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An $n$-gonal antiprism is a polyhedron with two parallel congruent $n$-gon bases, one rotated relative to the other, and a band of $2n$ triangular side faces joining them. For the usual convex regular realization, both bases are regular and all side triangles are equilateral triangles.

Its combinatorial counts are $V=2n$, $E=4n$, and $F=2n+2$: the bases contribute $2n$ edges and the zigzag side band another $2n$. Hence $V-E+F=2$, agreeing with Euler characteristic. A triangular antiprism is an octahedron; a square antiprism has eight vertices, two square bases and eight triangular side faces.

A prism instead joins corresponding base vertices with a band of quadrilateral faces. The relative rotation and triangular band are the defining differences in the antiprism construction.

A polyhedron formed by joining two pyramids along a common polygonal base on opposite sides of its plane. The common base is internal; the external boundary consists of the two sets of triangular side faces. A triangular bipyramid has five vertices of a polyhedron, nine edges and six triangular faces.

For a closed polyhedron, let $A_F$ be a face area and ${\mathbf{n}}_F$ its outward unit normal vector. The sum of the face area vectors is zero. To prove it, fix any constant vector $\mathbf{q}$ and apply the divergence theorem to the constant vector field $\mathbf{q}$: its divergence vanishes, while its outward flux is $\mathbf{q}\cdot {\sum }_F{A}_F{\mathbf{n}}_F$. This is zero for every $\mathbf{q}$, so the sum vanishes. For a tetrahedron, the same identity follows directly by expanding the four oriented face cross products; multiplying every face vector by two leaves the identity intact.

A polyhedron that is also a convex set. A bounded full-dimensional convex hull of finitely many points is a convex polyhedron; each face lies in a supporting plane, with the whole solid in its inward half-space.

David A. Cox coauthored a systematic text on toric varieties with John B. Little and Hal Schenck. The subject connects algebraic geometry with the geometry of polyhedra: convex cones and a fan in toric geometry encode affine pieces and their gluings. The book develops this combinatorial description alongside divisors, homogeneous coordinates and line bundles.^[ref]

A line segment along which two faces of a polyhedron meet. A rectangular parallelepiped has twelve edges, four parallel to each of its three side directions.

Francis Bach developed a convex-analytic treatment of submodular set functions and their applications to machine learning. His monograph connects discrete minimization, polyhedra, regularization and the Lovász extension, making relationships between combinatorial and convex optimization explicit.^[ref]

For a finite ground set, submodularity is equivalent to diminishing marginal returns: adding an element to a smaller set gives at least as large a gain as adding it to a containing set. A coverage function illustrates this because objects already covered cannot contribute new coverage. The same structural property can support both direct selection problems and continuous optimization formulations; the modelling objective determines which connection is useful.

Hal Schenck coauthored a text on toric varieties with David A. Cox and John B. Little. It develops the bridge between the geometry of polyhedra and algebraic geometry, including the construction of varieties from a fan in toric geometry and the study of their divisors and line bundles. Combinatorial data thereby become tools for geometric calculations.^[ref]

At an edge of a convex polyhedron, its interior dihedral angle is the angle of the wedge inside the solid, measured in a plane perpendicular to the edge. If the two incident faces have outward unit normal vectors ${\hat{\mathbf{n}}}_1,{\hat{\mathbf{n}}}_2$, then their normal angle is supplementary to the interior angle. Thus $\cos \theta =-{\hat{\mathbf{n}}}_1\cdot {\hat{\mathbf{n}}}_2$. Equivalently, subtract from each vector pointing into one face its orthogonal projection onto the common edge; the angle between the resulting inward face directions is the interior angle. This distinguishes an interior angle from the acute angle convention for two unoriented planes.

For a toric divisor $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$ on a normal split toric variety, define the rational polyhedronIts integral points index the torus character basis of global sections of the divisorial sheaf ${\mathcal{O}}_X(D)$. If $D$ is a Cartier divisor, this is an invertible sheaf. For a general Weil divisor it is a rank-one reflexive sheaf.

The polyhedron can be unbounded. For affine space ${\mathbb{A}}^1$ and $D=0$, its inequality is $m\ge 0$, and the integral points give the basis $1,t,t^2,\dots$ of $k[t]$. If the toric fan is complete, the recession toric cone is zero, so a nonempty $P_D$ is bounded. Even then its vertices need not be integral for a non-Cartier Weil divisor. ^[ref]

Leonhard Euler (1707–1783) developed analysis as a working language for mechanics, geometry and number theory. His 1736–1737 treatment of Newtonian mechanics put motion into the methods of differential equations, while his work on surfaces and polyhedra connected geometric quantities with structural relations such as the Euler characteristic.^[ref]

The Euler's formula $e^{i\theta }=\cos \theta +i\sin \theta$ exemplifies the resulting unity: the complex exponential describes rotation, so oscillations can be manipulated through multiplication and differentiation. The Euler characteristic of a convex polyhedron, $V-E+F=2$, concerns a different invariant altogether. Distinguishing these meanings makes his many eponyms useful connections rather than interchangeable formulas.

