
\bigbreak
{\nobigbf 2. Archimedean solids I}

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In this section, we compute the spectra of seven
Archimedean solids:
$\langle 3,6,6\rangle$, $\langle 3,4,3,4\rangle$,
$\langle 4,6,6\rangle$, $\langle 3,8,8\rangle$,
$\langle 3,4,4,4\rangle$, $\langle 4,6,8\rangle$
and $\langle 3,5,3,5\rangle$ as well as the spectra of the antiprisms,
by using mirrors ([PS]) and relations between adjacency matrices.
We illustrate this method by computing
the spectrum of $\langle 3,4,4,4\rangle$.

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\centerline{\psfig{file=fig21.ps,width=3.0in}}

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\centerline{\eightbf Figure 2.1}
%Figura 2.1 do 3444, com espelhos

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In Figure 2.1,
full lines form the Schlegel diagram ([C])
of $\langle 3,4,4,4\rangle$, i.e.,
the stereographic projection of the polyhedron to the plane. Dotted
horizontal, vertical and round
lines indicate three planes of symmetry associated to three commuting
involutions $A, B$ and $C$.
Let $V$ be the vector space of complex valued functions
on the set of vertices of $\langle 3,4,4,4\rangle$.
Let $X$ be the $24 \times 24$ adjacency matrix of the polyhedron
(i.e., of its graph).
The involutions $A$, $B$ and $C$, as well as the matrix $X$,
can be interpreted as commuting linear transformations from $V$ to $V$.
Thus, the eigenspaces $V_+^A$ and $V_-^A$ associated to the eigenvalues
$1$ and $-1$ of $A$ are invariant under $B$, $C$ and $X$.
In a similar fashion, $V$ splits as a direct sum of eight subspaces
(which a priori could be trivial) of the form $V_{s_A,s_B,s_C} =
V_{s_A}^A \cap V_{s_B}^B \cap V_{s_C}^C$, where subscripts denote signs.
These subspaces are invariant under $X$ and the problem of computing the
spectrum of $X$ reduces to the same problem for the restriction
$X_{s_A,s_B,s_C}$ of $X$ to $V_{s_A,s_B,s_C}$.

Each invariant subspace can be coordinatized by the values $r$, $s$ and
$t$ of each vector on the three vertices indicated in Figure 2.1:
mirroring by $A$, $B$ and $C$ prescribes the values at the remaining
vertices. In particular, in this example, the eight invariant subspaces
are of dimension $3$. The adjacency matrices for
$V_{++-}, V_{+-+}$ and $V_{-++}$ are trivially conjugate (by renaming
the dotted lines), and therefore have the same spectrum; the same
happens to $V_{+--}, V_{-+-}$ and $V_{--+}$. We have
$$X_{+++} = \pmatrix{2 & 1 & 1 \cr 1 & 2 & 1 \cr 1 & 1 & 2 \cr}, \qquad
  X_{++-} = \pmatrix{2 & 1 & 1 \cr 1 & 0 & 1 \cr 1 & 1 & 0 \cr},$$
$$X_{+--} = \pmatrix{0 & 1 & 1 \cr 1 & 0 & 1 \cr 1 & 1 & -2 \cr}, \qquad
  X_{---} = \pmatrix{-2 & 1 & 1 \cr 1 & -2 & 1 \cr 1 & 1 & -2 \cr},$$
and the spectrum of $X$ is thus immediately computed.

For $\langle 3,6,6\rangle$, there are two commuting involutions
corresponding to reflections with respect to the two orthogonal planes
indicated by dotted lines in the Schlegel diagram in Figure 2.2.
The values $r$, $s$, $t$ and $u$ in the vertices marked in the figure
determine a unique vector in $V_{++}$.
Similarly, $r$, $s$, $t$ provide a basis for $V_{+-}$
and $s$, $t$ a basis for $V_{--}$
($V_{-+}$ is equivalent to $V_{+-}$ by renaming planes).

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\centerline{\psfig{file=fig22.ps,width=3.0in}}

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\centerline{\eightbf Figure 2.2}
%Figura 2.2 do 366, com espelhos

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The same three mirrors used for $\langle 3,4,4,4\rangle$
work for $\langle 3,4,3,4\rangle$,
reducing the problem to that of finding the spectra of a
$3 \times 3$ matrix in $V_{+++}$, a $2 \times 2$ matrix in $V_{++-}$
and a $1 \times 1$ matrix in $V_{+--}$
($V_{---}$ is 0-dimensional).
Again, the mirrors used for $\langle 3,4,4,4\rangle$
convert the computation
of the spectrum of $\langle 3,8,8\rangle$ into the study of
the restriction of the adjacency matrices to invariant subspaces
of dimension 3.
For $\langle 4,6,8\rangle$, however,
these mirrors produce $6 \times 6$ matrices.
A fourth mirror, indicated in Figure 2.3,
can be used to further split the invariant subspaces.
The reflection $D$ on the fourth mirror
does not commute with either $A$ or $B$.
Still, $V_{+++}, V_{++-}, V_{--+}$ and $V_{---}$
are invariant under $D$ and can thus be decomposed
into the two eigenspaces for the restrictions of $D$.
Notice that the four remaining subspaces are not invariant
under $D$. This is no serious problem, however:
for example, the restriction of the adjacency matrix to $V_{-++}$
has the same spectrum as the restriction to $V_{++-}$,
and this last space can be split by $D$.
The values $r, s, t$ in the figure again provide a basis
for each of the eight relevant invariant subspaces
and the spectrum of $\langle 4,6,8\rangle$ is now easily obtained.
The solid $\langle 4,6,6\rangle$ can be handled in the same fashion;
in section 5 we compute its spectrum using other methods.

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\centerline{\psfig{file=fig23.ps,width=3.0in}}

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\centerline{\eightbf Figure 2.3}
%Figura 2.3 do 468, com espelhos

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We finish this section with three examples of computations of spectra
based on algebraic relations between adjacency matrices.
Let $Y$ be the adjacency matrix of a $2n$-gon:
then $X = Y^2 + Y - 2I$ is the adjacency matrix of the antiprism
with $n$-gonal basis 
and an eigenvalue $\lambda$ of $Y$ corresponds to an eigenvalue
$\lambda^2 + \lambda - 2$ of $X$.
As another example, the spectrum of $X_{3434}$,
the adjacency matrix of $\langle 3,4,3,4\rangle$,
can be computed in terms of the spectrum of $X_{444}$,
the adjacency matrix of the cube.
Indeed, let $Y_{v,e}$ be a $8 \times 12$ incidence matrix obtained
by numbering vertices and edges of the cube and setting
$y_{i,j} = 1$ if the $i$-th vertex belongs to the $j$-th edge.
Dually, let $Y_{e,v} = {(Y_{v,e})}^T$.
Notice that $X_{444} + 3 I = Y_{v,e} Y_{e,v}$
and that $X_{3434} + 2 I = Y_{e,v} Y_{v,e}$:
thus, the spectra of the left hand sides are equal
except for 4 extra zero eigenvalues in the second one.
The same method yields the spectrum of $\langle 3,5,3,5\rangle$
from the spectrum of the dodecahedron.
The spectrum of a regular polygon,
the cube and the dodecahedron are given in [CDS].

