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5-simplex

A 5-simplex is a five-dimensional polytope in geometry, defined as the convex hull of six affinely independent points in \mathbb{R}^5.^[1] It generalizes lower-dimensional simplices such as the triangle (2-simplex) and tetrahedron (3-simplex), serving as the simplest convex polytope in five-dimensional space. Also known as a hexateron, the regular 5-simplex is self-dual.^[2] It consists of six vertices, fifteen edges, twenty equilateral triangular faces, fifteen regular tetrahedral cells, and six regular 4-simplex (pentachoral) facets, with all elements being regular simplices of the appropriate dimension.^[2] The 5-simplex can be coordinatized in \mathbb{R}^6 as the standard 5-simplex, the convex hull of the points where each has a single coordinate equal to 1 and the rest 0, then embedded into \mathbb{R}^5 via a suitable projection while preserving affine independence.^[2] Its dihedral angle is \cos^{-1}(1/5) \approx 78.46^\circ^[3], and it satisfies Euler's polytope formula with characteristic 1, confirming its topological structure as a 5-ball with boundary homeomorphic to a 4-sphere. In higher-dimensional geometry, the 5-simplex plays a foundational role in simplicial complexes, convex optimization, and the study of uniform polytopes, including its rectifications, truncations, and sterications.^[1]

Definition and Properties

Geometric Definition

A 5-simplex is the five-dimensional analogue of lower-dimensional simplices, such as the line segment (1-simplex), equilateral triangle (2-simplex), regular tetrahedron (3-simplex), and pentachoron (4-simplex). These figures represent the simplest polytopes in their respective dimensions, generalizing the concept of a convex polytope formed by connecting vertices without intersections other than at shared faces.^[2] Geometrically, a 5-simplex is defined as the convex hull of six affinely independent points in 5-dimensional Euclidean space \mathbb{R}^5. Affine independence of points v_0, v_1, \dots, v_5 requires that the vectors v_1 - v_0, v_2 - v_0, \dots, v_5 - v_0 are linearly independent, ensuring the points span the full 5-dimensional affine subspace without redundancy. The convex hull consists of all points expressible as convex combinations \sum_{i=0}^5 \lambda_i v_i where \lambda_i \geq 0 and \sum_{i=0}^5 \lambda_i = 1, forming a bounded, convex 5-polytope with these six vertices.^[4] The standard 5-simplex is constructed as the convex hull in \mathbb{R}^6 of the six standard basis vectors e_1, \dots, e_6, where each e_i has a 1 in the i-th coordinate and 0 elsewhere. This yields the set of points (x_1, \dots, x_6) \in \mathbb{R}^6 satisfying x_i \geq 0 for all i and \sum_{i=1}^6 x_i = 1, embedded within the 5-dimensional hyperplane defined by the summation constraint. As the basic unit, the 5-simplex serves as a prerequisite element in simplicial complexes, enabling the triangulation and modeling of higher-dimensional geometric spaces through collections of such simplices glued along shared faces.^[2]^[5]

Combinatorial Elements

The 5-simplex, as an n-simplex with n=5, is combinatorially defined as the convex hull of 6 affinely independent points in 5-dimensional space, where its k-faces for $0 \leq k \leq 5 are the convex hulls of all subsets of k+1 of these vertices. The number of such k-faces is given by the binomial coefficient \binom{6}{k+1}, yielding 6 vertices (k=0), 15 edges (k=1), 20 triangular faces (k=2), 15 tetrahedral cells (k=3), 6 pentachoric hypercells (k=4), and 1 5-cell itself (k=5).^[6]^[7] The 1-skeleton of the 5-simplex, consisting of its vertices and edges, forms the complete graph K_6, where every pair of the 6 vertices is connected by an edge.^[8] As a self-dual polytope, the 5-simplex is combinatorially equivalent to its dual, resulting in matching numbers of k-faces and (5-k)-faces for each k.^[9]^[10] Incidence relations among these elements follow from the combinatorial structure: each vertex is incident to 5 edges, as it connects to each of the other 5 vertices; each edge is incident to 4 triangular faces, determined by choosing one additional vertex from the remaining 4 to form a triangle.^[6]^[11]

