Polyhedron

In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from el πολύ (poly-) 'many', and ἕδρον (-hedron) 'base, seat') is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts.[1] Also, the term polyhedron is often used to refer implicitly to the whole structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices.

There are many definitions of polyhedra, not all of which are equivalent. Under any definition, polyhedra are typically understood to generalize two-dimensional polygons and to be the three-dimensional specialization of polytopes (a more general concept in any number of dimensions). Polyhedra have several general characteristics that include the number of faces, topological classification by Euler characteristic, duality, vertex figures, surface area, volume, interior lines, Dehn invariant, and symmetry. A symmetry of a polyhedron means that the polyhedron's appearance is unchanged by a transformation, such as rotating and reflecting.

Convex polyhedra are a well-defined class of polyhedra with several equivalent standard definitions. Every convex polyhedron is the convex hull of its vertices, and the convex hull of a finite set of points is a polyhedron. Many common families of convex polyhedra, with cubes and pyramids are familiar examples of such.

There exist many miscellaneous families of polyhedra. Space-filling polyhedra are those that can be packed together with copies of themselves or with other different types of polyhedra in three-dimensional space. Flexible polyhedra are the polyhedra that can change their overall shape while preserving the shapes of their faces; some have such. Ideal polyhedron is a convex polyhedron defined in three-dimensional hyperbolic space. Lattice polyhedra are the convex polyhedra that can be constructed with integers coordinates. Orthogonal polyhedra are polyhedra where all edges are orthogonal, parallel to all three axes of Cartesian coordinate system. Copies of polyhedra can share a centre, which is known as polyhedral compounds. Polyhedra can be generalized into infinitely many faces called apeirohedra, the underlying space of which is a complex Hilbert space known as complex polyhedra, as well as allowing curved faces and edges.

The origin of polyhedra dates back to the ancient era. Ancient Egypt's four-sided Egyptian pyramids are known for the pyramidal structure, with the study of calculating their volume, specifically the volume of a frustum, in Moscow Mathematical Papyrus. In Ancient Greek, the Etruscan dodecahedron was discovered at Etruscan civilization made of soapstone on Monte Loffa, and Platonic solids were discovered and studied by Ancient Greek mathematicians, with Plato describes its association to the natures for each in his Timaeus, later soon treatment studied in Euclid's Elements. In Renaissance, toroidal polyhedra were used for sketching on polyhedral's perspective views, skeletal models, and nets appearance. Leonhard Euler worked on the polyhedral characteristics and the solution for the Seven Bridges of Königsberg's problem, underlying the field of topology. Johannes Kepler discovered two non-convex regular polyhedra, extended by Louis Poinsot prepending two more remaining polyhedra, known as Kepler-Poinsot polyhedra. Many results on polyhedral concepts, like Hilbert's third problem, Steinitz's theorem, and stellation of Platonic solids. Polyhedra are used in many fields, as well as appearing in biological creatures, nature, and modern computational geometry.

## Contents

* 1 Definition
* 2 General characteristics
* 2.1 Number of faces
* 2.2 Topological classification
* 2.3 Duality
* 2.4 Vertex figures
* 2.5 Surface area and lines inside polyhedra
* 2.6 Volume
* 2.7 Dehn invariant
* 2.8 Net
* 3 Symmetries
* 3.1 By elements of polyhedron
* 3.2 By point group in three dimensions
* 4 Convex polyhedra
* 5 Other families of polyhedra
* 5.1 Space-filling polyhedra
* 5.2 Flexible polyhedra
* 5.3 Ideal polyhedron
* 5.4 Lattice polyhedron
* 5.5 Polyhedral compound
* 5.6 Zonohedron
* 5.7 Orthogonal polyhedron
* 6 Generalisations
* 6.1 Apeirohedra
* 6.2 Complex polyhedra
* 6.3 Curved polyhedra
* 7 Higher-dimensional polyhedra
* 8 History
* 8.1 Before the Greeks
* 8.2 Ancient Greece
* 8.3 Ancient China
* 8.4 Medieval Islam
* 8.5 Renaissance
* 8.6 17th–19th centuries
* 8.7 20th–21st centuries
* 9 See also
* 10 Notes
* 11 References
* 12 External links
* 12.1 General theory
* 12.2 Lists and databases of polyhedra
* 12.3 Free software
* 12.4 Resources for making physical models

## Definition

There are several standard definitions of convex polyhedron, but they are equivalent, so the notion of convex polyhedron is agreed upon.

