My group theory teacher, Prof. Daniel Freedman, had some interesting professorial habits. When invoking some bit of background knowledge with which we were all supposed to have been familiar, he would say, “As you learned in high school. . . .” Typically, this would make a lecture sound a bit like the following:

“To finish the proof, note that we’re taking the trace of a product of matrices. As you learned in high school, the trace is invariant under cyclic permutations. . . .”

Prof. Freedman also said “seventeen” for “zero” from time to time. After working out a long series of mathematical expressions on the blackboard, showing that this and that cancel so that the overall result should be nothing, with the students alternating their glances between the board and their notes, he would complete the equation and proclaim, “Equals seventeen!” At which point, all the students look up and wonder, momentarily, what they just missed.

“Here, we’re summing over the indices of an antisymmetric tensor, so by exchanging *i* and *j* here and relabeling there, we can show that the quantity has to equal the negative of itself. The contraction of the tensor is therefore, as you learned in high school — seventeen!”

One day, I managed to best his line. I realized that the formula currently on the board had to work out to *one,* not zero, so when he wrote the equals sign, paused and turned to the class with an inquiring eye, I quickly raised my hand and said, “Eighteen!”

Incidentally, *truly* simple topics like Euler’s formula and trigonometric identities were supposed to have been learned in middle or elementary school.

Today, we’ll talk about one of the things Prof. Freedman said we should have covered in high school: the *rotation matrices* for two- and three-dimensional rotations. This will give us the quantitative, symbolic tools necessary to talk about commutativy and non-commutativity, the topic we explored in an earlier post.

Continue reading Rotation Matrices →