Polychoir

A polychōr or polychōron ( that , from ancient Greek πολύς polýs 'much' and χῶρος chōros 'space'; plural polychōra ) is a 4-dimensional polytope (4-polytope). Polychora are bounded by polyhedra .

The simplest example is the pentachoron , another well-known example is the tesseract , a generalization of the classic cube to four dimensions.

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

Platonic polychora

As in the 3-dimensional, the regular (= regular) among the Polychora are called Platonic. A polytop is regular if it is delimited by regular polytopes in a regular manner. In fact, there is a 4-dimensional Platonic polychoir for each of the five 3-dimensional Platonic polyhedra : ${\displaystyle n}$ ${\displaystyle (n\!\!-\!\!1)}$

• to the tetrahedron the pentachoron (5-cell) with Schläfli symbol {3,3,3}, a simplex with 5 cells ( tetrahedra ),
• to the cube the tesseract (8-cell) with Schläfli symbol {4,3,3}, the hypercubus (dimension polytope) with 8 cells ( cubes ),
• to the octahedron the 24-cell (Ikositetrachor) with 24 cells ( octahedron ) (Schläfli-Symbol {3,4,3}),
• to the dodecahedron the 120-cell with 120 cells ( dodecahedra ) (Schläfli symbol {5,3,3}) and
• to the icosahedron the 600-cell with 600 cells ( icosahedra ) (Schläfli symbol {3,3,5}).

There is also a sixth

• the four-dimensional cross polytope (Orthoplex), the 16-cell , with 16 cells (tetrahedra) (Schläfli symbol {3,3,4}),

so that there are six Platonic polychora (4-polytopes) in four-dimensional space .

8-cell and 16-cell are dual to one another , and 120-cell to 600-cell (and vice versa). The dual polytope for the 5-cell is the 5-cell and for the 24-cell the 24-cell - they are dual to themselves, in short: self-dual.

If the number of dimensions is greater than 4, then in the -dimensional Euclidean space there are only the -Simplex, the -Mass polytope and the -Cross polytope as regular polytopes . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

See also

• Polychory
• Polychoral correlation

Web links

• Eric W. Weisstein : Polychoron . In: MathWorld (English).
• Uniform Polychora and Other Four Dimensional Shapes. At: polytope.net.