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The cube can also be dissected into 48 smaller instances of this same characteristic 3-orthoscheme (just one way, by all of its symmetry planes at once). The characteristic tetrahedron of the cube is an example of a Heronian tetrahedron. Every regular polytope, including the regular tetrahedron, has its characteristic orthoscheme. There is a 3 ...
Triangular pyramid: Y 3 (A tetrahedron is a special pyramid) T = Y 3; O = aT (ambo tetrahedron) C = jT (join tetrahedron) I = sT (snub tetrahedron) D = gT (gyro tetrahedron) Triangular antiprism: A 3 (An octahedron is a special antiprism) O = A 3; C = dA 3; Square prism: P 4 (A cube is a special prism) C = P 4; Pentagonal antiprism: A 5. I = k ...
4-dimensional hyperpyramid with a cube as base. The hyperpyramid is the generalization of a pyramid in n-dimensional space. In the case of the pyramid, one connects all vertices of the base, a polygon in a plane, to a point outside the plane, which is the peak. The pyramid's height is the distance of the peak from the plane.
Square pyramids are added to the six square faces of the snub cube, and triangular pyramids are added to the eight triangular faces that do not share an edge with a square. The pyramid heights are adjusted to make them coplanar with the other 24 triangular faces of the snub cube. The result is the pentagonal icositetrahedron.
The square pyramid caps have shortened isosceles triangle faces, with six of these pyramids meeting together to form a cube. The dual of this honeycomb is composed of two kinds of octahedra (regular octahedra and triangular antiprisms), formed by superimposing octahedra into the cuboctahedra of the rectified cubic honeycomb.
2-dimensional hyperpyramid with a line segment as base 4-dimensional hyperpyramid with a cube as base. In geometry, a hyperpyramid is a generalisation of the normal pyramid to n dimensions. In the case of the pyramid one connects all vertices of the base (a polygon in a plane) to a point outside the plane, which is the peak. The pyramid's ...
Building a pyramid on each face of a regular tetrahedron, using six units, results in a cube (the central fold of each module lays flat, creating square faces instead of isosceles right triangular faces, and changing the formula for the number of faces, edges, and vertices), or triakis tetrahedron.
A double symmetry construction can be constructed by placing a small cube into each large cube, resulting in a nonuniform honeycomb with cubes, square prisms, and rectangular trapezoprisms (a cube with D 2d symmetry). Its vertex figure is a triangular pyramid with its lateral faces augmented by tetrahedra. Dual cell