|
! ! ! ! ! ! |- !r !r !r !r !r !r !r |- ! ! ! ! ! ! ! |- align=center valign=top |60px |60px |60px |60px |60px |60px |60px |- |colspan=7|A quasiregular polyhedron or tiling has exactly two kinds of regular face, which alternate around each vertex. Their vertex figures are rectangles. |} In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive and hence a step closer to regular polyhedra than the semiregular which are merely vertex-transitive. There are only two convex quasiregular polyhedra, the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing their faces contain all the faces of the dual-pair cube and octahedron, in the first, and the dual-pair icosahedron and dodecahedron in the second case. These forms representing a pair of a regular figure and its dual can be given a vertical Schläfli symbol or r to represent their containing the faces of both the regular and dual regular . A quasiregular polyhedron with this symbol will have a vertex configuration p.q.p.q (or (p.q)2). More generally, a quasiregular figure can have a vertex configuration (p.q)r, representing ''r'' (2 or more) instances of the faces around the vertex. Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, with vertex configuration (3.6)2. Other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling, (3.7)2. Or more generally, (p.q)2, with 1/p+1/q<1/2. ! r ! r ! r ! r ! r ! r |- ! ! ! ! ! ! ! |- align=center valign=top !(3.3)2 !(4.4)2 !(5.5)2 !(6.6)2 !(7.7)2 !(8.8)2 !(∞.∞)2 |- ! ! ! ! ! ! ! |- align=center valign=top |60px |60px square tiling |60px order-4 pentagonal tiling |60px order-4 hexagonal tiling |60px order-4 heptagonal tiling |60px order-4 octagonal tiling |60px Order-4 apeirogonal tiling |- !colspan=8| General triangles (p p 3) |- align=center valign=top ! ! ! ! ! ! ! |- align=center valign=top !(3.3)3 !(4.4)3 !(5.5)3 !(6.6)3 !(7.7)3 !(8.8)3 !(∞.∞)3 |- ! ! ! ! ! ! ! |- align=center valign=top |60px |60px |60px |60px |60px |60px |60px |- !colspan=8| General triangles (p p 4) |- align=center valign=top ! ! ! ! ! ! ! |- align=center valign=top !(3.3)4 !(4.4)4 !(5.5)4 !(6.6)4 !(7.7)4 !(8.8)4 !(∞.∞)4 |- ! ! ! ! ! ! ! |- align=center valign=top |60px |60px |60px |60px |60px |60px |60px |- |colspan=7|A regular polyhedron or tiling can be considered quasiregular if it has an even number of faces around each vertex (and thus can have alternately colored faces). |} Some regular polyhedra and tilings (those with an even number of faces at each vertex) can also be considered quasiregular by differentiating between faces of the same number of sides, but representing them differently, like having different colors, but no surface features defining their orientation. A regular figure with Schläfli symbol can be quasiregular, with vertex configuration (p.p)q/2, if q is even. The octahedron can be considered quasiregular as a ''tetratetrahedron'' (2 sets of 4 triangles of the tetrahedron), (3a.3b)2, alternating two colors of triangular faces. Similarly the square tiling (4a.4b)2 can be considered quasiregular, colored as a ''checkerboard''. Also the triangular tiling can have alternately colored triangle faces, (3a.3b)3. == Wythoff construction == Coxeter defines a ''quasiregular polyhedron'' as one having a Wythoff symbol in the form ''p | q r'', and it is regular if q=2 or q=r.〔Coxeter, H.S.M., Longuet-Higgins, M.S. and Miller, J.C.P. Uniform Polyhedra, ''Philosophical Transactions of the Royal Society of London'' 246 A (1954), pp. 401–450. (Section 7, The regular and quasiregular polyhedra ''p | q r'')〕 The Coxeter-Dynkin diagram is another symbolic representation that shows the quasiregular relation between the two dual-regular forms: ||||||''q | 2 p'' |- align=center ||||| ||''p | 2 q'' |- align=center | ||r|| ||''2 | p q'' |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quasiregular polyhedron」の詳細全文を読む スポンサード リンク
|
