MAM2000 (Essays/SciAm)

# The Rotation Matrix

Suppose one wants to confine the rotation of the hypercube to the third and fourth dimensions, the most mysterious rotation of all. The rotation matrix rot34 is applied. Its entries are O's, 1's and three other numbers distributed according to the following pattern:

 1 0 0 0 0 1 0 0 0 0 a b 0 0 -b a

The angle of the desired rotation in degrees is selected and stored in the variable ang, and the numbers a and b depend on ang: a is equal to cos(ang) and b is equal to sin(ang), where cos and sin are the trigonometric functions sine and cosine.

The rule for generating the other five rotation matrixes is simple. The a's appear on the main diagonal of each matrix in positions that correspond to the dimensions affected by the rotation. The b's appear at all the other intersections of rows and columns that correspond to the rotating dimensions. All other entries on the main diagonal are 1's, and the rest of the entries in the matrix are O's. For example, rot13 is the following matrix:

 a 0 b 0 0 1 0 0 -b 0 a 0 0 0 0 1
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