### M Theory Lesson 169

Thanks to Steven H. Cullinane for this diagram of Hamilton's quaternions, the whirligigs. Recall that this plane over the three element finite field $\mathbb{F}_3$ arises in the $d = 3^{n}$ MUB problem, associated to the Fourier transform of the $p = 3$ mass matrices.

## 3 Comments:

Okay, I get it. The 2x2 matrices, acting on the 2x1 vectors, do the arrow transformations (when working in the field with three elements 0, 1, -1).

I fear my title for this picture, "Hamilton's whirligigs," was misleading. The quaternions are, of course, Hamilton's, but the pictures of the quaternion actions on 3x3 arrays are mine.

Oh, thanks Steven! All the better.

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