In contrast, for the definition of polyhedron in general, no universal agreement exists. As Branko Grünbaum observed,

> The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... at each stage ... the writers failed to define what are the polyhedra.[2]

Inequivalent definitions of "polyhedron" have been given in different contexts.[3] Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyhedra (such as solids whose boundaries are not manifolds). And although several of these definitions explicitly require the number of faces of a polyhedron to be finite, shapes with infinitely many faces such as the skew apeirohedra have also been called polyhedra.[4] Nevertheless, there is general agreement that a polyhedron is a solid or surface that can be described by its vertices (corner points), edges (line segments connecting certain pairs of vertices), faces (two-dimensional polygons), and that it sometimes can be said to have a particular three-dimensional interior volume. One can distinguish among these different definitions according to whether they describe the polyhedron as a solid, whether they describe it as a surface, or whether they describe it more abstractly based on its incidence geometry.[5]

Two polycubes (unions of finitely many unit cubes) whose boundary can be covered by finitely many planes but that do not meet some of the more restrictive definitions of polyhedra: The solid on the left has a face that is not a simple polygon, and the solid on the right has a pair of faces that meet in more than one edge.

* A common and somewhat naive definition of a polyhedron is that it is a solid whose boundary can be covered by finitely many planes[6][7] or that it is a solid formed as the union of finitely many convex polyhedra.[8] Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. The faces of such a polyhedron can be defined as the connected components of the parts of the boundary within each of the planes that cover it, and the edges and vertices as the line segments and points where the faces meet. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, whose faces may not form simple polygons, and some of whose edges may belong to more than two faces.[9]
* Definitions based on the idea of a bounding surface rather than a solid are also common.[10] Cromwell's Polyhedra gives a similar definition but without the restriction of at least three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra.[10] Similar notions form the basis of topological definitions of polyhedra, as subdivisions of a topological manifold into topological disks (the faces) whose pairwise intersections are required to be points (vertices), topological arcs (edges), or the empty set. However, there exist topological polyhedra (even with all faces triangles) that cannot be realized as acoptic polyhedra.[11]

A square pyramid and the associated abstract polytope. Here, the elements of a square pyramid can be defined as the partially ordered set.

* One modern approach is based on the theory of abstract polyhedra. These can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element (in this partial order) when the vertex or edge is part of the edge or face. Additionally, one may include a special bottom element of this partial order (representing the empty set) and a top element representing the whole polyhedron. If the sections of the partial order between elements three levels apart (that is, between each face and the bottom element, and between the top element and each vertex) have the same structure as the abstract representation of a polygon, then these partially ordered sets carry exactly the same information as a topological polyhedron.[citation needed] However, these requirements are often relaxed, to instead require only that sections between elements two levels apart have the same structure as the abstract representation of a line segment.[12] (This means that each edge contains two vertices and belongs to two faces, and that each vertex on a face belongs to two edges of that face.) Geometric polyhedra, defined in other ways, can be described abstractly in this way, but it is also possible to use abstract polyhedra as the basis of a definition of geometric polyhedra. A realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron.[13] Realizations that omit the requirement of face planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered.[12] Unlike the solid-based and surface-based definitions, this works perfectly well for star polyhedra. However, without additional restrictions, this definition allows degenerate or unfaithful polyhedra (for instance, by mapping all vertices to a single point) and the question of how to constrain realizations to avoid these degeneracies has not been settled.

In all of these definitions, a polyhedron is typically understood as a three-dimensional example of the more general polytope in any number of dimensions. For example, a polygon has a two-dimensional body and no faces, while a 4-polytope has a four-dimensional body and an additional set of three-dimensional "cells". However, some of the literature on higher-dimensional geometry uses the term "polyhedron" to mean something else: not a three-dimensional polytope, but a shape that is different from a polytope in some way. For instance, some sources define a convex polyhedron to be the intersection of finitely many half-spaces, and a polytope to be a bounded polyhedron.[14][15] The remainder of this article considers only three-dimensional polyhedra.

## General characteristics

### Number of faces

Polyhedra may be classified and are often named according to the number of faces. The naming system is based on Classical Greek, and combines a prefix counting the faces with the suffix "hedron", meaning "base" or "seat" and referring to the faces. For example a tetrahedron is a polyhedron with four faces, a pentahedron is a polyhedron with five faces, a hexahedron is a polyhedron with six faces, etc.[16] For a complete list of the Greek numeral prefixes see Numeral prefix § Table of number prefixes in English, in the column for Greek cardinal numbers. The names of tetrahedra, hexahedra, octahedra (eight-sided polyhedra), dodecahedra (twelve-sided polyhedra), and icosahedra (twenty-sided polyhedra) are sometimes used without additional qualification to refer to the Platonic solids, and sometimes used to refer more generally to polyhedra with the given number of sides without any assumption of symmetry.[17]

### Topological classification

The tetrahemihexahedron, a non-orientable self-intersecting polyhedron with four triangular faces (red) and three square faces (yellow). As with a Möbius strip or Klein bottle, a continuous path along the surface of this polyhedron can reach the point on the opposite side of the surface from its starting point, making it impossible to separate the surface into an inside and an outside. (Topologically, this polyhedron is a real projective plane.)

