|
" TITLE="2">)〔Regular polytopes, pp.48-50, p.98〕 |- |bgcolor=#e7dcc3|Schläfli symbol|| |- |bgcolor=#e7dcc3|Coxeter diagram|| ∪ = |- |bgcolor=#e7dcc3|Stellation core||Octahedron |- |bgcolor=#e7dcc3|Convex hull||Cube |- |bgcolor=#e7dcc3|Index||UC4, W19 |- |bgcolor=#e7dcc3|Polyhedra||2 tetrahedra |- |bgcolor=#e7dcc3|Faces||8 triangles |- |bgcolor=#e7dcc3|Edges||12 |- |bgcolor=#e7dcc3|Vertices||8 |- |bgcolor=#e7dcc3|Dual||Self-dual |- |bgcolor=#e7dcc3|Symmetry group Coxeter group||octahedral (''O''h) () or |} The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's ''Divina Proportione,'' 1509.〔.〕 It is the simplest of five regular polyhedral compounds. It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. It can also be seen as one of the stages in the construction of a 3D Koch Snowflake, a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure. The first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron. ==Construction== The stellated octahedron can be constructed in several ways: *it is a stellation of the regular octahedron, sharing the same face planes.. The stellation facets are very simple: 60px (See Wenninger model W19.) *It is also a regular polyhedron compound, when constructed as the union of two tetrahedra (a tetrahedron and its dual tetrahedron). *It can be obtained as an augmentation of the regular octahedron, by adding tetrahedral pyramids on each face. In this construction it has the same topology as the convex Catalan solid, the triakis octahedron, which has much shorter pyramids. *It is a facetting of the cube, sharing the same vertices. 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「stellated octahedron」の詳細全文を読む スポンサード リンク
|
