Polyhedron

A polyhedron is, in elementary geometry, a three-dimensional solid bounded by finitely many flat polygonal faces, straight edges, and vertices. Familiar examples include cubes, tetrahedra, and pyramids. The plural is polyhedra or polyhedrons. The meaning varies by mathematical context: elementary geometry emphasizes polygonal surfaces, whereas convex mathematics defines polyhedra through linear inequalities and allows unbounded objects and dimensions other than three. A polytope is the corresponding bounded convex object in arbitrary dimension. (mathworld.wolfram.com)

Faces, edges, and vertices

The boundary of an ordinary, non-self-intersecting polyhedral solid is assembled from polygonal faces. Two adjacent faces meet along an edge, and edges meet at vertices. In a closed polyhedral surface that is a two-dimensional manifold, each edge belongs to exactly two faces, and the faces around each vertex form a single cyclic arrangement. A surface with a boundary, or one with branching or singularities, need not satisfy these conditions. The distinction matters when deciding whether a collection of polygons constitutes an ordinary solid boundary or a more general polyhedral object. (mathworld.wolfram.com)

For a convex three-dimensional polytope, vertices, edges, and polygonal faces are respectively its zero-, one-, and two-dimensional faces. The two-dimensional faces are also called facets. More generally, a facet of a $d$-dimensional polytope has dimension $d-1$. Thus, in higher-dimensional mathematics, face is a broader term than its usual elementary meaning. (ti.inf.ethz.ch)

Two different kinds of information describe a polyhedron:

• Geometric information: the positions of vertices and the resulting lengths, angles, and shapes.
• Combinatorial information: which vertices lie on which edges and faces, and which faces are adjacent.

Computer representations commonly store vertex coordinates together with these incidence and adjacency relationships. Objects can have the same combinatorial structure without having the same geometry: changing coordinates need not change which parts are connected. (groups.csail.mit.edu)

Convex polyhedra and linear inequalities

A solid is convex when the line segment joining any two of its points lies entirely within it. In the language of convex sets, a convex polyhedron is an intersection of finitely many closed half-spaces. In Euclidean space $\mathbb{R}^{n}$, it can be written as

where $A$ is a real matrix, $b$ is a vector, and the inequality is interpreted componentwise. Each row specifies a linear inequality whose boundary, when its normal vector is nonzero, is a hyperplane. (math.mit.edu)

This definition does not require boundedness or nonempty interior. For example, the region $x,y,z\geq0$ is an unbounded polyhedron, while a square lying in a plane is a lower-dimensional polyhedron in three-dimensional space. Equalities can be expressed as pairs of opposite inequalities. These examples illustrate why the convex mathematical definition is broader than “a three-dimensional solid with polygonal faces.” (courses.csail.mit.edu)

For nonempty bounded objects, two descriptions are equivalent:

• An H-representation gives a finite intersection of half-spaces.
• A V-representation gives the convex hull of finitely many points.

In particular, a bounded convex polyhedron is the convex hull of its vertices. Converting between these descriptions is a central problem of polyhedral computation, known as vertex enumeration or facet enumeration, depending on the direction of conversion. (people.inf.ethz.ch)

Principal families

Convex regular polyhedra

The five convex regular polyhedra are the Platonic solids. Each has congruent regular polygonal faces and a uniform arrangement of faces at every vertex:

Their classification follows from restrictions on how regular polygons can meet at a convex vertex. If each face has $p$ sides and $q$ faces meet at each vertex, the sum of the face angles must be less than $2\pi$, giving

For integers $p,q\geq3$, the only possibilities are $(3,3)$, $(4,3)$, $(3,4)$, $(5,3)$, and $(3,5)$. These correspond to the five solids listed above. (math.mit.edu)

Uniform and other regular-faced polyhedra

A uniform polyhedron has regular polygonal faces and a symmetry group that acts transitively on its vertices: any vertex can be carried to any other by a symmetry of the whole object. The convex uniform polyhedra comprise the Platonic solids, the thirteen Archimedean solids, and the families of uniform prisms and antiprisms. The Archimedean solids use more than one face type and conventionally exclude prisms and antiprisms. (math.harvard.edu)

Global symmetry is stronger than merely having the same local sequence of polygons at every vertex. The elongated square gyrobicupola, for example, has the same local face arrangement at each vertex but is not an Archimedean solid. This distinction prevents a purely local description from being mistaken for vertex transitivity. (pi.math.cornell.edu)