Some polyhedra have two distinct sides to their surface. For example, the inside and outside of a convex polyhedron paper model can each be given a different colour (although the inside colour will be hidden from view). These polyhedra are orientable. The same is true for non-convex polyhedra without self-crossings. Some non-convex self-crossing polyhedra can be coloured in the same way but have regions turned "inside out" so that both colours appear on the outside in different places; these are still considered to be orientable. However, for some other self-crossing polyhedra with simple-polygon faces, such as the tetrahemihexahedron, it is not possible to colour the two sides of each face with two different colours so that adjacent faces have consistent colours. In this case the polyhedron is said to be non-orientable. For polyhedra with self-crossing faces, it may not be clear what it means for adjacent faces to be consistently coloured, but for these polyhedra it is still possible to determine whether it is orientable or non-orientable by considering a topological cell complex with the same incidences between its vertices, edges, and faces.[18]

A more subtle distinction between polyhedron surfaces is given by their Euler characteristic, which combines the numbers of vertices V, edges E, and faces F of a polyhedron into a single number χ defined by the formula

χ=V−E+F.

The same formula is also used for the Euler characteristic of other kinds of topological surfaces. It is an invariant of the surface, meaning that when a single surface is subdivided into vertices, edges, and faces in more than one way, the Euler characteristic will be the same for these subdivisions. For a convex polyhedron, or more generally any simply connected polyhedron with the surface of a topological sphere, it always equals 2. For more complicated shapes, the Euler characteristic relates to the number of toroidal holes, handles or cross-caps in the surface and will be less than 2.[19] All polyhedra with odd-numbered Euler characteristics are non-orientable. A given figure with even Euler characteristic may or may not be orientable. For example, the one-holed toroid and the Klein bottle both have χ=0, with the first being orientable and the other not.[18]

For many (but not all) ways of defining polyhedra, the surface of the polyhedron is required to be a manifold. This means that every edge is part of the boundary of exactly two faces (disallowing shapes like the union of two cubes that meet only along a shared edge) and that every vertex is incident to a single alternating cycle of edges and faces (disallowing shapes like the union of two cubes sharing only a single vertex). For polyhedra defined in these ways, the classification of manifolds implies that the topological type of the surface is completely determined by the combination of its Euler characteristic and orientability. For example, every polyhedron whose surface is an orientable manifold and whose Euler characteristic is 2 must be a topological sphere.[18]

A toroidal polyhedron is a polyhedron whose Euler characteristic is less than or equal to 0, or equivalently whose genus is 1 or greater. Topologically, the surfaces of such polyhedra are torus surfaces having one or more holes through the middle.[20] A notable example is the Szilassi polyhedron, which geometrically realizes the Heawood map. A polyhedron with the symmetries of a regular polyhedron and with genus more than one is a Leonardo polyhedron.[21]

### Duality

Main article: Dual polyhedron

The octahedron is dual to the cube

For every convex polyhedron, there exists a dual polyhedron having

* faces in place of the original's vertices and vice versa, and
* the same number of edges.

The dual of a convex polyhedron can be obtained by the process of polar reciprocation.[22] Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron.[23]

Abstract polyhedra also have duals, obtained by reversing the partial order defining the polyhedron to obtain its dual or opposite order.[13] These have the same Euler characteristic and orientability as the initial polyhedron. However, this form of duality does not describe the shape of a dual polyhedron, but only its combinatorial structure. For some definitions of non-convex geometric polyhedra, there exist polyhedra whose abstract duals cannot be realized as geometric polyhedra under the same definition.[24]

### Vertex figures

Main article: Vertex figure

For every vertex one can define a vertex figure, which describes the local structure of the polyhedron around the vertex. Precise definitions vary, but a vertex figure can be thought of as the polygon exposed where a slice through the polyhedron cuts off a vertex.[10] For the Platonic solids and other highly-symmetric polyhedra, this slice may be chosen to pass through the midpoints of each edge incident to the vertex,[25] but other polyhedra may not have a plane through these points. For convex polyhedra, and more generally for polyhedra whose vertices are in convex position, this slice can be chosen as any plane separating the vertex from the other vertices.[26] When the polyhedron has a center of symmetry, it is standard to choose this plane to be perpendicular to the line through the given vertex and the center;[27] with this choice, the shape of the vertex figure is determined up to scaling. When the vertices of a polyhedron are not in convex position, there will not always be a plane separating each vertex from the rest. In this case, it is common instead to slice the polyhedron by a small sphere centered at the vertex.[28] Again, this produces a shape for the vertex figure that is invariant up to scaling. All of these choices lead to vertex figures with the same combinatorial structure, for the polyhedra to which they can be applied, but they may give them different geometric shapes.

