5-cell
In geometry, the 5-cell, also known as the pentachoron or four-simplex, is the simplest regular convex four-dimensional polytope, consisting of five vertices connected by ten edges and bounded by five regular tetrahedral cells.[1] It serves as the four-dimensional analog of the regular tetrahedron in three dimensions and the equilateral triangle in two dimensions, formed as the convex hull of five mutually equidistant points in four-dimensional Euclidean space.[2] With the Schläfli symbol {3,3,3}, it exemplifies a regular simplex and belongs to the family of Platonic solids extended to higher dimensions.[1] The 5-cell possesses a self-dual symmetry, meaning its dual polytope is congruent to itself, and its skeleton forms the complete graph K_5 with 60 rotational symmetries (alternating group A_5) and a full symmetry group of order 120 arising from the Coxeter group A_4.[1][3][4] It is one of six regular convex polychora in four dimensions, distinguished by its uniform edge lengths and the fact that three tetrahedral cells meet at each edge, preventing a full tessellation of four-dimensional space unlike in lower dimensions.[2] Projections of the 5-cell into three-dimensional space typically reveal configurations such as four outer tetrahedra surrounding an inner one or symmetric arrangements with displaced vertices to equalize edges, aiding visualization of its higher-dimensional structure.[2] These properties make the 5-cell a foundational object in the study of polytopes, Coxeter groups, and multidimensional geometry.[3]Definition and Elements
Vertices, edges, faces, and cells
The regular 5-cell, or pentachoron, possesses a simple combinatorial structure as the 4-dimensional simplex. It consists of 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells, with its 1-skeleton forming the complete graph K_5.[1] Each cell is a regular tetrahedron, and the Schläfli symbol {3,3,3} denotes this uniform composition of triangular elements at each level.[1] The cells meet along shared elements according to the incidence structure of the simplex: 4 cells meet at each vertex, 3 cells meet at each edge (also called a ridge in 4D terminology), and 2 cells meet at each triangular face.[5] These incidences can be derived combinatorially; for instance, given 5 vertices labeled 1 through 5, the number of tetrahedral cells containing a fixed edge (say between vertices 1 and 2) is the number of ways to choose the remaining two vertices from the other three, yielding \binom{3}{2} = 3. Similarly, for a fixed triangular face (vertices 1, 2, 3), there are \binom{2}{1} = 2 ways to choose the fourth vertex for a cell, and for a vertex (say 1), \binom{4}{3} = 4 cells include it by selecting three others.[6] The boundary of the 5-cell, formed by gluing the 5 tetrahedral cells along their 10 faces (with each boundary face belonging to exactly one cell), yields a triangulation of the 3-sphere S^3.[7] This is topologically equivalent to the boundary of the 4-dimensional ball. The Euler characteristic \chi = V - E + F - C = 5 - 10 + 10 - 5 = 0 verifies this, as the boundary of any convex 4-polytope is homeomorphic to S^3, which has \chi = 0.[8] To compute \chi, sum over the alternating number of k-faces: \sum_{k=0}^{3} (-1)^k f_k = f_0 - f_1 + f_2 - f_3, where f_k = \binom{5}{k+1} gives the face counts, resulting in 0 for the closed 3-manifold boundary.[6]Schläfli symbol and Wythoff construction
The 5-cell possesses the Schläfli symbol \{3,3,3\}, which specifies that its bounding cells are regular tetrahedra \{3,3\} with three such cells meeting at each triangular face and three meeting at each edge.[1][9] This notation encapsulates the regularity and combinatorial structure of the polytope, where the successive entries describe the density of elements at lower-dimensional facets: triangular faces \{3\} with three meeting at each vertex within a cell, extended to the 4-dimensional arrangement.[10] As a consequence of this symbol, the 5-cell comprises five tetrahedral cells in total.[10] The 5-cell admits a representation via the Wythoff symbol $3 \mid 3\, 3\, 3, which corresponds to the linear Coxeter-Dynkin diagram consisting of four nodes interconnected by single bonds, with the vertical bar denoting the initial active mirror in the reflection sequence.[11] This symbol arises from the Wythoff construction within the framework of the A_4 Coxeter group, where reflections across the hyperplanes defined by the diagram generate the symmetry.[9] In this construction, the 5-cell emerges as the convex hull of the orbit of a generic point under the action of the A_4 reflection group in 4-dimensional Euclidean space, ensuring all vertices lie at equal distance from the origin and yield the regular tetrahedral arrangement.[9] The 5-cell thereby stands as the regular 4-simplex, the simplest convex regular polytope in four dimensions and the fourth member of the simplex family after the point, segment, triangle, and tetrahedron.[1]Geometric Properties
