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The 5 Platonic solids are called a tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron with 4, 6, 8, 12, and 20 sides respectively. The regular hexahedron is a cube . Table of polyhedra
The Dalí cross, a net of a tesseract The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space.. In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. [1]
There are 34 topologically distinct convex heptahedra, excluding mirror images. [2] ( Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
By a theorem of Descartes, this is equal to 4 π divided by the number of vertices (i.e. the total defect at all vertices is 4 π). The three-dimensional analog of a plane angle is a solid angle . The solid angle, Ω , at the vertex of a Platonic solid is given in terms of the dihedral angle by
All vertices are valence-6 except the 12 centered at the original vertices which are valence 5. A geodesic polyhedron is a convex polyhedron made from triangles. They usually have icosahedral symmetry, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles.
The tesseract is one of 6 convex regular 4-polytopes. In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope.They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex) and angles /, /, and / Thus its sides coincide with one side and two particular diagonals of the regular heptagon. [7]
In the image above, the long body diagonals are those between opposite red vertices and between opposite blue vertices, and the short body diagonals are those between opposite yellow vertices. Cartesian coordinates for the vertices of the deltoidal icositetrahedron centered at the origin and with long body diagonal length 2 are:
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