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The exterior face is a 90-60-30 triangle which is one-sixth of an octahedron face. The three faces interior to the octahedron are: a 45-90-45 triangle with edges , , , a right triangle with edges , , , and a right triangle with edges , , .
Faces are 8-by-8 combined to larger faces for a = b = 0 (cube) and 6-by-6 for a = b = c (octahedron). The 9 mirror lines of full octahedral symmetry can be divided into two subgroups of 3 and 6 (drawn in purple and red), representing in two orthogonal subsymmetries: D 2h , and T d .
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6 ...
Geodesic subdivisions can also be done from an augmented dodecahedron, dividing pentagons into triangles with a center point, and subdividing from that Chiral polyhedra with higher order polygonal faces can be augmented with central points and new triangle faces. Those triangles can then be further subdivided into smaller triangles for new ...
A Goldberg polyhedron is one whose faces are 12 pentagons and some multiple of 10 hexagons. There are three classes of Goldberg polyhedron, one of them is constructed by truncating all vertices repeatedly, and the truncated icosahedron is one of them, denoted as GP ( 1 , 1 ) {\displaystyle \operatorname {GP} (1,1)} .
The convex shapes in this family range from the octahedron itself through the regular icosahedron to the cuboctahedron, with its square faces subdivided into two right triangles in a flat plane. Extending the range of the parameter past the proportion that gives the cuboctahedron produces non-convex shapes, including Jessen's icosahedron.
The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers (upper left to lower right) say how many of each element occur in the whole 4-polytope. The non-diagonal numbers say how many of the column's element occur in or at the row's element.
The remaining octahedron edges cross each other in pairs, within the interior of the compound; their crossings are at their midpoints and form right angles. The compound of three octahedra can also be formed from three copies of a single octahedron by rotating each copy by an angle of π /4 around one of the three symmetry axes that pass ...