Polyhedra and Geodesics
Matrices you should know
Symmetries of the Cube
Let R1,1,1 be that symmetry which rotates the cube 120◦ clockwise around the axis through
(1, 1, 1) and (−1, −1, −1). Let R1,y,1 = R−1,y,−1 be that symmetry whose axis of rotation
joins the midpoints of (1, 1, 1) and (1, −1, 1), and (−1, 1, −1) and (−1, −1, −1).
1.
2.
3.
4.
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6.
7.
8.
9.
10.

0

= 0
1

0

= 1
0


1 0
0 1: Clockwise 120◦ rotation about the axis.
R1,1,1
0 0

0 1
0 0: Counterclockwise 120◦ rotation about the axis.
R21,1,1
1 0

0 0 −1
0 : Clockwise 120◦ rotation about the axis.
R1,1,−1 = 1 0
0 −1 0


0 1 0
R21,1,−1 =  0 0 −1: Counterclockwise 120◦ rotation about the axis.
−1 0 0


0
0 1
R1,−1,1 = −1 0 0: Clockwise 120◦ rotation about the axis.
0 −1 0


0 −1 0
R21,−1,1 = 0 0 −1: Counterclockwise 120◦ rotation about the axis.
1 0
0


0 0 −1
R−1,1,1 = −1 0 0 : Clockwise 120◦ rotation about the axis.
0 1 0


0 −1 0
0 1: Counterclockwise 120◦ rotation about the axis.
R2−1,1,1 =  0
−1 0 0


1 0 0
Rx = 0 0 1: Clockwise 90◦ rotation about the x-axis.
0 −1 0


1 0
0
R2x = 0 −1 0 : 180◦ rotation about the x-axis.
0 0 −1
RotMat.1
11 March 2009
Polyhedra and Geodesics
Matrices you should know

11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.

1 0 0
R3x = 0 0 −1: Counterclockwise 90◦ rotation about the x-axis.
0 1 0


0 0 −1
Ry = 0 1 0 : Clockwise 90◦ rotation about the y-axis.
1 0 0


−1 0 0
R2y =  0 1 0 : 180◦ rotation about the y-axis.
0 0 −1


0 0 1
R3y =  0 1 0: Counterclockwise 90◦ rotation about the y-axis.
−1 0 0


0 1 0
Rz = −1 0 0: Clockwise 90◦ rotation about the z-axis.
0 0 1


−1 0 0
R2z =  0 −1 0: 180◦ rotation about the z-axis.
0
0 1


0 −1 0
R3z = 1 0 0: Counterclockwise 90◦ rotation about the z-axis.
0 0 1


−1 0 0
Rx,1,1 =  0 0 1: 180◦ rotation about the axis.
0 1 0


−1 0
0
0 −1: 180◦ rotation about the axis.
Rx,1,−1 =  0
0 −1 0


0 0 1
R1,y,1 = 0 −1 0: 180◦ rotation about the axis.
1 0 0


0
0 −1
R1,y,−1 =  0 −1 0 : 180◦ rotation about the axis.
−1 0
0


0 1 0
R1,1,z = 1 0 0 : 180◦ rotation about the axis.
0 0 −1
RotMat.2
11 March 2009
Polyhedra and Geodesics
Matrices you should know


0 −1 0
0 : 180◦ rotation about the axis.
23. R1,−1,z = −1 0
0
0 −1


1 0 0
24. I = 0 1 0: The identity.
0 0 1
RotMat.3
11 March 2009