Part 2: Symmetries of the square

1:Rigid motions of the square 

 There are four rigid motions of the square, namely anti-clockwise rotations by 90o, 180o, 270o and 0o. If we call the first one r, they form a cyclic group of order 4: 

{r, r2, r3, r4 = I}, 

where I means the identity map. Any other rigid motion, for example a clockwise rotation is actually one of these.

 If we label the corners in anticlockwise order {1, 2, 3, 4}, these rotations induce the permutations (1234), (13)(24), (1432) and (1). Note that the anti-clockwise order of the corners is preserved by rigid motions.

 

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2: Other symmetries of the square 

 But there are other symmetries of the square, namely flips or reflections about horizontal, vertical or diagonal axes, and these reverse the anti-clockwise order of the corners. For example, let f be reflection in the vertical axis. Then f has order 2, and we find four more symmetries: f, r.f, r2.f and r3.f.  Every other symmetry of the square is actually one of these eight, for example, reflection in the diagonal axis joining 1 and 3 is r.f and f.r  = r 3.f.   To see that there cannot be any more than 8 symmetries, note that each symmetry determines exactly one position for each corner. There are 4 possible positions for corner 1, and once one has been chosen, there are 2 possible positions for corner 2, and once one has been chosen the positions of corners 3 and 4 are determined. So there are at most 8 possibilities altogether.

For a multiplication table of symmetries of the square, see here and the following two pages.

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3: Linear representation 

 If we consider the square sitting in the Euclidean plane with horizontal and vertical sides and the centre at the origin, each symmetry can be realised by a linear transformation corresponding to a non-singular 2x2 matrix with all entries 0 or 1 or -1 having exactly one 0 in each row or column.
 
 
1 0 -1 0 1 0 -1 0
0 1 0 1 0 -1 0 -1
.
0 1 0 1 0 -1 0 -1
1 0 -1 0 1 0 -1 0

There are 8 such matrices. Since the area of the square is unchanged by a symmetry, the determinants are all +1 or -1. The rigid motions correspond to the four with determinant 1, and the others to the 4 with determinant -1. You can check that that these eight matrices are orthogonal, that is their inverse is their transpose, and every orthogonal matrix has determinant 1 or -1.

 The group of symmetries of the square is called the dihedral group D8. It can be used to exemplify: 

  1. the group axioms
  2. normal subgroups
  3. the fundamental homomorphism theorems
  4. cosets and Lagrange's Theorem
  5. group action on a set 
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Part 3: Symmetries of the cube

4:Rigid motions of the cube 

 First let's look at rigid motions of the cube. They form a group under composition, called the octahedral group, denoted O3.
 
 
Type Map  Number  Order
1  Identity map 1  1
2  Rotation round face-centered axis by p/2 or 3p/2 6  4 
3  Rotation round face-centered axis by p 3  4 
4  Rotation round corner-centered axis 8  3
5  Rotation round edge-centered axis 6  2
TOTAL 24 

There can be no more rigid motions of the cube: there are six ways to choose which face is at the bottom, and when chosen, it can be in any of 4 positions. Thus there are 24 possibilities and we have found 24 different rigid motions.
 

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5: Other symmetries of the cube 

 There is another group of order 24, the symmetric group S4 on four letters. If we can find a set of four "elements" of the cube that are permuted by the rigid motions, we will have "almost proved" that O3 is isomorphic to S4. Well, there are four pairs of opposite corners, so four long diagonals, and each rigid motion maps a long diagonal into a long diagonal, and hence permutes the set of long diagonals. Furthermore, for each pair L and M of long diagonals there is a rigid motion mapping L into M. To complete the proof that O3 is isomorphic to S4, we just have to show that the mapping from O3 to S4 which takes each rigid motion into the permutation of the long diagonals that it induces is a group homomorphism.

 The rigid motions of a given type seem to be more closely related to each other than to those of a different type. That is because they are conjugates. If a and b are elements of a group G, we say that a conj b if there exists c in G such that b = c-1ac, or equivalently, if there exists c in G such that cb = ac. Conjugacy is an equivalence relation on a group and hence partitions G into disjoint equivalence classes. The connection with the types of rigid motions of the cube is that if a is a rigid motion of the cube of a certain type, and c is any rigid motion, then c-1ac is a rigid motion of the same type. Conversely, if a and b are rigid motions of the same type, then there is a rigid motion c such that b = c-1ac.

 Just as there are symmetries of the square that are not rigid motions, there are symmetries of the cube that are not rigid motions. They are called the indirect symmetries of the cube. That is, there are distance preserving mappings of the cube onto itself that cannot be realised by any rigid motion in 3-dimensional space. To find one, take a plane parallel to two opposite sides of the cube passing through the centre. and reflect each point of the cube through this plane. It is clear that if two points have distance d then their images also have distance d, so this is a distance preserving transformation of the cube. Call this transformation j.j reverses the orientation of the edges and corners of each face. This is easy to see for the faces bisected by the reflection plane. For the other two, you must visualize the clockwise order of the edges and vertices as seen from the outside and then note that after reflection this order has reversed. Each square face has undergone an operation like a flip of the square in D8! 

Just as you get all the symmetries of the square by composing each rigid motion with the flip f, so you get all the symmetries of the cube by composing each rigid motion with the operation j To see this, note that there are at most 48 symmetries, since you can first choose one of two orientations for the corners of a face, then one of 24 rigid motions. But we have already found 48 different symmetries, so that's all there are folks.

 

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6: Linear representations 

 Just like in the case of D8, the symmetries of the cube can be realized by non-singular linear transformations in R3 which are represented by 3x3 orthogonal matrices which have 2 zeros in each row and column, the other entry being +1 or -1. There are 48 such matrices, 24 of determinant 1 representing the rigid motions and 24 of determinant -1 representing the indirect symmetries. .

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 Last update: 10 October, 1999

 Author: Phill Schultz, schultz@maths.uwa.edu.au