occasional meanderings in physics' brave new world

Name:
Location: New Zealand

Marni D. Sheppeard

## Wednesday, September 26, 2007

### M is for Magic

As we have seen, Carl Brannen's QFT uses circulant matrices. By resetting a mass scale, one may renormalise a 1-circulant

XYZ
ZXY
YZX

by a constant $\lambda = \frac{1}{Y + Z - 2X}$ so that the resulting circulant obeys the condition $2X = Y + Z$. This turns the circulant into a magic square. For 2-circulants the condition is instead $2Z = X + Y$. Although perhaps not a very useful observation, it is certainly entertaining! The total number of $5 \times 5$ normal (ie. matrices built from the first few ordinals) magic squares was only computed in 1973, and the number of $6 \times 6$ ones is still unknown. There is only one $3 \times 3$ normal square, up to rotation and reflection.

A paper by A. Adler uses circulants to find an algorithm for generating higher order normal magic n-cubes, by playing with p-adic L functions. For $p = 3$, Adler constructs two cute normal magic cubes: a $3 \times 3 \times 3$ cube and a $27 \times 27 \times 27$ cube. I was further intrigued by this paper of Adler's, containing the conjecture that magic n-cubes always form a free monoid. It shows first that sets of magic squares contain prime squares, out of which all others are constructed, and then that generating functions built from cardinalities for magic cubes have the remarkable property of being everywhere divergent!