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Concept
Hemipolyhedron
Résumé
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 polyhedron – hence the "hemi" prefix.
The prefix "hemi" is also used to refer to certain projective polyhedra, such as the hemi-cube, which are the image of a 2 to 1 map of a spherical polyhedron with central symmetry.
Their Wythoff symbols are of the form p/(p − q) p/q | r; their vertex figures are crossed quadrilaterals. They are thus related to the cantellated polyhedra, which have similar Wythoff symbols. The vertex configuration is p/q.2r.p/(p − q).2r. The 2r-gon faces pass through the center of the model: if represented as faces of spherical polyhedra, they cover an entire hemisphere and their edges and vertices lie along a great circle. The p/(p − q) notation implies a {p/q} face turning backwards around the vertex figure.
The nine forms, listed with their Wythoff symbols and vertex configurations are:
Note that Wythoff's kaleidoscopic construction generates the nonorientable hemipolyhedra (all except the octahemioctahedron) as double covers (two coincident hemipolyhedra).
In the Euclidean plane, the sequence of hemipolyhedra continues with the following four star tilings, where apeirogons appear as the aforementioned equatorial polygons:
Of these four tilings, only 6/5 6 ∞ is generated as a double cover by Wythoff's construction.
Only the octahemioctahedron represents an orientable surface; the remaining hemipolyhedra have non-orientable or single-sided surfaces. This is because proceeding around an equatorial 2r-gon, the p/q-gonal faces alternately point "up" and "down", so any two consecutive ones have opposite senses. This is equivalent to demanding that the p/q-gons in the corresponding quasiregular polyhedra below can be alternatively given positive and negative orientations.
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