Measures and metrics
The regular 5-cell, or pentachoron, possesses several key geometric measures in its Euclidean realization with unit edge length a = 1. These include distances from the center to various structural elements, as well as volumetric and angular properties derived from its symmetry as a regular 4-simplex. These metrics are fundamental to understanding its size and shape, and they scale with powers of the edge length in general cases. The circumradius R, the distance from the center to a vertex, is \sqrt{\frac{2}{5}} \approx 0.632. The midradius (or edge radius), the distance from the center to the midpoint of an edge, is \sqrt{\frac{3}{20}} \approx 0.387. The face radius, the distance from the center to the centroid of a triangular face, is \frac{1}{\sqrt{15}} \approx 0.258. The inradius r, the distance from the center to a bounding hyperplane (cell facet), is \frac{1}{\sqrt{40}} = \frac{\sqrt{10}}{20} \approx 0.158. These radii follow from the inner product structure of the vertex vectors in the regular simplex, where the Gram matrix has diagonal entries R^2 and off-diagonal entries -\frac{R^2}{4}, leading to the edge length relation a^2 = 2R^2 \left(1 + \frac{1}{4}\right) = \frac{5}{4} R^2, solved for R and extended analogously for other elements.[Coxeter (1973)] The hypervolume (4-dimensional content) of the regular 5-cell is V = \frac{\sqrt{5}}{96} a^4 \approx 0.0233 a^4. This formula arises from the recursive construction of the simplex volume, V_n = \frac{1}{n} V_{n-1} h_n, where V_3 = \frac{\sqrt{2}}{12} a^3 is the tetrahedral cell volume and h_4 = \sqrt{\frac{5}{8}} a \approx 0.791 a is the height to a bounding cell, yielding V_4 = \frac{1}{4} \cdot \frac{\sqrt{2}}{12} a^3 \cdot \sqrt{\frac{5}{8}} a = \frac{\sqrt{5}}{96} a^4. The 3-dimensional surface content, comprising the total volume of the five bounding tetrahedral cells, is $5 \cdot \frac{\sqrt{2}}{12} a^3 = \frac{5 \sqrt{2}}{12} a^3 \approx 0.590 a^3. These volumetric measures highlight the compact scaling of the 5-cell relative to its edge length, consistent with the general pattern for regular simplices.[Coxeter (1973)] The dihedral angle between two adjacent cells is \arccos\left(\frac{1}{4}\right) \approx 75.52^\circ. To derive this from normal vectors, consider the center O of the 5-cell and two adjacent tetrahedral cells sharing a triangular face. The outward unit normal \mathbf{n}_1 to the first cell is the direction from O to the centroid C_1 of that cell, normalized: \mathbf{n}_1 = \frac{\overrightarrow{OC_1}}{|\overrightarrow{OC_1}|}. Similarly for \mathbf{n}_2 to the second cell. The centroids C_1 and C_2 are averages of their respective four vertices each. Due to the symmetry, the angle \phi between \mathbf{n}_1 and \mathbf{n}_2 satisfies \cos \phi = -\frac{1}{4}, as the inner product \langle \mathbf{n}_1, \mathbf{n}_2 \rangle = -\frac{1}{4} follows from the simplex's vertex Gram matrix projected onto the facet centroids (the shared face contributes symmetrically, while the differing vertices adjust the cosine to -\frac{1}{4}). The dihedral angle \theta, the internal angle between the cells, is then \pi - \phi, so \cos \theta = -\cos \phi = \frac{1}{4}, yielding \theta = \arccos\left(\frac{1}{4}\right).[Coxeter (1973)]Dihedral angles and edge lengths
In the regular 5-cell, all edges are of equal length, ensuring uniformity across its structure. The two-dimensional faces are equilateral triangles, each with interior angles of 60°. The dihedral angle, which is the angle between two adjacent three-dimensional cells (regular tetrahedra), is \arccos\left(\frac{1}{4}\right), approximately 75.52°. This value arises from a computation using vector inner products in the centered simplex: place the vertices such that their position vectors sum to zero and have equal norms, then the normal to a facet (spanned by n-1 vertices) is the negative of the omitted vertex's position vector; the cosine of the dihedral angle is then the negative of the cosine of the angle between these normals, yielding \cos \theta = -\frac{\mathbf{v}_i \cdot \mathbf{v}_j}{|\mathbf{v}_i| |\mathbf{v}_j|} = \frac{1}{n} for the regular n-simplex with n=4.