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\section*{Angles in Four Dimensions}
\label{Life in Four Dimensions}

\large


Life in 4 dimensions is different from 3 dimensions.  There are two
angles between two two-dimensional subspaces through the origin, which
we will from now on simply call \emph{planes}, not just one.  Let $P_1$ be
the projection to the first plane $\pi_1$ and let $P_2$ be the
projection to the second plane $\pi_2$.  The matrix $P_1P_2P_1$ does
the following: First project to $\pi_1$, then project to $\pi_2$, then
project to $\pi_1$.  The directions perpendicular to $\pi_1$ are
annihilated by the first step.  Therefore, two eigenvalues are 0.  The
projection $P_1$ leaves a vector in $\pi_1$ unchanged, then $P_2$ maps
it to $\pi_2$.  then $P_1$ maps it back to $\pi_1$.  Vectors from
$P_1$ are thus mappped to $P_1$ by the matrix $P_1P_2P_1$, and it is a
self-adjoint operator.  Therefore it has two real eigenvalues with
perpendicular eigenvectors in $P_1$, and these eigenvalues are
$\cos^2\alpha_1$ and $\cos^2\alpha_2$, where $\alpha_1$ and $\alpha_2$
are the angles between $\pi_1$ and~$\pi_2$.  Project the unit circle
in $P_1$ to $P_2$. The result will be a (possibly) degenerate
ellipse. We denote its major and minor axis by $\cos\alpha_1$ and
$\cos\alpha_2$, because the angle between the point of $P_1$ that is
mapped to the major axis forms an angle $\alpha_1$ between itself and
its image, and similarly for $\alpha_2$.  The role of these angles is
to fix the relative position between two planes up to congruence.


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