Maths - Cube Rotation

As an example of a permutation group (for an introduction to permutation groups see this page) we will look at a finite subset of the 3D rotation group SO(3), so we will look at all the rotation transforms of a cube that map it to itself. For information about 3D rotations see this page.

We can study cube rotation using the various algebras associated with 3D rotations such as:

but the aim on this page is to separate the properties of this rotation in a way that is not dependant on the properties of any specific algebra and instead brings out the higher level symmetries.

Lets take as an example the rotation of a cube in ways that does not change its shape (i.e. multiples of 90° rotations about x, y or z) although we will keep track of the rotations possibly by markings on the faces of the cube. We could have chosen other platonic solids, other then a cube, such as a dodecahedron. These other platonic solids are described on this page.

There are a number of ways to analyise this, one way is to number the vertices and track the effect of each rotation on these vertices.

cube verticies

So a 90° rotation about each of the x, y or z axis could be defined as follows (as a cycle decomposition described on this page)

x = rotation about x = (1, 4, 8, 5) (2, 3, 7, 6)
y =rotation about y = (1, 5, 6, 2) (4, 8, 7, 3)
z = rotation about z = (1, 2, 3, 4) (5, 6, 7, 8)

We now need to work out all the permutations of these rotations.

There are 24 made up of 1 identity element, 9 rotations about opposite faces, 8 rotations about opposite vertices and 6 rotations about opposite lines.

    number of rotations
rotate about opposite faces rotate cube about face

9

(3 for each pair of faces)

rotate about opposite vertices rotate cube about vertex

8

(2 for each pair of vertices)

rotate about opposite lines rotate cube about lines

6

(one for each pair of lines)

This gives 9 + 8 + 6 = 23 possible rotations of the cube, plus the identity element (leave it where it is giving 24 possible rotations in total.

There is a quicker way to discover that there are 24 rotations. If we realise that there is a one to one correspondence between rotations and possible orientations of the cube, then all we have to do is count the possible orientations, for instance we could:

Permutation Group of Cube

This has 24 possible rotations, we can generate these by starting with the identity element and the 90° rotations about the x, y and z axis, then by combining these in different sequences we can generate all 24 permutations. In fact, as we shall see below, we only need to start with 2 permutations to generate all possible rotations.

  identity (0°) about x (90°) about y (90°) about z (90°)
permutation i c d d
cycle notation   (1, 4, 8, 5) (2, 3, 7, 6) (1, 5, 6, 2) (4, 8, 7, 3) (1, 2, 3, 4) (5, 6, 7, 8)

We will denote these permutations by the letters i ,x ,y and z. The composition of x and y (y o x) will be denoted xy:

c

o

d

=

i

To be consistent with other literature on this, we need to reverse the order of the functions, that is the right hand function is applied first then the left hand function. So, for example, xy = (y o x).

We can draw up a complete table of the composition operation which defines this example completely. It is a bit of a drudge to work out all these permutations by hand so I have written a program to do this automatically - This is still work in progress - see this page.

Generating rotations from i and x

In order to try out the program with a simple example, I supplied the program with 2 permutations ('i' and 'x') to start it going and it generated two more to complete the group:

  identity (0°) about x (90°) about x (180°) about x (270°)
permutation i c xx xxx
cycle notation   (1, 4, 8, 5) (2, 3, 7, 6) (8 1)(7 2)(6 3)(5 4) (5 8 4 1)(6 7 3 2)

So from the generators: 'i' and 'x' the program has generated two other permutations 'xx' (apply x twice) and 'xxx' (apply x three times).

The complete multiplication (Cayley) table for this group is then:

a*bb.ib.xb.xxb.xxx
a.iixxxxxx
a.xxxxxxxi
a.xxxxxxxix
a.xxxxxxixxx

Generating rotations from i, x and y

This time I supplied the program with 3 permutations ('i' , 'x' and 'y') to start it going and it generated 21 more to complete the group of 24:

permutation i c d xx
cycle notation   (1, 4, 8, 5) (2, 3, 7, 6) (1, 5, 6, 2) (4, 8, 7, 3) (8 1)(7 2)(6 3)(5 4)
permutation xy yx yy xxx
cycle notation (8 6 1)(4 7 2)(3)(5) (1)(4 5 2)(8 6 3)(7) (6 1)(5 2)(8 3)(7 4) (5 8 4 1)(6 7 3 2)
permutation xxy xyx xyy yxx
cycle notation (7 1)(3 2)(6 4)(8 5) (5 1)(8 2)(7 3)(6 4) (7 1)(8 2)(4 3)(6 5) (4 1)(8 2)(5 3)(7 6)
permutation yyx yyy xxxy xxyx
cycle notation (2 1)(5 3)(6 4)(8 7) (2 6 5 1)(7 8 4 3) (6 3 1)(2)(5 7 4)(8) (6 8 1)(7 4 2)(3)(5)
permutation xxyy xxyx xyyy yxxx
cycle notation (3 1)(4 2)(7 5)(8 6) (1)(5 4 2)(6 8 3)(7) (3 8 1)(7 5 2)(4)(6) (8 3 1)(5 7 2)(4)(6)
permutation yyyx xxxyx xyxxx xyyyx
cycle notation (3 6 1)(2)(7 5 4)(8) (2 3 4 1)(6 7 8 5) (4 3 2 1)(8 7 6 5) (7 1)(6 2)(5 3)(8 4)

I don't know if these are the shortest sequences, perhaps I should get the program to check all combinations and see if these are any shorter sequences? At the moment the program allocates the name from the first case of a particular permutation it finds and then subsequent cases are given the original name.

So we don't need to use a third generator 'z' and we already have all 24 possible rotations.