Metric Properties

The regular 5-simplex is the five-dimensional analog of the tetrahedron, possessing uniform edge lengths and symmetric metric properties that generalize those of lower-dimensional simplices. Its measurable attributes, such as volume and angles, are determined by the edge length a, providing insight into its size and shape in Euclidean 5-space. These properties are derived from the geometry of regular polytopes and can be expressed in closed form, facilitating comparisons across dimensions. The volume V of a regular 5-simplex with edge length a is given by

V = \frac{\sqrt{6}}{480 \sqrt{2}} \, a^5 \approx 0.00361 a^5.

This formula arises from integrating over the simplex or using the Cayley-Menger determinant for polytopal volumes, scaled to the edge length. More generally, the volume of a regular n-simplex follows

V_n = \frac{\sqrt{n+1}}{n! \, 2^{n/2}} \, a^n,

which for n=5 yields the specific expression above; the factorial term accounts for the n! pyramidal decompositions, while the square root and power of 2 reflect the orthogonal projections and symmetries in the construction.^[12] The dihedral angle \theta between two adjacent 4-simplex facets measures the internal angle at the shared 3-simplex ridge and is

\theta = \arccos\left(\frac{1}{5}\right) \approx 78.46^\circ.

This angle is constant across all facet pairs due to regularity and can be derived from the normals to the facets or using tangent sphere configurations at the ridges, generalizing to \arccos(1/n) for the n-simplex.^[3] All vertex-to-vertex distances in the regular 5-simplex are equal to the edge length a, reflecting its complete graph K_6 connectivity with uniform metrics. The circumradius R, the radius of the unique hypersphere passing through all six vertices, is

R = a \sqrt{\frac{5}{12}} \approx 0.645 a.

This follows from the geometry of the vertex positions relative to the centroid, where the squared distance yields the factor n/(2(n+1)) for dimension n=5. The inradius r, the radius of the insphere tangent to all six facets, satisfies r = R/n and is thus

r = a \sqrt{\frac{1}{60}} \approx 0.129 a.

The midradius \rho, the radius of the intersphere tangent to all 15 edges at their midpoints, is obtained from the distance to edge midpoints and given by

\rho = a \sqrt{\frac{1}{6}} \approx 0.408 a.

These radii establish the scale of the insphere, circumsphere, and midsphere, with the midradius bridging the vertex and facet metrics through edge tangency.^[13]

Nomenclature

Standard Name

The conventional designation for the regular polytope in five-dimensional Euclidean space, analogous to the triangle in two dimensions and the tetrahedron in three dimensions, is the 5-simplex or 5-dimensional simplex, reflecting its role as the simplest convex polytope with six vertices in five dimensions. This naming generalizes the concept of an n-simplex as the convex hull of n+1 affinely independent points, establishing a consistent terminology across higher-dimensional geometry.^[14] As a regular 5-polytope, the 5-simplex is represented by the Schläfli symbol \{3,3,3,3\}, which recursively describes its structure: each vertex figure is a regular tetrahedron \{3,3,3\}, building iteratively from lower-dimensional simplices.^[15] In the context of simplicial complex theory, the 5-simplex is standardly denoted as \Delta^5, the convex hull of six points in \mathbb{R}^6 satisfying \sum_{i=0}^5 t_i = 1 with t_i \geq 0, serving as the fundamental building block for triangulations and homology computations.^[16] This nomenclature was formalized and popularized by H.S.M. Coxeter in his 1948 treatise Regular Polytopes, which systematically classified and named higher-dimensional regular polytopes, including the simplex family, influencing subsequent literature on convex geometry and Coxeter groups.^[17]

