Polychoron

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In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells.

The two-dimensional analogue of a polychoron is a polygon, and the three-dimensional analogue is a polyhedron.

The term polychoron (plural polychora), from the Greek roots poly ('many') and choros ('room' or 'space') and has been advocated by Norman Johnson and George Olshevsky, but it is little known in general polytope theory. Other names for polychoron include: polyhedroid and polycell.

Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space.

Contents

Definition

Polychora are closed four-dimensional figures. We can describe them further only through analogy with such three dimensional polyhedron counterparts as pyramids and cubes.

The most familiar example of a polychoron is the tesseract or hypercube, the 4d analogue of the cube. A tesseract has vertices, edges, faces, and cells. A vertex is a point where four or more edges meet. An edge is a line segment where three or more faces meet, and a face is a polygon where two cells meet. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Furthermore, the following requirements must be met:

Euler characteristic

The Euler characteristic for 4-polytopes that are topological 3-spheres (including all convex 4-polytopes) is zero. χ=V-E+F-C=0.

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