Spinning Compounds
When a compound has rotational freedom,
we can animate it by slowly changing the rotation angle. The idea was detailed
on the page about compounds of cubes,
and this page collects a wide range of models of this sort. For the cube
compounds, the notation used in Verheyen's Symmetry Orbits, listed
in the references, is given here [in square
brackets].
Compounds with rotational freedom often incorporate pairs of counter-rotating
cubes to maintain a plane of symmetry. This technique was first used by
Skilling in certain uniform compounds,
but many of the compounds on this page are not uniform. In compounds with
rotational freedom around the tetrahedral 3-fold axes, the components need
not come in such pairs.
In each case, as the components spin, you can see a full range of shapes
that the compound can take on. There are also certain special angles at
which various alignments occur and additional symmetry occurs: At the initial
zero angle, all the components overlap exactly in many of these examples.
At the halfway point, n components often overlap into n/2.
Some special configurations to look for are listed in parentheses after
each entry.
Note that every component is moving at a continuous steady rotation
rate. They never stop or reverse direction, but when two components pass
through each other it may appear that they bounce.
(This list is not complete yet...)
Fifteen compounds of cubes:
-
Skilling's uniform compound of 6
cubes, spinning about the 4-fold axes [6 | S4 x I / C4 x I] (halfway:
uniform compound of 3 cubes)
-
6 cubes, all spinning with
a 4-fold axis aligned aligned on prism's 12-fold axis, 3 clockwise, 3 counterclockwise,
[2n | D4n x I / C4 x I; n=3] (initially: 12-toothed "gear"; halfway: 24-toothed
"gear")
-
6 cubes, all spinning with
a 3-fold axis aligned aligned on prism's 9-fold axis, 3 clockwise, 3 counterclockwise,
[2n | D3n x I / C3 x I; n=3] (initially: 9-fold; halfway: 18-fold)
-
6 cubes, all spinning with
a 2-fold axis aligned aligned on prism's 6-fold axis, 3 clockwise, 3 counterclockwise,
[2n | D2n x I / D1 x I; n=3] (initially: 6-fold; halfway: 12-fold)
-
5 cubes, each with a 2-fold
axis aligned to a prism's 2-fold axis [nA | Dn x I / C2 x I; n=5] (initially:
five cubes
with a common 4-fold axis; partway: five
cubes with a common 3-fold axis; halfway: five
cubes with a common 2-fold axis)
-
5 cubes, each with a 4-fold
axis aligned to a prism's 2-fold axis [nB | Dn x I / C2 x I; n=5] (initially:
five cubes
with a common 4-fold axis; partway: uniform
comound of five cubes; halfway: five
cubes with a common 2-fold axis)
-
4 cubes, spinning on tetrahedral
3-fold axes, [4 | A4 x I / C3 x I] (partway: compound
of 4 of the 5 cubes; halfway: Bakos's
compound of 4 cubes)
-
6 cubes, with 4-fold axes
on tetrahedral 2-fold axes [6| A4 x I / C2 x I] (initially: uniform
compound of 3 cubes; halfway: the rigid
octahedral compound of six cubes)
-
8 cubes, spinning about
the 3-fold axes [8 | S4 x I / C3 x I] (halfway: Bakos's
compound of 4 cubes)
-
12 | S4 x I / D1 x I (varying angles: 1,
2, 3,
4, 5)
-
12 cubes, spinning about
the 2-fold axes [12A | S4 x I / C2 x I] (partway: Bakos's
compound of 4 cubes; halfway: the rigid
octahedral compound of six cubes)
-
12 cubes, spinning on
4-fold axes aligned to the octahedral 2-fold axes [12B | S4 x I / C2 x
I] (initially: uniform compound of 3 cubes;
halfway: the rigid octahedral compound of
six cubes)
-
20 | A5 x I / C3 x I (varying angles: 1,
2, 3)
(dual to 1: the rigid uniform
compound of 20 octahedra meeting 2 per vertex), (duals to 2
and 3 are uniform compounds
of 20 octahedra with rotational freedom)
-
30A | A5 x I / C2 x I (varying angles: 1,
2, 3,
4, 5)
-
30B | A5 x I / C2 x I (varying angles: 1,
2, 3,
4, 5)
Fifteen compounds of octahedra (repsectively dual to the above):
-
6 octahedra, spinning
about 4-fold axes [6 | S4 x I / C4 x I] (halfway: 3
octahedra)
-
6 octahedra, all spinning
with a 4-fold axis aligned on prism's 12-fold axis, 3 clockwise, 3 counterclockwise,
[2n | D4n x I / C4 x I; n=3] (initially: 12-toothed "gear"; halfway: 24-toothed
"gear")
-
6 octahedra, all spinning
with a 3-fold axis aligned aligned on prism's 9-fold axis, 3 clockwise,
3 counterclockwise, [2n | D3n x I / C3 x I; n=3] (initially: 9-fold; halfway:
18-fold)
-
6 octahedra, all spinning
with a 2-fold axis aligned aligned on prism's 6-fold axis, 3 clockwise,
3 counterclockwise, [2n | D2n x I / D1 x I; n=3] (initially: 6-fold; halfway:
12-fold)
-
5 octahedra, each
with a 2-fold axis aligned to a prism's 2-fold axis [nA | Dn x I / C2 x
I; n=5] (initially: five octahedra
with a common 4-fold axis; partway: five
octahedra with a common 3-fold axis; halfway: five
octahedra with a common 2-fold axis)
-
5 octahedra, each
with a 4-fold axis aligned to a prism's 2-fold axis [nB | Dn x I / C2 x
I] (initially: five
octahedra with a common 4-fold axis; partway: uniform
comound of five octahedra; halfway: five
octahedra with a common 2-fold axis)
-
uniform compound of 4 octahedra,
spinning on tetrahedral 3-fold axes, [4 | A4 x I / C3 x I] (halfway: rigid
uniform compound of 4 octahedra)
-
6 octahedra, with 4-fold
axes on tetrahedral 2-fold axes [6| A4 x I / C2 x I] (initially: 3
octahedra; halfway: the rigid compound
of six octahedra)
-
uniform compound of 8 octahedra
[8 | S4 x I / C3 x I] (halfway: uniform
compound of 4 octahedra)
-
12 | S4 x I / D1 x I (varying angles: 1,
2, 3,
4, 5)
-
12 octahedra, spinning
about the 2-fold axes [12A | S4 x I / C2 x I] (partway: uniform
compound of 4 octahedra; halfway: the
rigid compound of six octahedra)
-
12 octahedra, spinning
on 4-fold axes aligned to the octahedral 2-fold axes [12B | S4 x I / C2
x I] (initially: 3 octahedra; halfway:
the rigid compound of six octahedra)
-
20 | A5 x I / C3 x I (varying angles: 1,
2, 3)
(dual to 1: the rigid uniform
compound of 20 octahedra meeting 2 per vertex), (duals to 2
and 3 are uniform compounds
of 20 octahedra with rotational freedom)
-
30A | A5 x I / C2 x I (varying angles: 1,
2, 3,
4, 5)
-
30B | A5 x I / C2 x I (varying angles: 1,
2, 3,
4, 5)
Compounds of icosahedra or dodecahedra:
Compounds of tetrahedra: