Polyhedron

A polyhedron is a three-dimensional solid assembled from polygonal faces meeting along edges and vertices.

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 uniform polyhedron has regular polygonal faces and a symmetry group acting transitively on its vertices. Thus every vertex has the same surrounding face pattern, but that pattern may use more than one regular polygon type. The definition includes both convex examples and, under the broader convention, star-faced or self-intersecting examples.

A cube is uniform: each vertex meets three congruent squares, and its rotational symmetries carry any vertex to any other. A cuboctahedron has triangles and squares, alternating around every vertex, and is also uniform. Uniformity does not require every pair of faces to be related by a symmetry; that stronger face condition distinguishes regular solids.

The requirement is geometric. Merely drawing a polyhedron with the same number of incident edges at every vertex does not ensure regular faces or a vertex-transitive symmetry group. Regular prisms and regular antiprisms supply infinite families of convex uniform polyhedra.

A cuboctahedron is a convex uniform polyhedron with eight equilateral triangular faces and six square faces. Around each of its twelve vertices the faces alternate triangle, square, triangle, square. It has twenty-four edges, and the Euler characteristic check is $12-24+14=2$.

A concrete model is the convex hull of all coordinate permutations of $(\pm 1,\pm 1,0)$. The six planes $x=\pm 1$, $y=\pm 1$, $z=\pm 1$ cut out the square faces; the eight planes $\pm x\pm y\pm z=2$ cut out the triangular faces. All edges have length $\sqrt{2}$. Equivalently, join the midpoints of neighbouring edges of a cube: its vertices become triangular faces, and its faces become squares. The construction makes the name and the shared symmetries with the cube and octahedron visible.

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.

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.

For an external viewpoint in general position, the horizon is the cycle of edges separating visible and nonvisible faces of a convex polyhedron. In the incremental three-dimensional convex hull algorithm, retain these edges, remove visible faces, and join the viewpoint to each horizon edge to form the new faces. Edges incident with two removed faces disappear.

A corner of a polyhedron, stored with its spatial coordinates and incidences with boundary edges and faces. In a closed manifold boundary representation, the incident faces form a single cyclic fan. The vertices of a polyhedron and edges define a graph, but geometric embedding and valid face incidence require additional information.

A ten-vertex cage formed by adding one cap over each square face of a square antiprism. Each axial cap is joined to four vertices in its nearer square; adjacent squares are rotated relative to each other. It is the usual idealized ten-vertex closo cluster skeleton.

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.

The Archimedean solid obtained by truncating an octahedron, with eight hexagonal faces and six square faces. In an fcc nanocrystal the analogous shape can expose both $\{111\}$ and $\{100\}$ facets; the relative facet areas need not equal those of the regular Archimedean solid.

A polyhedron with two congruent parallel polygonal bases whose corresponding vertices are joined by parallel edges. A right prism has edges between the bases perpendicular to their planes. The optical use of a transparent prism is a separate application of this geometric shape.

Consider a homogeneous optical prism with apex angle $0<A<\pi$, in the same exterior medium on both sides. Let $n>1$ be the relative refractive index, the inside refractive index divided by the exterior refractive index. For a ray crossing both faces in the usual principal section, the internal angles obey $r_1+r_2=A$. Snell's law gives external incidence and emergence angles $i=f(r_1)$ and $e=f(r_2)$, whereThe deviation is $\Delta =i+e-A$. On the interior of this interval,Thus convexity gives $f(r_1)+f(r_2)\ge 2f(A/2)$. The minimum occurs for the symmetric path $r_1=r_2=A/2$, yieldingA path with nongrazing entry and exit requires $n\sin (A/2)<1$. Equality describes only a grazing limit; a larger value prevents transmission across both faces in this section. The condition $n>1$ matters: if $n<1$, the sign of $f^{\prime \prime }$ reverses, so the symmetric path maximizes this signed deviation instead. Measuring the minimum deviation therefore determines $n=\sin ((A+{\Delta }_{\min })/2)/\sin (A/2)$ for an ordinary glass-in-air optical prism.