A linear polyhedron is an intersection of finitely many closed affine half-spaces in finite-dimensional real space. It may be empty, unbounded, or lower-dimensional. This usage differs from a three-dimensional geometric polyhedron. The strict separation of disjoint linear polyhedra follows from linear programming duality.

Matthias Beck coauthored a text on integer-point enumeration in polyhedra with Sinai Robins. It connects generating functions, Ehrhart polynomials and Ehrhart reciprocity. Counting lattice points in dilated shapes thereby reveals geometric information such as volume, while reciprocity relates counts in the interior to counts in the closed shape.^[ref]

An octahedron is a polyhedron with eight faces. The regular octahedron has six vertices, twelve equal edges and eight equilateral triangular faces. One realization has vertices $\pm {\mathbf{e}}_1,\pm {\mathbf{e}}_2,\pm {\mathbf{e}}_3$; two vertices are adjacent exactly when they are not an opposite pair. Its edge graph is the octahedral graph.

A parallelepiped is a polyhedron with six parallelogram faces in three opposite pairs. Three generating vectors determine its edges; the absolute value of their determinant gives its volume in orthonormal Cartesian coordinates.

For solid modelling, a polyhedron is a solid whose boundary consists of finitely many planar polygons, called faces, meeting along straight edges and at vertices of a polyhedron. A face is not merely a list of arbitrary spatial points: its boundary must lie in one plane and bound a valid planar region. The boundary is closed and consistently oriented so that inside and outside are defined.

For example, a tetrahedron has four triangular faces, six edges and four vertices of a polyhedron. A more general polyhedron need not be convex and can have handles or cavities. The concise definition is

For an ordinary closed manifold boundary representation, three requirements are:

Together these support the answerFor a connected orientable closed boundary of genus $g$, the Euler characteristic provides an additional check, $V-E+F=2-2g$, when faces are disk-like. The familiar value $2$ applies to a sphere-like shell; it must not be imposed blindly on every polyhedron. Correct counts alone do not exclude intersections or invalid local incidence.

For an open cover $X=A\cup B$, the Mayer–Vietoris theorem gives the long exact sequenceThe same sequence holds for a union of simplicial subcomplexes. All homology groups here have integer coefficients.

An explicit triangulation of $X_q$ is as follows. Take vertices $b_0,b_1,b_2$ forming a target circle, a circle of $3q$ vertices $a_0,\dots ,a_{3q-1}$, and a cone vertex $c$. With subscripts on $a$ interpreted modulo $3q$ and on $b$ modulo three, include trianglesfor every $j$, together with all their faces. The first two types triangulate the mapping cylinder of the degree-$q$ covering $a_j\mapsto b_j$; the last type fills the inner circle by a disc. A filled disc with an attached annular collar is still a disc, and identifying the outer collar boundary by this covering gives exactly the source quotient. Distinct inner vertices ensure this is a genuine simplicial complex, not just a one-cell presentation.

Let $A$ be the mapping cylinder and $B$ the coned disc. Then $A$ retracts to the target circle, $B$ is contractible and $A\cap B$ is the inner circle. Its inclusion into $A$ induces multiplication by $q$ on $H_1$. The exact sequence therefore reduces toConnectedness and the absence of higher-dimensional simplices giveThis is a two-dimensional Moore space with torsion in degree one.

The suspension of a connected complex is the union of two cones whose intersection retracts to the original complex. Reduced Mayer-Vietoris givesIn particular its $H_1$ vanishes. For the wedge of two connected polyhedra, choose neighborhoods retracting onto the two summands with contractible intersection at the wedge point. ThenDecompose each prescribed $G_i$ into cyclic summands. Realize each infinite cyclic summand in degree $i$ by $S^i$, and each finite nontrivial summand $\mathbb{Z}/q$ by $S^{i-1}{X}_q$, meaning $i-1$ successive suspensions. Take their finite wedge, or a point if there are no summands. The suspension formula shifts the torsion to degree $i$ and creates no other positive-degree homology, while the wedge formula adds summands. Thus the resulting connected polyhedron satisfies $\boxed{H_0(X)=\mathbb{Z},H_i(X)=G_i(1\le i\le n),H_i(X)=0(i>n)}$. The finite triangulations above, their simplicial suspensions and wedges provide an actual polyhedron.

For every nonempty simplex $\sigma$ of $K$, let $b_{\sigma }$ be its barycentre. The first barycentric subdivision $K^{\prime }$ has the $b_{\sigma }$ as vertices, andis a simplex exactly when, after reordering, ${\sigma }_0⊊\cdots ⊊{\sigma }_q$. Its realization is the same polyhedron as that of $K$. The iterated barycentric subdivision is defined by

A seven-vertex polyhedron consisting of a five-membered equatorial ring and two axial vertices, each joined to every ring vertex. Removing an equatorial vertex leaves the six-vertex pentagonal-pyramidal skeleton used for some nido clusters.