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Hexahedron

A hexahedron is a polyhedron with six faces.^[1] The most common and regular form of a hexahedron is the cube, in which all six faces are congruent squares meeting at right angles.^[1]^[2] Hexahedra can be convex or concave, though convex hexahedra are more commonly studied in geometry.^[3] There are exactly seven topologically distinct convex hexahedra: the triangular dipyramid, pentagonal pyramid, tetragonal antiwedge, hemiobelisk, hemicube, pentagonal wedge, and cube.^[1] These vary in the number of vertices and edges while satisfying Euler's formula for polyhedra (V - E + F = 2, where F = 6), resulting in vertex counts from 5 to 8 and edge counts from 9 to 12.^[1] For instance, the cube has 8 vertices and 12 edges, all of degree 3 (three edges meeting at each vertex).^[1]^[4] In addition to the regular cube, other notable hexahedra include rhombohedra, where opposite faces are congruent rhombi, and parallelepipeds, which generalize rectangular prisms with parallelogram faces.^[1]^[4] Hexahedra appear in various applications, such as finite element analysis in engineering, where they serve as 3D elements composed of tetrahedra for modeling complex structures.^[5] Their dual graphs correspond to hexahedral graphs, aiding in combinatorial geometry studies.^[1]

Definition

Formal Definition

A hexahedron is a three-dimensional polyhedron consisting of exactly six faces, each a planar polygon, with these faces meeting along edges to form a closed surface.^[1] A polyhedron, in turn, is defined as a solid figure bounded by a finite number of flat polygonal faces, straight edges, and vertices, distinguishing it from other geometric objects like spheres or curved surfaces.^[6] This face count sets the hexahedron apart from related polyhedra, such as the pentahedron with five faces or the heptahedron with seven faces.^[7]^[8] Hexahedra are classified into convex and non-convex types based on their geometric structure. A convex hexahedron is one where every line segment connecting two points in its interior lies entirely within the hexahedron, and all dihedral angles are less than 180 degrees, ensuring no indentations or reflex angles.^[9]^[10] In contrast, a non-convex hexahedron features at least one reflex interior angle exceeding 180 degrees or structural indentations that may cause some interior line segments to exit the figure temporarily.^[10] For simple convex hexahedra, Euler's formula applies, stating that the number of vertices V, edges E, and faces F satisfy the relation V - E + F = 2.^[11] For example, the cube has 8 vertices, 12 edges, and 6 faces.^[12] It serves as the archetypal example of a regular convex hexahedron.^[1]

Etymology and Terminology

The term "hexahedron" originates from the Ancient Greek words ἕξ (héx), meaning "six," and ἕδρα (hédra), meaning "seat," "base," or "face," literally denoting a figure with six bases or faces.^[13]^[14] This nomenclature reflects the polyhedron's defining characteristic of possessing six polygonal faces, a concept rooted in classical geometry. Historical references to hexahedra appear indirectly in Euclid's Elements (circa 300 BCE), where the cube—a specific regular hexahedron—is discussed extensively in Book XI as a solid with six square faces, though Euclid primarily uses the term κύβος (kubos) for the cube without generalizing to irregular forms.^[15] The broader term "hexahedron" (ἑξάεδρον, hexaedron) was employed by Heron of Alexandria in the 1st century CE to specifically denote the cube, distinguishing it from more general rectangular solids he termed "cubes" in a looser sense.^[15] During the Renaissance revival of geometry, the term gained prominence in European mathematics; Johannes Kepler referenced the hexahedron (as the cube among Platonic solids) in his Mysterium Cosmographicum (1596), integrating it into cosmological models while treating it as the regular case distinct from general hexahedra.^[16] The word entered English around 1571, formalizing its use in mathematical discourse to encompass any six-faced polyhedron beyond the regular cube.^[17] In modern terminology, "hexahedron" distinguishes the general polyhedron from the "cube," which refers exclusively to the regular hexahedron with congruent square faces. Related terms include "cuboid," denoting a hexahedron with six rectangular faces (not necessarily square), and "parallelepiped," describing a hexahedron bounded by six parallelograms.^[1] These distinctions highlight geometric variations, while topologically, all hexahedra are equivalent as genus-zero surfaces with six faces, regardless of their embedding in space.^[1]

Properties

Combinatorial Properties

A hexahedron is a polyhedron with six faces, and for simple convex examples like the cube, it possesses 8 vertices, 12 edges, and 6 faces.^[1] Convex hexahedra have between 5 and 8 vertices and 9 to 12 edges. These counts satisfy Euler's characteristic formula for polyhedra of spherical topology, \chi = V - E + F = 8 - 12 + 6 = 2.^[18] There are seven distinct combinatorial types of convex hexahedra, classified by their face adjacency graphs (or 1-skeleton duals), with one type comprising a chiral pair of enantiomorphs. These types arise from valid combinations of triangular, quadrilateral, and pentagonal faces, where no more than two pentagons are allowed to ensure convexity and topological validity. In convex hexahedra, every vertex has degree at least 3, as required for simple polyhedra without degenerate features. The most common type, with 8 vertices, features uniform 3-valent vertices (three edges meeting at each), though irregular variants may include vertices of degree 4 or higher.^[1] The dual graph of any hexahedron is a hexahedral graph consisting of 6 vertices (one per face) connected by edges representing face adjacencies.^[19] Non-convex hexahedra exhibit infinite topological diversity while maintaining genus 0 (spherical topology and Euler characteristic 2), arising from indentations that alter face connectivities without introducing holes.

