The Cuboctahedron
I have long been fascinated by the Cuboctahedron. It is visually very interesting because of
the way it combines Triangular and Square symmetries, which we normally don't see appearing
together. This beauty indicates something deeper. As you see in this site, this polyhedron just
keeps re-appearing over and over.
As we've seen in our study of the polyhedra,
it is the rectification of two dual regular polyhedra, the Cube and Octahedron, so it has the faces
of both. Like all other rectifications or regular polyhedra, it has an equatorial polygon,
the Hexagon. This implies that, like the Hexagon, it is radially equilateral. Among the Platonic and
Archimedean polyhedra, it is the only polyhedron with this characteristic. The Cuboctahedron has four
equatorial Hexagons located halfway between two opposite Triangular faces, each has 1/2 of its vertices
(the opposite faces account for the remaining vertices) and 1/4 of its edges.
Fig. 1: Cuboctahedra.
As we've seen our study of the regular convex polychora,
it is, like my favourite polychoron (the 24-cell) a rectification of a cross polytope.
However, it is also the equatorial polyhedron of the 24-cell, which implies that the latter
is also radially equilateral. Of all convex regular polytopes, only the Hexagon, Tesseract and
24-cell have this property. The 24-cell has 12 equatorial Cuboctahedra (as many as flank
the flight of stairs on the left in Figs. 1 and 2) located halfway between pairs of
opposite Octahedra, each Cuboctahedron has 1/2 of its vertices (the opposite cells account for the
remaining vertices), 1/4 of its edges and 1/12 of its Triangular faces. These proportions are
very similar to the relation between the Hexagon and Cuboctahedron.
Fig. 2: Cuboctahedra.
We have also seen in that page that the Cuboctahedron also generates my favourite Platonic solid,
the Icosahedron, by adding diagonals to its Squares and then enforcing equal length to all edges.
This is possible because of the unusual kinematics of the
cuboctahedron. Furthermore, as we've also seen in our study of the regular convex polychora, this transformation is an important element in
the derivation of my second-favourite regular polychoron (the 600-cell) from the 24-cell (Gosset's
construction).
Fig. 3: Cuboctahedra.
As we've seen our study of the star polychora,
a Cuboctahedral configuration can be used to describe the geometric relations between the 12 regular
polychora with the symmetry of the 600-cell. This connection is to me somewhat mysterious, but the
fact that the Cuboctahedron has 12 vertices in a highly symmetric arrangement is a clear
requirement.
Fig. 4: Cuboctahedra.
This musterious feature of the Cuboctahedron might be related to its
sphere packing properties.
In one of the two most compact configurations of equal-sized spheres, the face centered cubic (FFC)
lattice, the centres of the spheres surrounding each sphere have the same arrangement as the
vertices of the Cuboctahedron. In the second such configuration, the Hexagonal close-packed (HCP)
packing, linking the centres of the spheres surrounding each sphere we obtain the triangular orthobicupola, one of
the Johnson solids.
This can be built by dividing the Cuboctahedron in two along one of its equatorial Hexagons and
rotating one half by 60 degrees.
Fig. 5: Cuboctahedra.
I don't know the reason why so many Cuboctahedra appear in this temple
(Nankunshen)
near Tainan, in
Taiwan. They certainlly suit the quiet, calm, meditative atmosphere of the place. But it is
not just that. I guess someone felt, like I do, that this apparently simple shape is
unusually interesting, full of hidden secrets, an inhabitant of a quiet, unchanging world beyond
our daily experience. Certainly something suitable for a temple!
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