Alternate Names

The 5-simplex is alternatively referred to as the hexateron in geometric literature on higher-dimensional polytopes. This name originates from the Greek prefix hexa- (six), denoting its six vertices or six bounding 5-cell facets, combined with the suffix -teron, a variant of tetra- adapted to signify the four-dimensional character of those facets. The term hexateron appears frequently in abstract polytope theory, where it emphasizes the figure's role as the simplest regular 5-polytope with Schläfli symbol {3,3,3,3}.^[18]

Construction

Cartesian Coordinates

The regular 5-simplex can be embedded in five-dimensional Euclidean space \mathbb{R}^5 using Cartesian coordinates for its six vertices, derived from the standard basis in the six-dimensional space \mathbb{R}^6 and projected onto the five-dimensional hyperplane where the coordinates sum to zero. This construction ensures the simplex is regular, with all edges of equal length and all vertices equidistant from the centroid at the origin.^[19]^[20] To obtain these coordinates, one standard method begins by defining five vertices as the standard basis vectors e_1, \dots, e_5 in \mathbb{R}^5 and the sixth vertex as v_6 = A (1,1,1,1,1), where A = \frac{1 - \sqrt{6}}{5}. The centroid is then subtracted from each to center at the origin, yielding vertices in \mathbb{R}^5 with edge length \sqrt{2}. This approach preserves the Euclidean metric and affine independence. The resulting vertices are:

\begin{align*} v_1 &= \left( \frac{24 + \sqrt{6}}{30}, \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30} \right), \\ v_2 &= \left( \frac{\sqrt{6} - 6}{30}, \frac{24 + \sqrt{6}}{30}, \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30} \right), \\ v_3 &= \left( \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30}, \frac{24 + \sqrt{6}}{30}, \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30} \right), \\ v_4 &= \left( \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30}, \frac{24 + \sqrt{6}}{30}, \frac{\sqrt{6} - 6}{30} \right), \\ v_5 &= \left( \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30}, \frac{\sqrt{6} - 6}{30}, \frac{24 + \sqrt{6}}{30} \right), \\ v_6 &= \left( -\frac{\sqrt{6}}{6}, -\frac{\sqrt{6}}{6}, -\frac{\sqrt{6}}{6}, -\frac{\sqrt{6}}{6}, -\frac{\sqrt{6}}{6} \right). \end{align*}

^[21] This set satisfies \|v_i - v_j\|^2 = 2 for all i \neq j, confirming regularity, and the centroid is at the origin since \sum_{i=1}^6 v_i = 0.^[20] These absolute Euclidean coordinates differ from barycentric coordinates, which represent positions as affine combinations relative to the vertices.^[19] For computational purposes, such as verifying distances or performing geometric calculations, this coordinate system allows direct use in \mathbb{R}^5 without further projection, as the embedding preserves the Euclidean metric.^[22]

Barycentric Coordinates

In a 5-simplex, barycentric coordinates provide an affine-invariant system for parameterizing points within the polytope as convex combinations of its six vertices, denoted v_1, v_2, \dots, v_6. Any point p inside or on the boundary of the 5-simplex can be uniquely expressed as p = \sum_{i=1}^6 \alpha_i v_i, where the barycentric coordinates (\alpha_1, \alpha_2, \dots, \alpha_6) satisfy \alpha_i \geq 0 for all i and \sum_{i=1}^6 \alpha_i = 1. This representation is unique for each point and extends naturally from the general theory of barycentric coordinates on convex sets, ensuring that the coordinates are non-negative and sum to unity, which aligns with the affine structure of the simplex.^[23] The vertices themselves correspond to the standard unit vectors in this coordinate system: for example, v_1 has coordinates (1, 0, 0, 0, 0, 0), v_2 has (0, 1, 0, 0, 0, 0), and so on up to v_6 with (0, 0, 0, 0, 0, 1). These basis-like representations facilitate computations independent of any specific embedding, such as in Euclidean space, making them particularly useful for intrinsic geometric operations on the 5-simplex.^[24] Barycentric coordinates are advantageous in simplicial decompositions of higher-dimensional spaces, where the 5-simplex serves as a fundamental building block for triangulations, allowing efficient subdivision into smaller simplices while preserving affine properties. In finite element methods (FEM) applied to 5-dimensional problems, such as those in computational physics or engineering simulations, these coordinates enable the construction of basis functions that are straightforward to evaluate and integrate over the simplex, supporting shape functions like the linear Lagrange elements where the barycentric weights directly yield the nodal values. This approach simplifies the assembly of stiffness matrices and ensures numerical stability in arbitrary dimensions, including 5D, without reliance on global coordinate systems.^[25]^[26] Furthermore, the 5-simplex in barycentric coordinates corresponds to the Dirichlet simplex in probability theory, where the coordinates (\alpha_1, \dots, \alpha_6) represent probability vectors over six categories, with the uniform measure on the simplex induced by the Dirichlet distribution with all parameters equal to 1. This connection is foundational in statistical modeling, as it parameterizes the space of multinomial distributions in 5 dimensions, facilitating Bayesian inference and sampling techniques on the simplex.^[27]