[12] Within each tetrahedral cell of the regular 5-cell, the dihedral angles are those of the regular tetrahedron, \arccos\left(\frac{1}{3}\right) \approx 70.53°, distinct from the larger polychoral dihedral angle of the 5-cell itself.[12] In irregular 5-cells, edge lengths vary, resulting in non-equilateral triangular faces whose angles are determined by the planar cosine rule: for a triangle with sides a, b, c, the angle \gamma opposite c satisfies \cos \gamma = \frac{a^2 + b^2 - c^2}{2ab}. Dihedral angles consequently differ across facets, bounded such that the minimum \beta and maximum \alpha satisfy \beta \leq \arccos\left(\frac{1}{4}\right) \leq \alpha, with equality if and only if the simplex is regular; for convexity, all dihedral angles must be less than $180^\circ, ensuring the vertices form a convex hull without interior points.[13]Coordinates and Constructions
Cartesian coordinates
The regular 5-cell, or 4-simplex, can be embedded in 4-dimensional Euclidean space \mathbb{R}^4 using explicit Cartesian coordinates for its five vertices. A standard set of coordinates, up to rotation and translation, is given by the points \begin{align*} v_1 &= (1, 1, 1, 1), \\ v_2 &= (1, -1, -1, -1), \\ v_3 &= (-1, 1, -1, -1), \\ v_4 &= (-1, -1, 1, -1), \\ v_5 &= (-1, -1, -1, 1). \end{align*} These coordinates position the vertices such that the Euclidean distance between any pair of distinct vertices is $2\sqrt{3}. To achieve unit edge length, scale each coordinate by the factor s = 1/(2\sqrt{3}). The scaled vertices are then s \cdot v_i for i=1,\dots,5. To derive this scaling, compute the squared distance between, say, v_1 and v_2: \|v_1 - v_2\|^2 = \|(0, 2, 2, 2)\|^2 = 0 + 4 + 4 + 4 = 12, so \|v_1 - v_2\| = \sqrt{12} = 2\sqrt{3}. By symmetry of the construction, all pairwise distances are identical. For unit edge length, require s \cdot 2\sqrt{3} = 1, yielding s = 1/(2\sqrt{3}). The inner products between distinct unscaled vertices confirm the regularity: after centering at the origin (by subtracting the centroid (-1/5, -1/5, -1/5, -1/5)), the Gram matrix has diagonal entries $16/5 and off-diagonal entries -4/5, consistent with the geometry of a regular simplex. An alternative construction places the vertices in a 4-dimensional hyperplane embedded within \mathbb{R}^5. The five vertices are the standard basis vectors e_1 = (1,0,0,0,0), e_2 = (0,1,0,0,0), e_3 = (0,0,1,0,0), e_4 = (0,0,0,1,0), and e_5 = (0,0,0,0,1). The pairwise distance between distinct e_i and e_j is \sqrt{2}. For unit edge length, scale by t = 1/\sqrt{2}, giving vertices t \cdot e_i. These points lie in the affine hyperplane \sum_{k=1}^5 x_k = 1 (or t after scaling). To obtain explicit coordinates in \mathbb{R}^4, project orthogonally onto the subspace orthogonal to the all-ones vector (1,1,1,1,1)/\sqrt{5}, using any orthonormal basis for that 4-dimensional subspace; the resulting embedding is isometric to the scaled alternating-sign coordinates above, up to orthogonal transformation. This hyperplane embedding generalizes the coordinate representation of regular simplices and facilitates computations in higher dimensions. The alternating-sign coordinates in \mathbb{R}^4 and the basis-vector embedding in the hyperplane of \mathbb{R}^5 both arise from solving the system of equations for five points with equal mutual distances, ensuring the convex hull is regular. Specifically, the vertices u_1, \dots, u_5 \in \mathbb{R}^4 satisfy \|u_i - u_j\| = 1 for i \neq j, with the points affinely independent. Centering at the origin imposes \sum u_i = [0](/page/0) and equal norms r^2 = \|u_i\|^2, leading to inner products \langle u_i, u_j \rangle = -r^2/4 for i \neq j. Substituting into the distance equation gives $2r^2 - 2(-r^2/4) = 1, so (5/2)r^2 = 1 and r^2 = 2/5, confirming the geometry. These representations enable numerical simulations and projections of the 5-cell.Boerdijk–Coxeter helix and nets
The Boerdijk–Coxeter helix provides an alternative construction for the 5-cell by stacking regular tetrahedra in a linear chain, where each consecutive tetrahedron shares a face with the previous one and is rotated by the tetrahedral dihedral angle of \arccos\left(\frac{1}{3}\right) \approx 70.53^\circ.[14] This helical arrangement ensures that the vertices align in a twisted path, and the 5-cell emerges as the finite segment comprising exactly five such tetrahedra, connected in a closed 4-dimensional ring without intersections.[15] Nets for the 5-cell are 3D unfoldings of its five tetrahedral cells arranged in Euclidean space without overlap or self-intersection, allowing visualization and verification of the polytope's topology prior to 4D reassembly. There are three distinct net topologies for the 5-cell.