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Compound of eight octahedra with rotational freedom; Compound of eight triangular prisms; Compound of five cubes; Compound of five cuboctahedra; Compound of five cubohemioctahedra; Compound of five great cubicuboctahedra; Compound of five great dodecahedra; Compound of five great icosahedra; Compound of five great rhombihexahedra; Compound of ...
If the shape is considered as a union of five cubes yielding a simple nonconvex solid without self-intersecting surfaces, then it has 360 faces (all triangles), 182 vertices (60 with degree 3, 30 with degree 4, 12 with degree 5, 60 with degree 8, and 20 with degree 12), and 540 edges, yielding an Euler characteristic of 182 − 540 + 360 = 2.
The tangram (Chinese: 七巧板; pinyin: qīqiǎobǎn; lit. 'seven boards of skill') is a dissection puzzle consisting of seven flat polygons, called tans, which are put together to form shapes. The objective is to replicate a pattern (given only an outline) generally found in a puzzle book using all seven pieces without overlap.
If two regular tetrahedra are given the same orientation on the 3-fold axis, a different compound is made, with D 3h, [3,2] symmetry, order 12.. Other orientations can be chosen as 2 tetrahedra within the compound of five tetrahedra and compound of ten tetrahedra the latter of which can be seen as a hexagrammic pyramid:
The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. Its interior is an octahedron, and correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e., rectifying the tetrahedron).
In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts transitively on the compound's vertices.
The interior of the compound of two dual tetrahedra is an octahedron, and this compound—called the stella octangula—is its first and only stellation. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron).
This can be seen as the three-dimensional equivalent of the compound of two pentagons ({10/2} "decagram"); this series continues into the fourth dimension as the compound of 120-cell and 600-cell and into higher dimensions as compounds of hyperbolic tilings.
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