4 SimplexEdit

The 4-simplex, also known as the pentachoron or 5-cell, is the simplest convex polytope in four dimensions. It arises as the convex hull of five points in 4D that do not lie in any common 3D hyperplane, i.e., five affinely independent points. In its regular form, all edges are equal, and the five vertices are arranged so that the shape has maximal symmetry in four dimensions. In this sense, the 4-simplex is the four-dimensional analogue of the familiar triangle (the 2-simplex) and the tetrahedron (the 3-simplex). For anyone exploring higher-dimensional geometry or modern approaches to triangulations and discretizations, the 4-simplex serves as a fundamental building block. See also tetrahedron and 5-cell.

The 4-simplex occupies a central role in several branches of mathematics and related disciplines. It is the simplest nontrivial four-dimensional convex body, and its combinatorial structure—five vertices, ten edges, ten triangular faces, and five tetrahedral facets—provides a clean, highly symmetric model for exploring high-dimensional connectivity and topology. The arrangement of its vertices and faces is governed by the symmetry group isomorphic to the permutation group S5, reflecting the indistinguishability of the five corner points under rigid motions. Its boundary is a 3-manifold that is topologically a 3-sphere, giving a tangible four-dimensional object whose boundary has the familiar closed, smooth character of a sphere in higher dimensions.

Geometry and structure - Basic definition: A 4-simplex is the convex hull of five points in 4D that are affinely independent. Any selection of five such points defines a 4-simplex, and regular realizations exist where all edges have the same length. - Combinatorial data: The 4-simplex has f-vector (f0, f1, f2, f3, f4) = (5, 10, 10, 5, 1). This encodes five vertices, ten edges, ten triangular faces, five tetrahedral facets, and one 4-dimensional body. - Faces and facets: Each facet is a 3-simplex (a tetrahedron). Each 1-face (edge) joins two vertices, and each 2-face is a triangle formed by three of the vertices. The totality of these parts assembles into a cohesive four-dimensional object. - Boundary and topology: The boundary of a 4-simplex is homeomorphic to a 3-sphere, which places the 4-simplex among the simplest examples of a closed, orientable 3-manifold built from flat pieces. - Distances and coordinates: A convenient standard realization places the five vertices in a 4D hyperplane of a five-dimensional coordinate system. One explicit model takes the five vertices to be the five standard basis points in 5-space, restricted to the hyperplane x1 + x2 + x3 + x4 + x5 = 1. In this model, any two distinct vertices are at the same distance, illustrating maximal symmetry. This coordinate realization also underpins easy computation of barycentric coordinates and facilitates connections to linear algebra and optimization. See barycentric coordinates and distance for related concepts.

Regularity and symmetry - Regular 4-simplices exhibit a high degree of symmetry. The full symmetry group permutes the five vertices without changing the intrinsic shape, yielding a group isomorphic to S5. This means all five vertices are on equal footing, and every edge-length pattern present is invariant under these symmetries. - Generalizations: The 4-simplex is the simplest member of the family of n-simplices, and it serves as a template for understanding how higher-dimensional simplices behave in terms of faces, facets, and symmetries as dimension increases.

Realizations and applications - Coordinate realizations and computations: The barycentric representation allows every point in the 4-simplex to be written as a weighted average of the five vertices with weights that sum to one and are nonnegative. This is useful in interpolation, finite element methods, and computational geometry. For a broader treatment of such coordinates, see barycentric coordinates. - Topological and geometric uses: As the most elementary four-dimensional polytope, the 4-simplex is a natural building block in the construction of piecewise-linear manifolds and in the study of triangulations of 4-manifolds. It also appears in algorithms that triangulate spaces for numerical simulation, such as those used in the finite element method and in related discretization techniques. - Physics and discretized spacetime: In certain approaches to spacetime in theoretical physics, spacetime is modeled by simplicial complexes built from 4-simplices. Regge calculus, for example, uses sums over 4-simplices to approximate curvature and gravitational effects on a discretized manifold. See Regge calculus for more on this topic.

Variants and related objects - Other dimensional analogues: The concept generalizes to n-simplices, with an n-simplex in n-dimensional space having n+1 vertices. The 2-simplex is a triangle, the 3-simplex is a tetrahedron, and the 4-simplex is the subject here; together they illustrate a consistent pattern across dimensions. See simplex and tetrahedron for broader context. - Alternate names and synonyms: In some literature the 4-simplex is called the pentachoron or, in the language of polytopes, the 5-cell. See pentachoron and 5-cell for discussions of these terms.

See also - tetrahedron - triangle - simplex - pentachoron - 5-cell - Regge calculus - finite element method - 3-sphere - S5

Note: The article presents the 4-simplex in a purely geometric and mathematical light, focusing on its definitions, properties, and applications in higher-dimensional geometry and discrete models.