[16] A representative example is the triangular bipyramid net, in which a central tetrahedron is surrounded by three adjacent tetrahedra forming a belt, with the fifth tetrahedron capping one end to maintain balanced connectivity.[16] These nets facilitate conceptual understanding of the 5-cell's face-sharing structure, confirming that each tetrahedron bonds to the other four along their triangular faces.[15] The development of nets for polychora, including the 5-cell, originated with H. S. M. Coxeter's work in 1928, aimed at enhancing visualization of higher-dimensional regular polytopes.[15]Symmetry and Isometries
Rotation and reflection groups
The symmetry group of the regular 5-cell, also known as the 4-simplex or pentachoron, is the Coxeter group of type A_4, denoted in Coxeter notation as [3,3,3]. This group has order 120 and is isomorphic to the symmetric group S_5, which acts by permuting the five vertices of the 5-cell. The Coxeter diagram for [3,3,3] consists of four nodes connected in a linear chain by single edges, each labeled with the number 3, representing the relations among the generating reflections.[17] The full symmetry group includes both rotations and reflections, generated by four reflections corresponding to the hyperplanes bisecting the edges of the 5-cell. These reflections satisfy the Coxeter relations derived from the diagram, ensuring the group's finite order and transitive action on the flags of the polytope.[17] The orientation-preserving subgroup, or rotation group, is the even subgroup of index 2, isomorphic to the alternating group A_5 with order 60. This subgroup consists of all even permutations of the vertices and excludes improper isometries like reflections.[17] The rotational symmetries are generated by specific rotations that preserve the combinatorial structure: 120° and 240° rotations around axes passing through a vertex and the centroid of the opposite tetrahedral cell, corresponding to 3-cycles in A_5, and 180° rotations around axes through the midpoints of two non-adjacent edges, corresponding to double transpositions. These generators, along with order-5 rotations from 5-cycles in A_5, fully account for the 60 elements, stabilizing the set of 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells under the group's action.[17]Geodesics and rotational symmetries
In the 5-cell, geodesics on the 1-skeleton correspond to the shortest paths along the edges of its complete graph K_5 structure, where each pair of vertices is directly connected by an edge of equal length, making all such paths trivial single-edge segments. On the hypersurface, which forms a 3-sphere topologically, geodesics are the locally shortest curves on the boundary manifold composed of five regular tetrahedral cells; these paths unfold across multiple faces and can be computed using the intrinsic metric of the polytope's surface. A notable feature is the presence of great 2-spheres within the circumscribed 3-sphere that link pairs of opposite edges—edges that share no common vertices—with the 2-sphere arising as the intersection of the circumhypersphere with the 2-plane equidistant from the midpoints of such a pair, providing a symmetric "equator" that bisects the polytope orthogonally. The rotational symmetries of the 5-cell form the alternating group A_5 of order 60, consisting of even permutations of its five vertices, and include specific types of rotations that preserve the polytope's regularity. There are 15 axes of 180° rotations, each passing through the midpoints of a pair of opposite edges; such a rotation swaps the two edges while inverting the positions of the remaining three vertices in a coordinated manner, effectively acting as a double 180° rotation in two orthogonal planes perpendicular to the axis. Additionally, there are rotations of 120° and 240° around axes connecting a vertex to the centroid of its opposite tetrahedral cell, cycling the three adjacent cells around that vertex; with five vertices, this yields 20 elements of order 3 in the group (10 axes, two non-trivial rotations each). The group also includes order-5 rotations (72°, 144°, 216°, 288°) around 6 axes, corresponding to the 24 elements of order 5 from 5-cycles. To illustrate a 120° vertex rotation explicitly, consider the 5-cell with vertices centered at the origin in ℝ⁴ using the coordinates scaled to have edge length \sqrt{3/2}:- v₁ = (1, 1, 1, 1)/√8
- v₂ = (1, -1, -1, -1)/√8
- v₃ = (-1, 1, -1, -1)/√8
- v₄ = (-1, -1, 1, -1)/√8
- v₅ = (-1, -1, -1, 1)/√8