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Uniform polyhedra and duals
Uniform polyhedra {{polyhedron-stub
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 cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. It has 10 faces (6 squares and 4 regular hexagons
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A ''regular hexagon'' has ...
), 24 edges and 12 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.
It is given Wythoff symbol 4 , 3, although that is a double-covering of this figure.
A nonconvex polyhedron has intersecting faces which do not represent new edges or faces. In the picture vertices are marked by golden spheres, and edges by silver cylinders.
It is a hemipolyhedron
In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These "hemi" faces lie parallel to the faces of some other symmetrical polyhedron, and their count is half the number of faces of that other p ...
with 4 hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A ''regular hexagon'' h ...
al faces passing through the model center. The hexagons intersect each other and so only triangular portions of each are visible.
Related polyhedra
It shares the vertex arrangement and edge arrangement with thecuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
(having the square faces in common), and with the octahemioctahedron
In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as . It has 12 faces (8 triangles and 4 hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral.
It is on ...
(having the hexagonal faces in common).
Tetrahexagonal tiling
The ''cubohemioctahedron'' can be seen as a net on the hyperbolictetrahexagonal tiling
In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r.
Constructions
There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the ,4 kaleidos ...
with vertex figure 4.6.4.6.
Hexahemioctacron
The hexahemioctacron is the dual of the cubohemioctahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the octahemioctacron. Since the cubohemioctahedron has four hexagonal faces passing through the model center, thus it isdegenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descr ...
, and can be seen as having four vertices at infinity.
In Magnus Wenninger
Father Magnus J. Wenninger OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction.
Early life and education
Born to Ge ...
's ''Dual Models'', they are represented with intersecting infinite prisms passing through the model center, cut off at a certain point that is convenient for the maker.
See also
* Hemi-cube - The four vertices at infinity correspond directionally to the four vertices of this abstract polyhedron.References
* (Page 101, Duals of the (nine) hemipolyhedra)External links
* *Uniform polyhedra and duals
Uniform polyhedra {{polyhedron-stub