Geometric Properties

The surface area S of a hexahedron is the sum of the areas of its six faces, given by S = \sum_{i=1}^{6} A_i, where A_i is the area of the i-th face.^[20] The volume V of a general hexahedron can be computed by decomposing it into tetrahedra and summing their volumes using determinant formulas, such as V = \frac{1}{6} \sum \det(\mathbf{v}_i - \mathbf{v}_0, \mathbf{v}_j - \mathbf{v}_0, \mathbf{v}_k - \mathbf{v}_0) for appropriate vertex vectors in the decomposition.^[21] Alternatively, the divergence theorem provides V = \frac{1}{3} \iint_{\partial P} \mathbf{r} \cdot \mathbf{n} \, dA, where P is the hexahedron, \partial P its boundary, \mathbf{r} the position vector, and \mathbf{n} the outward normal. For an orthogonal hexahedron (cuboid) with dimensions length l, width w, and height h, the volume simplifies to V = l \times w \times h.^[22] For a parallelepiped defined by edge vectors \mathbf{a}, \mathbf{b}, and \mathbf{c}, the volume is V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|, the absolute value of the scalar triple product.^[23] In convex hexahedra, dihedral angles range from greater than 0° to less than 180°, ensuring the faces fold inward without intersection.^[24] Regular forms, such as the cube, have dihedral angles of exactly 90°.^[25] For a hexahedron of uniform density, the center of mass is the centroid of its volume. For symmetric cases like the cube, this is located at the average of its vertex coordinates.^[26] The moment of inertia tensor for symmetric cases, such as the cuboid aligned with coordinate axes, has diagonal elements I_{xx} = \int (y^2 + z^2) \, dm, and similarly for others, computable via integration over the volume.^[27] Isoperimetric inequalities for hexahedra bound the volume relative to surface area; among convex hexahedra with fixed surface area, the regular hexahedron (cube) maximizes the volume, as established by results showing optimality for six faces.^[28] A general bound follows from polyhedral isoperimetric inequalities, such as $36\pi V^2 \leq S^3, adapted from spherical cases but with equality approached by regular forms.^[20]

Convex Hexahedra

Regular Hexahedron

The regular hexahedron, commonly known as the cube, is the only convex polyhedron with six faces where all faces are congruent regular polygons, specifically squares, all edges are of equal length, and all vertices are identical in that exactly three faces meet at each vertex.^[25] This configuration satisfies the strict criteria for regularity, distinguishing it from other hexahedra. As one of the five Platonic solids, the cube represents the highest degree of symmetry among polyhedra with square faces, and its dual is the regular octahedron, where vertices of the cube correspond to faces of the octahedron and vice versa.^[25]^[29] In standard positioning, the vertices of a regular hexahedron with edge length a are located at coordinates \left(\pm \frac{a}{2}, \pm \frac{a}{2}, \pm \frac{a}{2}\right), providing a centered alignment along the coordinate axes.^[25] Key geometric formulas for the cube include a surface area of $6a^2, a volume of a^3, a face diagonal of a\sqrt{2}, and a space diagonal of a\sqrt{3}; additionally, the dihedral angle between adjacent faces is exactly $90^\circ or \pi/2 radians.^[25] The symmetry group of the regular hexahedron is the full octahedral group O_h, which encompasses 48 elements including 24 proper rotations and 24 improper isometries such as reflections and inversion.^[30] This group acts transitively on the flags of the cube, ensuring maximal uniformity. The cube possesses 9 planes of symmetry: three passing through opposite faces and six bisecting opposite edges.^[31]