For a prism in air with fixed apex angle $A$, Snell's law gives $r_1=\arcsin (\sin i/n)$, $r_2=A-r_1$ and $e=\arcsin (n\sin r_2)$. The exit angle $e$ is relative to the moving exit-face normal; the deviation $D$ is relative to the incoming beam. Rotating the prism while keeping the incident beam fixed changes the laboratory outgoing direction by $\Delta D$. Steering the incoming beam while fixing the prism instead changes it by $\Delta e$. These are distinct experiments; emergence and total-internal-reflection conditions must also be checked.

A prism with two hexagonal faces and six quadrilateral side faces. Its twelve vertices and eighteen edges provide an alternating aluminium–oxygen cage model when the graph is coloured bipartitely.

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.

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.

A pyramid is a polyhedron with a polygon as its base and triangular faces joining the base edges to one apex outside the base plane. If its base area is $A$ and perpendicular height is $h$, its volume is $Ah/3$. At height $z$ above the base, similarity gives cross-sectional area $A(1-z/h{)}^2$. Integrating this area from $0$ to $h$ proves the formula.

A square pyramid has a square base. If the base edge has length $a$ and all four lateral edges have length $\ell >a/\sqrt{2}$, the apex lies above the base center. The center-to-vertex distance is $a/\sqrt{2}$, so the Pythagorean theorem gives $h=\sqrt{{\ell }^2-a^2/2}$. Its volume is $a^{2}h/3$. When all eight edges have the same length $a$, this reduces to $a^3/(3\sqrt{2})$.

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.

A rectangular parallelepiped is a parallelepiped whose three edge directions at each vertex are mutually perpendicular. Its faces are rectangles; if its edge lengths are $a,b,c$, its volume is $abc$. A cube is the special case $a=b=c$.

A pentagonal prism has two pentagonal bases and five quadrilateral side faces, with three edges meeting at every vertex. Radial projection from an interior point gives a spherical polygon decomposition with five quadrilaterals and two pentagons.

A tetrahedron is the three-dimensional simplex, with four vertices, six edges, and four triangular faces.

Let $D=\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})>0$. The reciprocal basis gives the coordinates of any position vector asThe point lies in the closed tetrahedron with vertices $0,\mathbf{a},\mathbf{b},\mathbf{c}$ exactly when $\alpha ,\beta ,\gamma \ge 0$ and $\alpha +\beta +\gamma \le 1$. Indeed, these conditions express it as a convex combination of the four vertices, with fourth weight $1-\alpha -\beta -\gamma$. They also describe the intersection of the four inward half-spaces bounded by its faces. Strict inequalities characterize its interior.

Place the centroid of a tetrahedron at the origin, so its four vertex vectors satisfy $r_1+r_2+r_3+r_4=0$. Midpoints of opposite edges have opposite vectors, hence equal distances from the centroid. For the three opposite-edge pairs, choose midpoint vectors $(r_1+r_j)/2$, $j=2,3,4$. Expanding their squared lengths and using ${\sum }_{j=2}^4{r}_j=-r_1$ givesThe argument uses no regularity or equal-edge assumption on the tetrahedron.

The center of the sphere tangent to all four faces is the positive face-area-weighted combination of the opposite vertices. If $V$ is the volume and $S$ the total face area, its distance from each face is $3V/S$. Indeed its weight for vertex $A$ is $S_A/S$ and the corresponding altitude is $3V/S_A$, so their product is the common distance. Positive weights place it inside the tetrahedron.

For independent edge vectors $u=B-A$, $v=C-A$, $w=D-A$, the displacement $x=O-A$ solves $x\cdot u=|u{|}^2/2$, $x\cdot v=|v{|}^2/2$, $x\cdot w=|w{|}^2/2$. The reciprocal cross-product basis gives the displayed formula. Its squared distances to all four vertices are equal. It need not lie inside the tetrahedron.

A tetrahedron whose six edges have equal length; its four faces are equilateral triangles.

