|
In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in n-dimensions with the Schläfli symbols and containing the symmetry of Coxeter group Rn (or B~n-1) for n>=3. The tessellation is constructed from 4 n-hypercubes per ridge. The vertex figure is a cross-polytope . The hypercubic honeycombs are self-dual. Coxeter named this family as δn+1 for an n-dimensional honeycomb. == Wythoff construction classes by dimension == There are two general forms of the hypercube honeycombs, the ''regular'' form with identical hypercubic facets and one ''semiregular'', with alternating hypercube facets, like a checkerboard. A third form is generated by an expansion operation applied to the regular form, creating facets in place of all lower-dimensional elements. For example, an ''expanded cubic honeycomb'' has cubic cells centered on the original cubes, on the original faces, on the original edges, on the original vertices, creating 3 colors of cells around in vertex in 1:3:3:1 counts. The orthotopic honeycombs are a family topologically equivalent to the cubic honeycombs but with lower symmetry, in which each of the three axial directions may have different edge lengths. The facets are hyperrectangles, also called orthotopes; in 2 and 3 dimensions the orthotopes are rectangles and cuboids respectively. (1 color, n colors) !''Checkerboard'' (2 colors) |- | δ2 | Apeirogon | | | | |- |δ3 | Square tiling |2 | | | |- | δ4 | Cubic honeycomb |3 | | | |- |δ5 | ''4-cube honeycomb'' |4 | | | |- | δ6 | ''5-cube honeycomb'' |5 | | | |- | δ7 | ''6-cube honeycomb'' |6 | | | |- | δ8 | ''7-cube honeycomb'' |7 | | | |- | δ9 | ''8-cube honeycomb'' |8 | | | |- |colspan=6| |- | δn | ''n-hypercubic honeycomb'' |n |colspan=2|... |} 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「hypercubic honeycomb」の詳細全文を読む スポンサード リンク
|