Irregular Convex Hexahedra

In addition to the regular hexahedron, there are six other topologically distinct convex hexahedra, each defined by unique adjacency patterns of vertices, edges, and faces, and all realizable with convex polygonal faces that meet properly along edges.^[32]^[1] These forms differ from the cube in their combinatorial structure and allow for greater geometric flexibility, including mixed face shapes such as triangles, quadrilaterals, and pentagons. One of these, the tetragonal antiwedge, is chiral and exists as a pair of enantiomorphic realizations that are non-superimposable mirror images.^[32]^[33] The triangular dipyramid consists of six triangular faces and can be constructed by joining two tetrahedra along a common face, resulting in five vertices and nine edges.^[32] The pentagonal pyramid features one pentagonal face and five triangular faces, with six vertices and ten edges, formed as a pyramid over a pentagonal base.^[1] The tetragonal antiwedge has four triangular faces and two quadrilateral faces, also with six vertices and ten edges, resembling a twisted quadrilateral prism.^[32] The hemiobelisk includes three triangular faces, two quadrilateral faces, and one pentagonal face, possessing seven vertices and eleven edges, and can be visualized as half of an elongated square pyramid cut by a plane.^[1] The hemicube comprises two triangular faces and four quadrilateral faces, with seven vertices and eleven edges, akin to a cube divided by a plane through two opposite edges.^[32] Finally, the pentagonal wedge has two triangular faces, two quadrilateral faces, and two pentagonal faces, featuring eight vertices and twelve edges, obtainable by truncating two corners of a tetrahedron.^[1] Realizations of these topologies often exhibit lower symmetry than the cube and variable dihedral angles to maintain convexity. For instance, the all-quadrilateral topology (shared with the cube) includes the parallelepiped, formed by six parallelogram faces defined by three vectors from one vertex, where dihedral angles depend on the angles between those vectors and can deviate from 90° while keeping all faces convex.^[34] A specific example is the rectangular cuboid with six rectangular faces and all dihedral angles at 90°, possessing D_{2h} point group symmetry.^[35]^[36] Another realization is the trigonal trapezohedron, with six congruent rhombic faces, where dihedral angles vary based on the rhombus geometry, typically between acute and obtuse values to ensure convexity.^[37] For the mixed-face topologies, dihedral angles similarly range, often from approximately 60° to 120° in symmetric realizations, reflecting reduced symmetry groups such as C_{2v} for the hemicube.^[32] All such hexahedra can be embedded in Euclidean space with strictly convex positions for their vertices.^[1]

Non-Convex Hexahedra

Concave Hexahedra

A concave hexahedron is a non-self-intersecting, non-convex polyhedron with exactly six faces, distinguished by the presence of at least one reflex interior dihedral angle exceeding 180 degrees. This reflex angle arises at the edges adjacent to an indented region, allowing line segments connecting certain points on the surface to lie outside the polyhedron. Unlike convex hexahedra, the convex hull of a concave hexahedron fully encloses it, resulting in a reduced volume for the concave form relative to its hull. Concave hexahedra can be constructed from convex hexahedra through excavation or displacement methods that introduce indentations without altering the topological structure. For instance, starting from a cube, a pyramidal indentation can be formed on one face by displacing a vertex inward, creating a "dart" or reentrant vertex that induces the concavity. This process maintains planar faces while producing a reflex dihedral angle at the indentation's edges, as illustrated in finite element mesh transformations where displacing a single vertex turns a valid convex hexahedron into a concave one.^[5] A representative example is the dented cube, formed by indenting a smaller pyramidal volume into one face of a cube, yielding a structure with reflex angles but preserving six faces. Another example arises in mesh generation contexts, where vertex displacement creates local concavities for modeling complex geometries, ensuring the hexahedron remains topologically equivalent to a cube but geometrically non-convex. Topologically, there are ten distinct hexahedra: seven that can be realized convexly and three additional types that can only be realized concavely.^[19] The three concave types have face configurations 4.4.3.3.3.3 (two quadrilaterals and four triangles; 6 vertices, 10 edges), 5.5.3.3.3.3 (two pentagons and four triangles; 7 vertices, 11 edges), and 6.6.3.3.3.3 (two hexagons and four triangles; 8 vertices, 12 edges). All satisfy Euler's formula V - E + F = 2. Concave embeddings of the convex types are also possible, maintaining their combinatorial properties. There exist infinitely many geometric varieties, parameterized by the depth and position of the concavity, allowing for continuous deformation while preserving the face count and non-intersecting nature.

Self-Intersecting Hexahedra

Self-intersecting hexahedra are polyhedra with six faces in which the faces cross through each other, creating a non-convex structure where parts of the boundary penetrate the interior. Unlike concave hexahedra, which feature indentations without crossings, self-intersecting forms allow faces to intersect, resulting in regions of higher topological density where the surface winds around the interior multiple times. Topologically, they can retain a genus-zero surface, satisfying Euler's formula V - E + F = 2, though the geometric embedding introduces intersections that complicate physical interpretations.^[38] Examples of self-intersecting hexahedra are rare and typically non-uniform, often constructed as compounds or stellations that approximate hexahedral symmetry. True single-component 6-faced realizations remain limited, with no uniform self-intersecting hexahedra among the complete set of 75 uniform polyhedra, which includes 53 star forms but none with exactly six faces.^[38] Construction of self-intersecting hexahedra often involves intersecting prisms or stellation processes, such as extending faces of a cube until they cross, but standard Schläfli symbols apply only to uniform cases, none of which fit six faces.^[38] Properties include complex winding numbers for determining enclosed regions and volume calculations via inclusion-exclusion principles to account for overlapping parts, often yielding effective symmetries lower than those of convex counterparts. While concave hexahedra number in the dozens combinatorially, self-intersecting variants exhibit fewer distinct topological types but allow infinite geometric variations through scaling and distortion; their study emerged in the 19th century alongside broader explorations of star polyhedra by mathematicians like Cauchy and Bertrand.^[38]

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