### Surface area and lines inside polyhedra

The surface area of a polyhedron is the sum of the areas of its faces, for definitions of polyhedra for which the area of a face is well-defined. The geodesic distance between any two points on the surface of a polyhedron measures the length of the shortest curve that connects the two points, remaining within the surface. By Alexandrov's uniqueness theorem, every convex polyhedron is uniquely determined by the metric space of geodesic distances on its surface. However, non-convex polyhedra can have the same surface distances as each other, or the same as certain convex polyhedra.[29]

When a line segment connect two vertices that are not in the same face, it forms a diagonal line of the polyhedron.[30] Not all polyhedra have diagonal lines, as in the family of pyramids[31]. The Schönhardt polyhedron has three diagonal lines, all of which lie entirely outside of it, and the Császár polyhedron has no diagonal lines (rather, every pair of vertices is connected by an edge).[32]

### Volume

Polyhedral solids have an associated quantity called volume that measures how much space they occupy. Simple families of solids may have simple formulas for their volumes; for example, the volumes of pyramids, prisms, and parallelepipeds can easily be expressed in terms of their edge lengths or other coordinates. (See Volume § Volume formulas for a list that includes many of these formulas.)

Volumes of more complicated polyhedra may not have simple formulas. The volumes of such polyhedra may be computed by subdividing the polyhedron into smaller pieces (for example, by triangulation). For example, the volume of a Platonic solid can be computed by dividing it into congruent pyramids, with each pyramid having a face of the polyhedron as its base and the centre of the polyhedron as its apex.

In general, it can be derived from the divergence theorem that the volume of a polyhedral solid is given by 13|∑F(QF⋅NF)area⁡(F)|, where the sum is over faces F of the polyhedron, QF is an arbitrary point on face F, NF is the unit vector perpendicular to F pointing outside the solid, and the multiplication dot is the dot product.[33] In higher dimensions, volume computation may be challenging, in part because of the difficulty of listing the faces of a convex polyhedron specified only by its vertices, and there exist specialized algorithms to determine the volume in these cases.[34]

### Dehn invariant

Main article: Dehn invariant

In two dimensions, the Bolyai–Gerwien theorem asserts that any polygon may be transformed into any other polygon of the same area by cutting it up into finitely many polygonal pieces and rearranging them. The analogous question for polyhedra was the subject of Hilbert's third problem. Max Dehn solved this problem by showing that, unlike in the 2-D case, there exist polyhedra of the same volume that cannot be cut into smaller polyhedra and reassembled into each other. To prove this Dehn discovered another value associated with a polyhedron, the Dehn invariant, such that two polyhedra can only be dissected into each other when they have the same volume and the same Dehn invariant. It was later proven by Sydler that this is the only obstacle to dissection: every two Euclidean polyhedra with the same volumes and Dehn invariants can be cut up and reassembled into each other.[35] The Dehn invariant is not a number, but a vector in an infinite-dimensional vector space, determined from the lengths and dihedral angles of a polyhedron's edges.[36]

Another of Hilbert's problems, Hilbert's eighteenth problem, concerns (among other things) polyhedra that tile space. Every such polyhedron must have Dehn invariant zero.[37] The Dehn invariant has also been connected to flexible polyhedra by the strong bellows theorem, which states that the Dehn invariant of any flexible polyhedron remains invariant as it flexes.[38]

### Net

A net of a regular dodecahedron

The faces of some polyhedra can be unfolded into an arrangement of non-overlapping edge-joined polygons in the plane. Such an arrangement is known as a net of the polyhedron. Nets can be used to construct polyhedron models from paper or other flexible materials.[39]

## Symmetries

File:Revolução de poliedros 03.webm Many of the most studied polyhedra are highly symmetrical. Their appearance is unchanged by some reflection by plane or rotation around the axes passing through two opposite vertices, edges, or faces in space. Each symmetry may change the location of a given element, but the set of all vertices (likewise faces and edges) is unchanged. The collection of symmetries of a polyhedron is called its symmetry group.[40]