Symmetry and Configuration

Coxeter-Dynkin Diagram

The Coxeter-Dynkin diagram of the regular 5-simplex consists of a linear chain of five nodes connected by single edges, denoted as o—o—o—o—o. This representation corresponds to the Schläfli symbol {3,3,3,3}, where the absence of numerical labels on the edges implies a branching factor of 3 at each connection.^[28] Each node in the diagram represents one of the five generating reflections of the associated Coxeter group, while each single edge signifies that the product of the two adjacent reflections has order 3. Geometrically, this order corresponds to a dihedral angle of 60° (or \pi/3 radians) between the reflecting hyperplanes (mirrors).^[28] The diagram defines the irreducible finite Coxeter group A_5, which is the full symmetry group of the regular 5-simplex. This group admits the presentation

\langle r_1, r_2, r_3, r_4, r_5 \mid r_i^2 = 1 \ (i=1,\dots,5), \ (r_i r_{i+1})^3 = 1 \ (i=1,\dots,4), \ (r_i r_j)^2 = 1 \ (|i-j|>1) \rangle,

where the relations encode the orders of products of the generators as determined by the diagram.^[29] More generally, the symmetry group of the regular n-simplex is the Coxeter group A_n, whose diagram is a linear chain of n nodes connected by single edges.^[29]

Full Symmetry Group

The full symmetry group of the regular 5-simplex is the finite Coxeter group of type A_5, which acts as the group of all isometries preserving the polytope, including reflections. This group is isomorphic to the symmetric group S_6, the group of all permutations of its 6 vertices, and has order $6! = 720.^[18] The Coxeter presentation is generated by 5 simple reflections corresponding to the nodes of its Coxeter-Dynkin diagram, with relations dictated by the diagram's edges labeled 3 for adjacent nodes.^[18] The rotational (orientation-preserving) subgroup is the even permutations within S_6, namely the alternating group A_6, which has index 2 in the full group and thus order 360.^[30] This subgroup consists solely of proper rotations and excludes reflections, preserving the handedness of the 5-simplex. The full group has 720 chambers in its Coxeter complex. Wythoff constructions utilize the A_5 Coxeter-Dynkin diagram to enumerate uniform 5-polytopes in the simplex symmetry family, where marking nodes with circles selects generators for vertex figures and facets. The regular 5-simplex arises from the unmarked diagram (all nodes as crosses), yielding the maximal symmetry case among the 6 uniform figures in this family.^[31]

