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In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid,〔Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68〕 octacube, or polyoctahedron, being constructed of octahedral cells. The boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24-cell is self-dual. In fact, the 24-cell is the unique convex self-dual regular Euclidean polytope which is neither a polygon nor a simplex. Due to this singular property, it does not have a good analogue in 3 dimensions, but in 2 dimensions the hexagon, along with all regular polygons, are self-dual. ==Constructions== A 24-cell is given as the convex hull of its vertices. The vertices of a 24-cell centered at the origin of 4-space, with edges of length 1, can be given as follows: 8 vertices obtained by permuting : (±1, 0, 0, 0) and 16 vertices of the form :(±, ±, ±, ±). The first 8 vertices are the vertices of a regular 16-cell and the other 16 are the vertices of the dual tesseract. This gives a construction equivalent to cutting a tesseract into 8 cubical pyramids, and then attaching them to the facets of a second tesseract. This is equivalent to the dual of a rectified orthoplex. The analogous construction in 3-space gives the rhombic dodecahedron which, however, is not regular. We can further divide the last 16 vertices into two groups: those with an even number of minus (−) signs and those with an odd number. Each of groups of 8 vertices also define a regular 16-cell. The vertices of the 24-cell can then be grouped into three sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract. The vertices of the dual 24-cell are given by all permutations of : (±1, ±1, 0, 0). The dual 24-cell has edges of length and is inscribed in a 3-sphere of radius . Another method of constructing the 24-cell is by the rectification of the 16-cell. The vertex figure of the 16-cell is the octahedron; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produce 8 octahedral cells. This process also rectifies the tetrahedral cells of the 16-cell which also become octahedra, thus forming the 24 octahedral cells of the 24-cell. 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「24-cell」の詳細全文を読む スポンサード リンク
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