### By elements of polyhedron

All the elements (vertex, face, and edge) that can be superimposed on each other by symmetries are said to form a symmetry orbit. If these elements lie in the same orbit, the figure may be transitive on the orbit. Individually, they are isohedral (or face-transitive, meaning the symmetry transformations involve the polyhedra's faces in orbit),[41][lower-alpha 1] isotoxal (or edge-transitive, which involves the edge's polyhedra),[42] and isogonal (or vertex-transitive, which involves the polyhedra's vertices). For example, the cube in which all the faces are in one orbit and involving the rotation and reflections in the orbit remains unchanged in its appearance; hence, the cube is face-transitive. The cube also has the other two such symmetries.[43]

The cube is a regular polyhedron, because its faces, edges, and vertices are transitive to another, and the appearance is unchanged.

When three such symmetries belong to a polyhedron, it is known as a regular polyhedron.[43] There are nine regular polyhedra: five Platonic solids (cube, octahedron, icosahedron, tetrahedron, and dodecahedron--all of which have regular polygonal faces) and four Kepler-Poinsot polyhedrons. Nevertheless, some polyhedrons may not possess one or two of those symmetries:

* A polyhedron that is vertex-transitive and edge-transitive is said to be quasiregular; they have regular faces, and their duals are face-transitive and edge-transitive.
* A vertex- but not edge-transitive polyhedron with regular polygonal faces is said to be semiregular.[lower-alpha 2] This includes the prisms and antiprisms. The dual of a semiregular polyhedron is face-transitive, and every vertex is regular.
* A vertex-transitive polyhedron with regular polygonal faces is said to be uniform. This class includes the regular, quasi-regular, and semi-regular polyhedra. The definition may be applied to both convex and star polyhedra. The dual of a uniform polyhedron is face-transitive and has regular vertices but is not necessarily vertex-transitive. The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are convex or not.
* A face- and vertex-transitive (but not necessarily edge-transitive) polyhedron is said to be noble. The regular polyhedra are also noble; they are the only noble uniform polyhedra. The duals of noble polyhedra are themselves noble.

Some polyhedra have no reflection symmetry, so that they have two enantiomorph forms, which are reflections of each other. Examples include the snub cuboctahedron and snub icosidodecahedron. In this case the polyhedron is said to be chiral.

### By point group in three dimensions

Main article: Point groups in three dimensions

The point group of polyhedra means a mathematical group endowed with its symmetry operations so that the appearance of polyhedra remains preserved while transforming in three-dimensional space. The indicated transformation here includes the rotation around the axes, reflection through the plane, inversion through a center point, and a combination of these three.[44]

The regular tetrahedron has full tetrahedral symmetry: three-fold rotation around axis passing both vertex and triangular face, and two-fold rotation around axis through two edges, as well as the reflection plane through two faces and one edge

The polyhedral group is the symmetry group originally derived from the three Platonic solids: tetrahedron, octahedron, and icosahedron. These three have point groups respectively known as tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry. Each of these focuses on the rotation group of polyhedra, known as the chiral polyhedral group, whereas the additional reflection symmetry is known as the full polyhedral group. One point group, pyritohedral symmetry, includes the rotation of tetrahedral symmetry and additionally has three planes of reflection symmetry and some rotoreflections. Overall, the mentioned polyhedral groups are summarized in the following bullets:[45]

* chiral tetrahedral symmetry T, the rotation group for a regular tetrahedron and has the order of twelve.
* full tetrahedral symmetry Td, the symmetry group for a regular tetrahedron and has the order of twenty-four.
* pyritohedral symmetry Th, the symmetry of a pyritohedron and has the order of twenty-four.
* chiral octahedral symmetry O, the rotation group of both cube and regular octahedron and has the order twenty-four.
* full octahedral symmetry Oh, the symmetry group of both cube and regular octahedron and has order forty-eight.
* chiral icosahedral symmetry I, the rotation group of both regular icosahedron and regular dodecahedron and has the order of sixty.
* full icosahedral symmetry Ih, the symmetry group of both regular icosahedron and regular dodecahedron and has the order of a hundred-twenty.

The square pyramid has pyramidal symmetry C4v. It shows the appearance is invariant by rotating every quarter of a full turn around its axis and possesses mirror symmetric relative to any perpendicular plane passing through its base's bisector

Point groups in three dimensions may also allow the preservation of polyhedra's appearance by the circulation around an axis. There are three various of these point groups:

* pyramidal symmetry Cnv, allowing rotate the axis passing through the apex and its base, as well as reflection relative to perpendicular planes passing through the bisector of a base. This point group symmetry can be found in pyramids,[46] cupolas, and rotundas.
* prismatic symmetry Dnh, similar to the pyramidal symmetry, but with additional transformation by reflecting it across a horizontal plane.[46] This may be achieved from the family of prisms and its dual bipyramids.[46]
* antiprismatic symmetry Dnv, which preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.[46] Examples can be found in antiprisms.