Isogonal Configuration

The 5-simplex exhibits an isogonal configuration as a vertex-transitive uniform 5-polytope, characterized by its symmetry group acting transitively on all flags. This transitivity ensures that any flag—a maximal chain of nested faces from vertex to the full polytope—can be mapped to any other via a symmetry, underscoring the highest degree of symmetry among simplices. The total number of flags is 720, equal to 6!, reflecting the combinatorial structure where each flag corresponds to a permutation of the six vertices.^[32]^[33] The vertex figure of the 5-simplex, obtained by intersecting with a hyperplane near a vertex, is a regular 4-simplex (pentachoron), preserving the regular tetrahedral cells adjacent to that vertex. Similarly, the edge figure is a regular 3-simplex (tetrahedron), the face figure (for a triangular 2-face) is a regular 2-simplex (equilateral triangle), and the cell figure (for a tetrahedral 3-cell) is a regular 1-simplex (line segment). These successive figures diminish dimensionally while maintaining regularity, embodying the self-dual simplicial uniformity.^[18] In enumerations of uniform 5-polytopes, the 5-simplex is the foundational regular member in systematic listings that extend Conway's polytope notation to higher dimensions. Its abstract configuration encapsulates the incidence relations among elements, with the f-vector (6 vertices, 15 edges, 20 triangular faces, 15 tetrahedral cells, 6 pentachoral facets, 1 overall 5-cell) denoting the transitive orbits under the symmetry action.^[34]

Variants and Visualizations

Lower Symmetry Forms

The 5-orthoscheme represents a lower symmetry variant of the 5-simplex, characterized by a chain of five mutually orthogonal edges emanating from a single vertex, with the remaining faces being lower-dimensional orthoschemes. This configuration inherently reduces the symmetry group compared to the full S_6 symmetry of the regular 5-simplex, as the orthogonal edge structure limits isometries to those preserving the right-angle chain, typically resulting in a discrete group of reflections and rotations along the coordinate axes. Such orthoschemes serve as fundamental domains for tiling spaces with regular simplices and are used in volume calculations and dissections in hyperbolic geometry.^[35] Truncated and bitruncated forms of the 5-simplex, while uniform in the regular case, become non-uniform when applied to irregular simplices due to the constraints of equal edge lengths in the original simplex, leading to facets with varying sizes and reduced symmetry groups that do not preserve all vertex transitivity. These variants are explored in dissections to analyze geometric properties under lower symmetry. Dissections of the 5-simplex into orthoschemes provide a method for such analysis.^[35]

Projected Representations

The 5-simplex, a five-dimensional regular polytope with six vertices, can be visualized through orthogonal projections onto lower-dimensional spaces, preserving parallelism but distorting distances and angles. An orthogonal projection to three dimensions typically renders the structure as a complex wireframe of its 15 edges, often appearing as a star-like figure formed by the complete graph K_6 of interconnected vertices, with tetrahedral cells projected as distorted polyhedra. Further orthogonal projection to two dimensions yields symmetrical patterns, such as arrangements within a unit disk that highlight the polytope's combinatorial skeleton without metric fidelity.^[36] Perspective and stereographic projections from a four-dimensional viewpoint provide more intuitive three-dimensional renderings by mapping points via inversion on a hypersphere, enabling the depiction of internal structures. These methods often show nested configurations of the 15 tetrahedral cells, with outer layers representing boundary facets and inner elements revealing connectivity, though occlusions limit full visibility of all cells simultaneously. Such projections emphasize the self-dual symmetry of the 5-simplex, using color-coding or transparency to differentiate cells.^[36] The four-dimensional analog of a Schlegel diagram projects the entire 5-simplex centrally onto one of its six pentachoral (4-simplex) facets, creating a configuration where one pentachoron encloses another, linked by edges and faces that subdivide the interior space without intersections. This representation, realized in four-dimensional Euclidean space, captures the full two-skeleton including all vertices, edges, and triangular faces, serving as a combinatorial blueprint for further three-dimensional visualization.^[37] Common images of the 5-simplex employ wireframe models to display the 15 edges as a projected complete graph K_6, facilitating study of its symmetry group. Ray-traced renders incorporate opacity and shading to simulate depth in three-dimensional views, distinguishing opaque outer tetrahedral cells from semi-transparent internal ones and enhancing perceptual understanding of the polytope's volume-filling properties in five dimensions.^[36]

