Small Cubicuboctahedron on:
[Wikipedia]
[Google]
[Amazon]
In
As the Euler characteristic suggests, the small cubicuboctahedron is a toroidal polyhedron of genus 3 (topologically it is a surface of genus 3), and thus can be interpreted as a (polyhedral) immersion of a genus 3 polyhedral surface, in the complement of its 24 vertices, into 3-space. (A neighborhood of any vertex is topologically a cone on a figure-8, which cannot occur in an immersion. Note that the Richter reference overlooks this fact.) The underlying polyhedron (ignoring self-intersections) defines a uniform tiling of this surface, and so the small cubicuboctahedron is a uniform polyhedron. In the language of abstract polytopes, the small cubicuboctahedron is a ''faithful realization'' of this abstract toroidal polyhedron, meaning that it is a nondegenerate polyhedron and that they have the same symmetry group. In fact, every automorphism of the abstract genus 3 surface with this tiling is realized by an isometry of Euclidean space.
Higher genus surfaces (genus 2 or greater) admit a metric of negative constant curvature (by the uniformization theorem), and the universal cover of the resulting 
Alternatively and more subtly, by chopping up each square face into 2 triangles and each octagonal face into 6 triangles, the small cubicuboctahedron can be interpreted as a non-regular ''coloring'' of the combinatorially ''regular'' (not just ''uniform'') tiling of the genus 3 surface by 56 equilateral triangles, meeting at 24 vertices, each with degree 7. Note each face in the polyhedron consist of multiple faces in the tiling, hence the description as a "coloring" – two triangular faces constitute a square face and so forth, as pe
this explanatory image
This regular tiling is significant as it is a tiling of the Klein quartic, the genus 3 surface with the most symmetric metric (automorphisms of this tiling equal isometries of the surface), and the orientation-preseserving automorphism group of this surface is isomorphic to the projective special linear group PSL(2,7), equivalently GL(3,2) (the order 168 group of all orientation-preserving isometries). Note that the small cubicuboctahedron is ''not'' a realization of this abstract polyhedron, as it only has 24 orientation-preserving symmetries (not every abstract automorphism is realized by a Euclidean isometry) – the isometries of the small cubicuboctahedron preserve not only the triangular tiling, but also the coloring, and hence are a proper subgroup of the full isometry group. The corresponding tiling of the hyperbolic plane (the universal covering) is the order-7 triangular tiling. The automorphism group of the Klein quartic can be augmented (by a symmetry which is not realized by a symmetry of the polyhedron, namely "exchanging the two endpoints of the edges that bisect the squares and octahedra) to yield the Mathieu group M24.
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the small cubicuboctahedron is a uniform star polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figure ...
, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons
In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon.
A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...
), 48 edges, and 24 vertices. Its vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
is a crossed quadrilateral.
The small cubicuboctahedron is a faceting of the rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at ...
. Its square faces and its octagonal faces are parallel to those of a cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the on ...
, while its triangular faces are parallel to those of an octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at e ...
: hence the name ''cubicuboctahedron''. The ''small'' suffix serves to distinguish it from the great cubicuboctahedron
In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14. It has 20 faces (8 triangles, 6 squares and 6 octagrams), 48 edges, and 24 vertices. Its square faces and its octagrammic faces are parallel to those of a ...
, which also has faces in the aforementioned directions.
Related polyhedra
It shares its vertex arrangement with thestellated truncated hexahedron
In geometry, the stellated truncated hexahedron (or quasitruncated hexahedron, and stellatruncated cube) is a uniform star polyhedron, indexed as U19. It has 14 faces (8 triangles and 6 octagrams), 36 edges, and 24 vertices. It is represented by ...
. It additionally shares its edge arrangement with the rhombicuboctahedron (having the triangular faces and 6 square faces in common), and with the small rhombihexahedron (having the octagonal faces in common).
Related tilings
As the Euler characteristic suggests, the small cubicuboctahedron is a toroidal polyhedron of genus 3 (topologically it is a surface of genus 3), and thus can be interpreted as a (polyhedral) immersion of a genus 3 polyhedral surface, in the complement of its 24 vertices, into 3-space. (A neighborhood of any vertex is topologically a cone on a figure-8, which cannot occur in an immersion. Note that the Richter reference overlooks this fact.) The underlying polyhedron (ignoring self-intersections) defines a uniform tiling of this surface, and so the small cubicuboctahedron is a uniform polyhedron. In the language of abstract polytopes, the small cubicuboctahedron is a ''faithful realization'' of this abstract toroidal polyhedron, meaning that it is a nondegenerate polyhedron and that they have the same symmetry group. In fact, every automorphism of the abstract genus 3 surface with this tiling is realized by an isometry of Euclidean space.
Higher genus surfaces (genus 2 or greater) admit a metric of negative constant curvature (by the uniformization theorem), and the universal cover of the resulting Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...
is the hyperbolic plane. The corresponding tiling of the hyperbolic plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive ( transitive on i ...
has vertex figure 3.8.4.8 (triangle, octagon, square, octagon). If the surface is given the appropriate metric of curvature = −1, the covering map is a local isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
and thus the ''abstract'' vertex figure is the same. This tiling may be denoted by the Wythoff symbol 3 4 , 4, and is depicted on the right.
Alternatively and more subtly, by chopping up each square face into 2 triangles and each octagonal face into 6 triangles, the small cubicuboctahedron can be interpreted as a non-regular ''coloring'' of the combinatorially ''regular'' (not just ''uniform'') tiling of the genus 3 surface by 56 equilateral triangles, meeting at 24 vertices, each with degree 7. Note each face in the polyhedron consist of multiple faces in the tiling, hence the description as a "coloring" – two triangular faces constitute a square face and so forth, as pethis explanatory image
This regular tiling is significant as it is a tiling of the Klein quartic, the genus 3 surface with the most symmetric metric (automorphisms of this tiling equal isometries of the surface), and the orientation-preseserving automorphism group of this surface is isomorphic to the projective special linear group PSL(2,7), equivalently GL(3,2) (the order 168 group of all orientation-preserving isometries). Note that the small cubicuboctahedron is ''not'' a realization of this abstract polyhedron, as it only has 24 orientation-preserving symmetries (not every abstract automorphism is realized by a Euclidean isometry) – the isometries of the small cubicuboctahedron preserve not only the triangular tiling, but also the coloring, and hence are a proper subgroup of the full isometry group. The corresponding tiling of the hyperbolic plane (the universal covering) is the order-7 triangular tiling. The automorphism group of the Klein quartic can be augmented (by a symmetry which is not realized by a symmetry of the polyhedron, namely "exchanging the two endpoints of the edges that bisect the squares and octahedra) to yield the Mathieu group M24.
See also
*Compound of five small cubicuboctahedra
Compound may refer to:
Architecture and built environments
* Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall
** Compound (fortification), a version of the above fortified with defensive struct ...
* List of uniform polyhedra
References
*External links
* {{mathworld2 , urlname = SmallCubicuboctahedron, title = Small cubicuboctahedron , urlname2 = UniformPolyhedron, title2 = Uniform polyhedron Toroidal polyhedra