A point group Cnh consists of rotating around the axis of symmetry and reflection on the horizontal plane. In the case of n=1, the symmetry group only preserves the symmetry by a full rotation solely, ordinarily denoting Cs.[47] Polyhedra may have rotation only to preserve the symmetry, and the symmetry group may be considered as the cyclic group Cn.[48] Polyhedra with the rotoreflection and the rotation by the cyclic group is the point group Sn.[49]

## Convex polyhedra

It has been suggested that this section be split out into another article titled Convex polyhedron. (Discuss) (November 2025)

|

Top left to bottom right: hexagonal pyramid (a prismatoid), truncated tetrahedron (an Archimedean solid), triakis icosahedron (a Catalan solid), and triaugmented triangular prism (a Johnson solid and deltahedron). All of these classes are convex polyhedra.

As mentioned above, the convex polyhedra are well-defined, with several equivalent standard definitions. They are often defined as bounded intersections of finitely many half-spaces,[14][15] or as the convex hull of finitely many points,[50] restricted in either case to intersections or hulls that have nonzero volume.

The classes of convex polyhedra include the following:

* the family of prismatoid, polyhedra whose vertices lie on two parallel planes and whose faces are likely to be trapezoids and triangles.[51] Examples of prismatoids are pyramids, wedges, parallelipipeds, prisms, antiprisms, cupolas, and frustums.
* Platonic solids are the five ancient polyhedra--tetrahedron, octahedron, icosahedron, cube, and dodecahedron--described by Plato in the Timaeus.[52] Archimedean solids are the class of thirteen polyhedra whose faces are all regular polygons and whose vertices are symmetric to each other;[lower-alpha 3] their dual polyhedra are the Catalan solids.[54]
* Johnson solids are a class of 92 convex polyhedra whose faces are all regular polygons, excluding the uniform polyhedra, namely the Platonic and Archimedean solids, as well as the infinite families of prisms and antiprisms.[55] These include the strictly convex deltahedra whose faces are all equilateral triangles.[56]
* Symmetrohedron is the family of infinitely many highly symmetric convex polyhedra, coined by Craig S. Kaplan and George W. Hart, containing regular polygonal faces on symmetry axes with gaps on the convex hull filled by irregular polygons. This involves three-dimensional symmetric groups of tetrahedral, octahedral, and icosahedral symmetry.[57]
* Near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons, but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces.[57]
* Goldberg polyhedron is a convex polyhedron with hexagonal and pentagonal faces. Described by (Goldberg 1937), it is obtained from regular octahedron, regular tetrahedron, and regular icosahedron.[58][59]

Convex polyhedra can be categorized into elementary polyhedra or composite polyhedra. Elementary polyhedra are convex, regular-faced polyhedra that cannot be produced into two or more polyhedra by slicing them with a plane.[60] Quite opposite to composite polyhedra, they can be alternatively defined as polyhedra constructed by attaching more elementary polyhedra. For example, triaugmented triangular prism is composite since it can be constructed by attaching three equilateral square pyramids onto the square faces of a triangular prism; the square pyramids and the triangular prism are elementaries.[61]

A midsphere

Some convex polyhedra possess a midsphere, a sphere tangent to each of their edges, which is intermediate in radius between the insphere and circumsphere for polyhedra in which all these spheres are present. Every convex polyhedron is combinatorially equivalent to a canonical polyhedron, a polyhedron that has a midsphere whose center coincides with the centroid of its tangent points with edges. The shape of the canonical polyhedron (but not its scale or position) is uniquely determined by the combinatorial structure of the given polyhedron.[62]

By forgetting the face structure, any polyhedron gives rise to a graph, called the skeleton of the polyhedron, with corresponding vertices and edges. Such figures have a long history: Leonardo da Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione, and similar wire-frame polyhedra appear in M.C. Escher's print Stars.[63] One highlight of this approach is Steinitz's theorem, which gives a purely graph-theoretic characterization of the skeletons of convex polyhedra: it states that the skeleton of every convex polyhedron is a planar graph with three-connected, and every such graph is the skeleton of some convex polyhedron. In other words, these are the connected graphs that can be drawn in the plane without any edges crossing, and that stay connected after removing any two of its vertices.[64]