For a positively oriented basis $\mathbf{a},\mathbf{b},\mathbf{c}$, put $\tau =\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})>0$. The center of an interior sphere at distance $r$ from each of the three tetrahedron face planes through the origin has the displayed expression. The three inward unit vectors are the normalized cross products $\mathbf{b}\times \mathbf{c}$, $\mathbf{c}\times \mathbf{a}$, $\mathbf{a}\times \mathbf{b}$. Taking inner products with these vectors solves three independent signed-distance equations. Reversing some signs instead produces centers on exterior sides. Existence of a sphere contained in the full tetrahedron also requires clearance from the fourth face.

Two edges of a tetrahedron are opposite when they share no vertex. The six edges split into three opposite pairs.

Place one pair of opposite tetrahedron vertices on the line $b+\lambda a$ and the other pair on $d+\mu c$, with fixed nonzero direction vectors $a,c$. Let $\Delta =d-b$ and let $\delta \lambda ,\delta \mu$ be the parameter differences within the two pairs. The scalar triple product of the three edge vectors from one vertex gives the displayed volume. Terms parallel to $a$ or $c$ vanish in the triple product, leaving only $\Delta \cdot (a\times c)$. Sliding either pair along its own line while preserving its separation leaves this volume unchanged. When the two lines are skew lines, it can also be written $V=d_{\perp }{\ell }_1{\ell }_2|\sin \alpha |/6$, where $d_{\perp }$ is their shortest distance, ${\ell }_1,{\ell }_2$ the two edge lengths and $\alpha$ their direction angle. Intersecting or parallel lines give zero volume because all four vertices are coplanar.

Place one vertex of a tetrahedron at the origin and denote the other vertex vectors by $a,b,c$. Orthogonality of two pairs of opposite edges of a tetrahedron forces $a\cdot b=a\cdot c=b\cdot c$. It follows that the third pair is orthogonal too. The three sums of squared lengths of opposite edges all equal $|a{|}^2+|b{|}^2+|c{|}^2-2a\cdot b$. This gives a dot product test for the special geometry without solving for the face planes.

A Platonic solid is a convex polyhedron whose faces are congruent regular polygons and whose vertex figures are all alike.

A regular icosahedron is a Platonic solid with twenty equilateral triangular faces, twelve vertices and thirty edges. Five faces meet at each vertex. Its orientation-preserving rotational symmetry has order sixty: the orbit-stabilizer theorem gives twenty faces times the three rotations stabilizing a chosen face.

A Platonic solid with six square faces, twelve edges and eight vertices. Its rotational symmetry group of a cube has order 24; including orientation-reversing isometries gives the order-48 symmetry group of a cube.

The regular dodecahedron is the Platonic solid with 12 regular pentagonal faces, 20 vertices, and 30 edges.

An Archimedean solid is a convex uniform polyhedron, excluding the Platonic solids and the infinite families of uniform prisms and antiprisms. Here uniform means that every face is a regular polygon and that the symmetry group acts transitively on the vertices of a polyhedron: a symmetry of the entire solid carries any chosen vertex of a polyhedron to any other. Merely having equal numbers of faces of each type at every vertex of a polyhedron is a weaker local condition.^[ref]

There are thirteen types under this convention, counting a mirror-image pair as one type. The infinite prism and antiprism families are excluded by convention even though their uniform members satisfy the same regular-face and global vertex-equivalence conditions.^[ref]

For example, the truncated octahedron has six square faces and eight regular hexagon faces. Counting face-edge incidences gives $(6\cdot 4+8\cdot 6)/2=36$ edge of a polyhedron; Euler characteristic then gives $36-14+2=24$ vertices of a polyhedron. At each vertex of a polyhedron, one square and two regular hexagons meet. The icosidodecahedron, with twenty equilateral triangles and twelve regular pentagons, gives a different regular-face combination.

The icosidodecahedron has 20 triangular and 12 pentagonal faces, with the faces alternating around each of its 30 vertices. It can be obtained by cutting a dodecahedron through the midpoints of its edges.