| 释义 |
OctahedronA Platonic Solid ( ) with six Vertices, 12 Edges, and eight equivalent Equilateral Triangular faces ( ), given bythe Schläfli Symbol  . It is also Uniform Polyhedron  with theWythoff Symbol  . Its Dual Polyhedron is the Cube. Like the Cube, it has the Octahedral Group of symmetries. The octahedron can be Stellated to give theStella Octangula. The solid bounded by the two Tetrahedra of the Stella Octangula (left figure) is anoctahedron (right figure; Ball and Coxeter 1987).
In one orientation (left figure), the Vertices are given by  ,  , . In another orientation (right figure), the vertices are  and  . In thelatter, the constituent Triangles are specified by
The face planes are  , so a solid octahedron is given by the equation
 | (1) |
A plane Perpendicular to a  axis of an octahedron cuts the solid in a regular HexagonalCross-Section (Holden 1991, pp. 22-23). Since there are four such axes, there are four possiblyHexagonal Cross-Sections. Faceted forms are the CuboctatruncatedCuboctahedron and Tetrahemihexahedron.
Let an octahedron be length  on a side. The height of the top Vertex from the squareplane is also the Circumradius
 | (2) |
 | (3) |
 | (4) |
so
 | (8) |
so the Inradius is
 | (12) |
 | (13) |
The Area of one face is the Area of an Equilateral Triangle
 | (14) |
 | (15) |
 | (16) |
References
Davie, T. ``The Octahedron.'' http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/octahedron.html.Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.
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