Prominent non-convex polyhedra include the star polyhedra. The regular star polyhedra, also known as the Kepler-Poinsot polyhedra, are constructible via stellation or faceting of regular convex polyhedra. Stellation is the process of extending the faces (within their planes) so that they meet. Faceting is the process of removing parts of a polyhedron to create new faces (or facets) without creating any new vertices.[65][66] A facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face;[65] for example, a polygon involving diagonals (face diagonals or space diagonals). The stellation and faceting are inverse or reciprocal processes: the dual of some stellation is a faceting of the dual to the original polyhedron.[67] Other examples are the Chazelle polyhedron[68] and toroidal polyhedra.[69]

## Other families of polyhedra

### Space-filling polyhedra

Paralellohedra, one of special kinds of space-filling polyhedra

Main articles: Space-filling polyhedron and Honeycomb (geometry)

A space-filling polyhedron is one that can be packed together with copies of itself to completely fill space. Such a close-packing or space-filling is often called a tessellation of space or a honeycomb. These include the parallelohedra (or Federov polyhedra), each of which forms a tessellation using copies of itself translated without rotation;[70] the plesiohedra, defined as the Voronoi cells of symmetric Delone sets; and the Hill tetrahedra, a family of space-filling tetrahedra. Parallelohedra and plesiohedra are examples of stereohedra, which tile space isohedrally.[71] Every space-filling polyhedron must have its Dehn invariant equal to zero.[72]

Some honeycombs involve more than one kind of polyhedron, such as a tiling of octahedra and tetrahedra and octahedra and cuboctahedra.[73]

### Flexible polyhedra

Main article: Flexible polyhedron

Steffen's polyhedron and one of the Bricard polyhedra's type

Some polyhedra can change their overall shape, while keeping the shapes of their faces the same, by varying the angles of their edges. A polyhedron that can do this is called a flexible polyhedron. By Cauchy's rigidity theorem, flexible polyhedra must be non-convex. The volume of a flexible polyhedron must remain constant as it flexes; this result is known as the bellows theorem.[74] Three examples are Steffen's polyhedron,[75] Bricard octahedron,[76] and Kokotsakis polyhedron.[77]

### Ideal polyhedron

Convex polyhedra can be defined in three-dimensional hyperbolic space in the same way as in Euclidean space, as the convex hulls of finite sets of points. However, in hyperbolic space, it is also possible to consider ideal points and the points within the space. An ideal polyhedron is the convex hull of a finite set of ideal points.[78] Its faces are ideal polygons, but its edges are defined by entire hyperbolic lines rather than line segments, and its vertices (the ideal points of which it is the convex hull) do not lie within the hyperbolic space.

### Lattice polyhedron

Reeve tetrahedra for different choices of the parameter

Convex polyhedra in which all vertices have integer coordinates are called lattice polyhedra or integral polyhedra. The Ehrhart polynomial of lattice a polyhedron counts how many points with integer coordinates lie within a scaled copy of the polyhedron, as a function of the scale factor. The study of these polynomials lies at the intersection of combinatorics and commutative algebra.[79] An example is Reeve tetrahedron.[80]

There is a far-reaching equivalence between lattice polyhedra and certain algebraic varieties called toric varieties.[81] This was used by Stanley to prove the Dehn–Sommerville equations for simplicial polytopes.[82]

### Polyhedral compound

A polyhedral compound is made of two or more polyhedra sharing a common centre. Symmetrical compounds often share the same vertices as other well-known polyhedra and may often also be formed by stellation. Some are listed in the list of Wenninger polyhedron models.

### Zonohedron

A zonohedron is a convex polyhedron in which every face is a polygon that is symmetric under rotations through 180°. Zonohedra can also be characterized as the Minkowski sums of line segments, and include several important space-filling polyhedra.[83]

### Orthogonal polyhedron

Some orthogonal polyhedra made of Soma cube pieces, themselves polycubes

Polyhedra are said to be orthogonal because all of their edges are parallel to the axes of a Cartesian coordinate system.[84] This implies that all faces meet at right angles, but this condition is weaker: Jessen's icosahedron has faces meeting at right angles, but does not have axis-parallel edges.[85] Aside from the rectangular cuboids, orthogonal polyhedra are nonconvex. They are the three-dimensional analogs of two-dimensional orthogonal polygons, also known as rectilinear polygons. Orthogonal polyhedra are used in computational geometry, where their constrained structure has enabled advances in problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net.[86] Polycubes are a special case of orthogonal polyhedra that can be decomposed into identical cubes, and are three-dimensional analogues of planar polyominoes.[87]

## Generalisations

The name 'polyhedron' has come to be used for a variety of objects having similar structural properties to traditional polyhedra.

