_{4} x I / D_{4}D_{2} | - |
8 | S_{4} / C_{3} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
24 | S_{4} / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in S_{4}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{3} |
with μ = μ_{0} | 2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{1} | 8 | S_{4} x I / D_{3}C_{3} | - |
24 | S_{4} x I / D_{1}C_{1} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
48 | S_{4} x I / E, for which the descriptive shares a mirror with a mirror in S_{4}xI, where the normal of mirror plane shares a 4-fold axis of the compound.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} |
with μ = μ_{0} | 6 | S_{4} x I / D_{4}D_{2} | - |
with μ = μ_{1} | 12 | S_{4} x I / D_{2}C_{2} | - |
24 | S_{4} x I / C_{2}C_{1} | μ
μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[ |
Example
Animation Interactive Model |
48 | S_{4} x I / E, for which the descriptive shares a mirror with a mirror in S_{4}xI, where the normal of mirror plane shares a 2-fold axis of the compound.
The special angles are: μ_{0} = 0 μ_{1} = asin(^{2√2}/_{3}) μ_{2} = ^{π}/_{2} |
with μ = μ_{0} | 2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{1} | 8 | S_{4} x I / D_{3}C_{3} | - |
with μ = μ_{2} | 12 | S_{4} x I / D_{2}C_{2} | - |
24 | S_{4} x I / C_{2} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
48 | S_{4} x I / E, for which the descriptive shares a half-turn with a half-turn in S_{4}xI.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} |
with μ = μ_{0} | 6 | S_{4} x I / D_{4}D_{2} | - |
with μ = μ_{1} | 12 | S_{4} x I / D_{2}C_{2} | - |
16 | S_{4} x I / C_{3} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
48 | S_{4} x I / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in S_{4}xI.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{3} |
with μ = μ_{0} | 2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{1} | 8 | S_{4} x I / D_{3}C_{3} | - |
12 | S_{4} x I / C_{4}C_{2} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
48 | S_{4} x I / E, for which the descriptive shares a half-turn with a 4-fold axis in S_{4}xI.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} |
with μ = μ_{0} | 2 | S_{4} x I / S_{4}A_{4} | - |
with μ = μ_{1} | 6 | S_{4} x I / D_{4}D_{2} | - |
30 | A_{5} / C_{2} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
60 | A_{5} / E, for which the descriptive shares a half-turn with a half-turn in A_{5}.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} |
with μ = μ_{0} | 30 | A_{5} x I / D_{2}C_{2} | - |
with μ = μ_{1} | 5 | A_{5} / A_{4} | - |
20 | A_{5} / C_{3} | μ
μ ∈ ]μ_{2}, μ_{0}[ ∪ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
60 | A_{5} / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in A_{5}.
The special angles are: μ_{0} = 0 μ_{1} = acos(^{√10}/_{4}) μ_{2} = -acos(^{√2(3+√5)}/_{8}) |
with μ = μ_{0} | 5 | A_{5} / A_{4} | - |
with μ = μ_{1} | 20A | A_{5} x I / D_{3}C_{3} | - |
with μ = μ_{2} | 20B | A_{5} x I / D_{3}C_{3} | - |
60 | A_{5} x I / C_{2}C_{1} | μ
μ ∈ ]μ_{0}, μ_{1}[ ∪ ]μ_{1}, μ_{2}[ ∪ ]μ_{2}, μ_{3}[ |
Example
Animation Interactive Model |
120 | A_{5} x I / E, for which the descriptive shares a mirror with a mirror in A_{5}xI.
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} |
with μ = μ_{0} | 30 | A_{5} x I / D_{2}C_{2} | - |
with μ = μ_{1} | 20A | A_{5} x I / D_{3}C_{3} | - |
with μ = μ_{2} | 20B | A_{5} x I / D_{3}C_{3} | - |
with μ = μ_{3} | 30 | A_{5} x I / D_{2}C_{2} | - |
60 | A_{5} x I / C_{2} | μ
μ ∈ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
120 | A_{5} x I / E, for which the descriptive shares a half-turn with a half-turn in A_{5}xI.
The special angles are: μ_{0} = 0 μ_{1} = ^{π}/_{4} |
with μ = μ_{0} | 30 | A_{5} x I / D_{2}C_{2} | - |
with μ = μ_{1} | 10 | A_{5} x I / A_{4} | - |
40 | A_{5} x I / C_{3} | μ
μ ∈ ]μ_{2}, μ_{0}[ ∪ ]μ_{0}, μ_{1}[ |
Example
Animation Interactive Model |
120 | A_{5} x I / E, for which the descriptive shares a 3-fold axis with a 3-fold axis in A_{5}xI.
The special angles are: μ_{0} = 0 μ_{1} = acos(^{√10}/_{4}) μ_{2} = -acos(^{√2(3+√5)}/_{8}) |
with μ = μ_{0} | 10 | A_{5} x I / A_{4} | - |
with μ = μ_{1} | 20A | A_{5} x I / D_{3}C_{3} | - |
with μ = μ_{2} | 20B | A_{5} x I / D_{3}C_{3} | - |
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:
The following table contains all (rigid) compounds that do not have a dihedral and cyclic symmetry:
^{1
}For available X3D / VRML plugins for your browser and OS check here.
^{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.
[HVerh00] Verheyen, Hugo F: Symmetry Orbits, Birkhauser; 1 edition (January 26, 1996)
[HVerh01] Verheyen, Hugo F:
Compound Lines of Polyhedra, unpublished
2007-03-12, 12:17