Higher ironing
The Coxeter group $\mathrm H_3$ has character table
| |
$C_{1}$ |
$C_{3}$ |
$C_{2,2}$ |
$C_{5}$ |
$'C_{5}$ |
| $\mathbf{1}$ |
$1$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $\chi_2$ |
$4$ |
$1$ |
$0$ |
$-1$ |
$-1$ |
| $\chi_3$ |
$5$ |
$-1$ |
$1$ |
$0$ |
$0$ |
| $\chi_4$ |
$3$ |
$0$ |
$-1$ |
$\varphi$ |
$\tilde{\varphi}$ |
| $\chi_5$ |
$3$ |
$0$ |
$-1$ |
$\tilde{\varphi}$ |
$\varphi$ |
Where $\varphi$ and $\tilde{\varphi}$ are the golden ratio and its conjugate, you may actually decide which is which :).
The two three-dimensional irreducible representations $\chi_4$ and $\chi_5$ differ by an exterior automorphism at the source.
However, their sum is integral, and it actually stabilizes a lattice of integral points whose sum is congruent to 0 mod 4:
| |
$C_{1}$ |
$C_{3}$ |
$C_{2,2}$ |
$C_{5}$ |
$'C_{5}$ |
| $\chi_4 \oplus \chi_5$ |
$6$ |
$0$ |
$-2$ |
$\varphi + \tilde \varphi$ |
$\tilde \varphi + \varphi$ |
Thus we may inscribe a dodecahedron and a great stellated dodecahedron in a (demi)-6 cube.
They will lie in two orthogonal 3-planes whose direct sum is Euclidean 6-space.
The set of $32=20+12$ vertices of the demi-cube is the orbit space of the representation whose character is $\chi_4 \oplus \chi_5$.
For the shadow we use the 5-fold symmetry of one of the generators of $\mathrm H_3$.
NB:
The ``explanation'' for this picture is actually closely related to that of the cover picture of Coxeter's best seller book Regular polytopes.
There one can see the shadow of a 5-cube and a stellated triacontahedron which ressemble to each other in an intriguing way.
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