Complete table:

a*bb.ib.xb.yb.xxb.xyb.yxb.yyb.xxxb.xxyb.xyxb.xyyb.yxxb.yyxb.yyyb.xxxyb.xxyxb.xxyyb.xyxxb.xyyyb.yxxxb.yyyxb.xxxyxb.xyxxxb.xyyyx
a.iixyxxxyyxyyxxxxxyxyxxyyyxxyyxyyyxxxyxxyxxxyyxyxxxyyyyxxxyyyxxxxyxxyxxxxyyyx
a.xxxxxyxxxxxyxyxxyyixxxyxxyxxxyyxyxxyyxyyyyxxxyxyyxyyyyxxxyxxxxyyyxyxyyyxyxxx
a.yyyxyyyxxxyxyyxyyyyxxxxxxyxxxxyxxxyyyyyxixyyyxxxxxxyxyxxxxxxyxxyyxxxxyxyxyyy
a.xxxxxxxxxyixxxyxxyxxxyyxyxxxyxyyxyyyxyyyxxxyyxyyxyyyxyxxyyyxyxxxxyxxyyyxxyxxx
a.xyxyxyxxyyxyxxxxyxyyxyyyxyxxxxxxyyyxxxyxyyxxyyyxxyxxxixxxyyyyxyxxxyyxxyxxyyxx
a.yxyxyxxxyxyxxxxxxyxxxxyxyxyyyxxxxxxyxyxxxyyyxxxyxyyxxxyyyyxixxyyxyxyyyyyxxxyy
a.yyyyyyxyyyxxyyxyxxyyyxixyyyxxxyxxxxxxxxyxyxyyyyxxxxxxyxxxyxxyxyxxyyyxxyxxxxyx
a.xxxxxxixxxyxyxxxyxyyxxxxyyxyyxyyyxxyyxyxxxxyxyxxyyyxxyyyxyyyxxyxxxxxyxyxxxyyyx
a.xxyxxyxxyxxxyyyyyxxxyxxyyyxxyyyxixyyyyxyyyxxxxxxyxxxxyxyyyxxyxyyxxxxxyxxxyxyxx
a.xyxxyxxyxxxxyxxyxxxxxxyyyxxxyxxyyxxxixxxyyyyxxyyyyxxyyyxyyyxxyyxxxyyxxyyxxxxyy
a.xyyxyyyyxyyyyyxyyyxyyyxxxxyyxyxxyyyxixxxyxxxyyxxxyxxxxxxxxyyxxxyxxyxyxxxxxyxyx
a.yxxyxxyxxxxxyxyyyxxxxxxyyxyyxxxyyyyxixxyxxxyyxyxyyxyyyxxxyxxyxxxxxyyxyxxxyyyxy
a.yyxyyxxxyyxyxxxyyyxxxyxxxxxxyyxyyyyxxxxxxyixxxyyyyxyyyxxyxxyxyxxxxyxyyyxyxxxyx
a.yyyyyyyyyxixxyxyxxxxyxxyxxxyyxyyxxxxxyxyyxxxyxxyyyxxxyxxyyyxxxxyyxxyyyxyxxxxxy
a.xxxyxxxyxxxyxyyxxyyyyxxxyyxyxxxyyyxxyxxxyxxyyxyxxxxxxyyyxxxxyyxxxxxyxyyixxyyyyy
a.xxyxxxyxyyyxxxyxyyyxixyyyyxxxyxyxxxxyxyyyxyxxxyxxxyyxyyxxxxxyyxxxyxyxxxyyxxxyyx
a.xxyyxxyyxyyyxxyyxyyyyxxxxxyyxyyyxyyyxxyxxxxxyxyxxyyyxixxxyxyyxxxyxxyxxxxxxyxxyx
a.xyxxxyxxxyxxxxxxxyyxxxixxxyxyxxyyyxyyyxxxxxyxyyxxxyxyyxyyyyxyyyxxxxxyyyyyyxxxxy
a.xyyyxyyyxyyyxxxxxyyyyxxxxyxxxyxyyxxxyxyxxxxxxxyxxyyyxxxyyxyxxxxyxyxxxiyyyxxyyyy
a.yxxxyxxxyxyyyxyxyyxxxyyyyxyxxxyxyyxyyyxxxyxxxyxyxxxxxxyxxxxyxxxyyixyyyxyxxxxyyx
a.yyyxyyyxxxyxyxxxxxyxxxyyxyyxxxyyyxxxyxxyyyxxxyxyxyyyxixyyyyxyyxxxyxxxxxyxyyxxxx
a.xxxyxxxxyxxyyyyxxyyyxxyxxxyxxxxyyyyxxxxyyxxxxyxxxxyxyyxxxyxyxxxxxxxyyyyyyxxyyiyy
a.xyxxxxyxxxxyyxxxxyxxyyyxyyyxxyxxxxyxyyxyyyyxxxxyyyyxxxxyyyxxxyxyyxxyxxxxyixxyyxx
a.xyyyxxyyyxxxxyyyyxxxxyxyyxxxyxyxxxxyyyyxxxyyxyxxxxyxxyyxxxxyxxxyxxyyxxxyyyyxxyyi

Cayley Graph

cayleygraph

Specific Algebras to Represent this Group


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Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Fearless Symmetry - This book approaches symmetry from the point of view of number theory. It may not be for you if you are only interested in the geometrical aspects of symmetry such as rotation groups but if you are interested in subjects like modulo n numbers, Galois theory, Fermats last theorem, to name a few topics the chances are you will find this book interesting. It is written in a friendly style for a general audience but I did not find it dumbed down. I found a lot of new concepts to learn. It certainly gives a flavor of the complexity of the subject and some areas where maths is still being discovered.

 

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