In geometry, a polychoron or 4polytope is a fourdimensional 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 twodimensional analogue of a polychoron is a polygon, and the threedimensional 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 4polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4polytopes can be cut and unfolded as nets in 3space.
Contents
Definition
Polychora are closed fourdimensional 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 threedimensional analogue of a face, and is therefore a polyhedron. Furthermore, the following requirements must be met:
Euler characteristic
The Euler characteristic for 4polytopes that are topological 3spheres (including all convex 4polytopes) is zero. χ=VE+FC=0.
Full article ▸