Simplex Compounds

In five-dimensional geometry, the principal uniform compound involving 5-simplices is the dual compound of two enantiomorphic 5-simplices, known as the stade, where the components interpenetrate while sharing a common center. This flag-transitive structure generalizes lower-dimensional analogs like the stella octangula and features a density of 2 due to the mutual intersection of the simplices. The symmetry group of this compound is the full A_5 \times \mathbb{Z}_2, reflecting the rotational symmetries of the icosahedral group augmented by an inversion. It consists of 12 vertices in total, with each 5-simplex contributing 6 distinct vertices positioned in dual configuration.^[38]^[17] Star compounds incorporating the regular 5-simplex \{3,3,3,3\} are constrained in five dimensions, with potential combinations such as those involving gyrochorons (uniform 4-polytopes like \{3,3,4\}) limited by the lack of non-degenerate star polytope constructions beyond lower-dimensional cases. These structures do not yield additional uniform compounds in 5D, as higher-dimensional stellations of simplices do not produce the necessary intersecting facets for density greater than 1 without degeneracy.^[38] The enumeration of such simplex compounds in five dimensions builds on extensions of uniform polychora studies, as conceptualized in John Skilling's framework for uniform polyhedra, adapted to higher dimensions.^[39] The 5-simplex, denoted by the Schläfli symbol {3,3,3,3}, is self-dual, with its polar reciprocal being combinatorially equivalent to itself.^[9] Among the three regular uniform 5-polytopes, it features 6 vertices and 6 4-simplex facets, in contrast to the 5-cube {4,3,3,3} with 32 vertices and 10 4-cube facets, and the 5-orthoplex {3,3,3,4}—the dual of the 5-cube—with 10 vertices and 32 4-simplex facets.^[40] The truncated 5-simplex, with Schläfli symbol t{3,3,3,3}, arises from vertex truncation of the 5-simplex under its A_5 symmetry, yielding a uniform 5-polytope with 30 vertices, 75 edges, and a mix of 4-simplex and truncated 4-simplex facets. This construction preserves the original symmetry while altering the facets to include both regular 4-simplices from the truncated vertices and truncated 4-simplices from the original facets.^[41] The A_5 Coxeter group, which realizes the full symmetry of the 5-simplex, supports additional uniform 5-polytopes via Wythoff constructions, such as the rectified 5-simplex (r{3,3,3,3}) and bitruncated 5-simplex (t_2{3,3,3,3}), each inheriting the simplex's vertex-transitive properties but with expanded vertex figures. The facets of the 5-simplex are themselves 4-simplices, uniform 4-polytopes with Schläfli symbol {3,3,3}, thereby connecting the 5-simplex's structure to the hierarchy of lower-dimensional uniform simplices.

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### Symmetry Groups of Regular Polytopes (n-Simplex)
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A regular polytope is a polytope whose symmetry group acts transitively on its flags, giving its symmetry group the highest possible order among polytopes ...
[37]
Uniform Polytera and Other Five Dimensional Shapes - polytope.net
Welcome to the all new polyteron web site. This web site will focus on the uniform polytopes of five dimensions. There are currently 1348 known uniform polytera ...
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https://bendwavy.org/klitzing/explain/compound.htm
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Stereographic Visualization of 5-Dimensional Regular Polytopes
The five-simplex { 3 , 3 , 3 , 3 } is self-dual and corresponds to the reflection symmetry group [ 3 , 3 , 3 , 3 ] , which is usually denoted as B 5 .
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[PDF] Old and new geometric polyhedra with few vertices - arXiv
R ), respectively the 5-simplex (a regular convex 5-polytope in. 5. R ). This ... Thus, the six vertices together form a Schlegel diagram of the 5-simplex in. 4.
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Compound
A compound can be defined to be regular, just like a polytope, when it is transitive on all sub-dimensional elements.
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Uniform compounds of uniform polyhedra
The work of Coxeter, Longuet-Higgins and Miller (1953) and of Skilling. (1975) is extended to give a complete list of uniform compounds of uniform polyhedra.