\chapter{Heat flow on the cube}
\section{Question}
Describe heat flow on a cube.
That is, given a (hollow) cube, where each face is a metal plate of homogeneous
temperature, and the edges are insulated (so heat flows through at some rate\footnote{Proportional to the difference between the temperatures of the plates,
for those who don't remember Fourier's law of heat conduction;
this is math, not physics}). Given initial heat distribution,
determine how the heat distribution evolves over time.
\newpage
\section{Solution}
Express the heat distribution in terms of \Def{cubical harmonics}
(eigenvectors for the heat operator on the cube).
This is cleaner if we assume that time is discrete,
and that heat flow is given by the discrete Laplacian.
This is diagonalizing the discrete Laplacian on the cube
(or octadehron, depending on your perspective);
draw a graph whose nodes are the faces of the cubes,
and there's an edge if they join in a side
(this is the vertex-and-edge graph of the octahedron,
the dual polyhedron).
techniques
* symmetry
* orthogonality
* discrete Laplacian
* cubical harmonics
(which are discrete analogs of the spherical harmonics, familiar
from atomic orbitals!)
Trick is to look for eigenvectors (not eigenvalues!),
starting with symmetric ones.
If you look for rotationally symmetric, it's quite quick.
An interest is that it's a *symmetric space* for a group,
not a group itself.
1: 1 1 1 1 1 1
0: 1 0 0 0 0 -1
-1/2: 1 -1/2 -1/2 -1/2 -1/2 1
You can visualize these as ``cubical harmonics'':
* a sphere
* a barbell (two ends opposite color because of different signs (``phases''))
* a barbell with a ring (ends of barbell same color, ring opposite color/sign/phase)
the -1/2 space has natural 3-element spanning set, which sum to zero: has a linear relation!
orthogonality helps with the computation:
given the first two, the third pops out immediately.
Notice that it also looks like eigenvectors for the heat equation in a (discrete) rod
(also for vibrations in a rod)
1 1 1
1 0 -1
1 -2 1