Nonconvex and star polyhedra

Nonconvex polyhedra may have indentations without their surfaces intersecting themselves. Broader definitions also admit star polyhedra, whose polygonal faces or surfaces can intersect. Under the classical definition allowing regular star polygons, there are four regular star polyhedra, called the Kepler–Poinsot polyhedra, in addition to the five convex regular solids. Such objects require care when interpreting their boundary and enclosed region; the elementary solid interpretation does not transfer automatically. (mathworld.wolfram.com)

Euler’s formula and topology

For a convex three-dimensional polyhedron, the numbers of vertices, edges, and faces satisfy

This is Euler’s polyhedron formula. A cube, for instance, gives $8-12+6=2$. The formula reflects the topology of the boundary rather than its precise dimensions: the boundary of a convex polyhedron has the topology of a sphere. (math.mit.edu)

The quantity $V-E+F$ is the Euler characteristic of the surface’s polygonal decomposition. Consequently, the formula also applies to nonconvex polyhedra whose boundaries are topological spheres. It is not valid without qualification for every object called a polyhedron. Surfaces with handles, boundaries, singularities, or self-intersections require the appropriate topological interpretation. For a connected closed orientable surface of genus $g$, the Euler characteristic is $2-2g$. (pages.vassar.edu)

Euler’s formula also supplies the classification of convex regular polyhedra with a combinatorial constraint. If each face has $p$ edges and each vertex meets $q$ edges, double-counting gives

Substitution yields

recovering the same five possible pairs as the face-angle argument. (math.mit.edu)

Duality

A dual polyhedron reverses the roles of vertices and facets while preserving the corresponding incidence relationships. In three dimensions, edges correspond to edges. A geometric construction for a full-dimensional convex polytope $P$ containing the origin in its interior is its polar:

Here $y^\mathsf{T}x$ is the standard inner product. A $k$-dimensional face of $P$ corresponds to an $(n-k-1)$-dimensional face of $P^\circ$. (www2.math.ethz.ch)

The cube and octahedron form a dual pair, as do the dodecahedron and icosahedron; the tetrahedron is combinatorially self-dual. Duality explains the exchange of vertex and face counts in these pairs. The combinatorial dual is determined by incidence structure, whereas its particular geometric realization depends on the construction used. (people.inf.ethz.ch)

Rigidity

Polyhedral rigidity concerns whether an object can change shape while its faces remain rigid and its edges behave as hinges. Cauchy’s rigidity theorem, proved in 1813, states that two convex polyhedra with corresponding congruent faces and the same face adjacencies are congruent. Thus a convex polyhedron cannot flex continuously while preserving its faces and their connections. (webspace.maths.qmul.ac.uk)

Convexity is essential. Flexible nonconvex polyhedra exist, so rigidity cannot be inferred solely from the fact that an object has polygonal faces. Face-preserving rigidity must also be distinguished from rigidity of an edge framework: fixing edge lengths alone may still allow a nontriangular face to change shape. (webspace.maths.qmul.ac.uk)

Historical development and applications

Regular polyhedra were studied in ancient Greece. Plato discussed the five regular solids in the Timaeus, while Euclid’s Elements gave their mathematical construction and classification. Later research extended the subject beyond regular solids to topology, rigidity, convexity, and algorithmic representation. (mathworld.wolfram.com)

In linear programming, the feasible set is a polyhedron defined by linear constraints. Polyhedral geometry provides the language for studying its vertices, faces, and unbounded directions, as well as for analyzing linear objectives on it. (courses.csail.mit.edu)

In computer graphics and solid modeling, a polygon mesh represents a surface through vertices, edges, and polygonal faces, commonly triangles. Such representations support rendering, collision detection, and rigid-body calculations. A surface mesh stores the boundary explicitly rather than the interior volume. Moreover, a collection of polygons is not necessarily a valid closed solid boundary: intersections, gaps, and nonmanifold connections must be distinguished from an ordinary polyhedral surface. (cs.cmu.edu)

References

1. Polyhedron — Wolfram MathWorldmathworld.wolfram.com
2. Linear Programming — MIT 6.854 Lecture Notescourses.csail.mit.edu
3. Frequently Asked Questions in Polyhedral Computationpeople.inf.ethz.ch
4. Geometry: C&A 2024 — Convex Polytopesti.inf.ethz.ch
5. MIT 6.838/4.214 — Meeting 5groups.csail.mit.edu
6. Homology — Chapter 11pages.vassar.edu
7. Triangles, Squares, Pentagonspi.math.cornell.edu
8. Kepler–Poinsot Polyhedron — Wolfram MathWorldmathworld.wolfram.com
9. Convex Bodieswww2.math.ethz.ch
10. Notes on the Rigidity of Graphswebspace.maths.qmul.ac.uk