### Apeirohedra

A classical polyhedral surface has a finite number of faces, joined in pairs along edges. The apeirohedra form a related class of objects with infinitely many faces. Examples of apeirohedra include:

* tilings or tessellations of the plane, and
* sponge-like structures called infinite skew polyhedra.

### Complex polyhedra

Main article: Complex polytope

There are objects called complex polyhedra, for which the underlying space is a complex Hilbert space rather than real Euclidean space. Precise definitions exist only for the regular complex polyhedra, whose symmetry groups are complex reflection groups. The complex polyhedra are mathematically more closely related to configurations than to real polyhedra.[88]

### Curved polyhedra

Some fields of study allow polyhedra to have curved faces and edges. Curved faces can allow digonal faces to exist with a positive area.

* When the surface of a sphere is divided by finitely many great arcs (equivalently, by planes passing through the center of the sphere), the result is called a spherical polyhedron. Many convex polytopes having some degree of symmetry (for example, all the Platonic solids) can be projected onto the surface of a concentric sphere to produce a spherical polyhedron. However, the reverse process is not always possible; some spherical polyhedra (such as the hosohedra) have no flat-faced analogue.[89]
* If faces are allowed to be concave as well as convex, adjacent faces may be made to meet together with no gap. Some of these curved polyhedra can pack together to fill space. Two important types are bubbles in froths and foams such as Weaire-Phelan bubbles,[90] and forms used in architecture.[91]

## Higher-dimensional polyhedra

From the latter half of the twentieth century, various mathematical constructs have been found to have properties also present in traditional polyhedra. Rather than confining the term "polyhedron" to describe a three-dimensional polytope, it has been adopted to describe various related but distinct kinds of structure.

A polyhedron has been defined as a set of points in real affine (or Euclidean) space of any dimension n that has flat sides. It may alternatively be defined as the intersection of finitely many half-spaces. Unlike a conventional polyhedron, it may be bounded or unbounded. In this meaning, a polytope is a bounded polyhedron.[14][15]

Analytically, such a convex polyhedron is expressed as the solution set for a system of linear inequalities. Defining polyhedra in this way provides a geometric perspective for problems in linear programming.[92]

## History

### Before the Greeks

Problem 14 of the Moscow Mathematical Papyrus, on calculating the volume of a frustum

Polyhedra appeared in early architectural forms such as cubes and cuboids, with the earliest four-sided Egyptian pyramids dating from the 27th century BC.[93] The Moscow Mathematical Papyrus from approximately 1800–1650 BC includes an early written study of polyhedra and their volumes (specifically, the volume of a frustum).[94] The mathematics of the Old Babylonian Empire, from roughly the same time period as the Moscow Papyrus, also included calculations of the volumes of cuboids (and of non-polyhedral cylinders), and calculations of the height of such a shape needed to attain a given volume.[95]

The Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery of an Etruscan dodecahedron made of soapstone on Monte Loffa. Its faces were marked with different designs, suggesting to some scholars that it may have been used as a gaming die.[96]

### Ancient Greece

Platonic solids assigned to the elements (drawings from Kepler's Harmonices Mundi)

Ancient Greek mathematicians discovered and studied the convex regular polyhedra, which came to be known as the Platonic solids. Their first written description is in the Timaeus of Plato (circa 360 BC), which associates four of them with the four elements and the fifth to the overall shape of the universe. A more mathematical treatment of these five polyhedra was written soon after in the Elements of Euclid. An early commentator on Euclid (possibly Geminus) writes that the attribution of these shapes to Plato is incorrect: Pythagoras knew the tetrahedron, cube, and dodecahedron, and Theaetetus (circa 417 BC) discovered the other two, the octahedron and icosahedron.[97] Later, Archimedes expanded his study to the convex uniform polyhedra which now bear his name. His original work is lost and his solids come down to us through Pappus.[98]

### Ancient China

Fourteen sided die from the Warring States period

Both cubical dice and 14-sided dice in the shape of a truncated octahedron in China have been dated back as early as the Warring States period.[99]

By 236 AD, Liu Hui was describing the dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids, using assemblages of these solids as the basis for calculating volumes of earth to be moved during engineering excavations.[100]

### Medieval Islam

After the end of the Classical era, scholars in the Islamic civilisation continued to take the Greek knowledge forward (see Mathematics in medieval Islam).[101] The 9th century scholar Thabit ibn Qurra included the calculation of volumes in his studies,[102] and wrote a work on the cuboctahedron. Then in the 10th century Abu'l Wafa described the convex regular and quasiregular spherical polyhedra.[103]

### Renaissance

Doppio ritratto, attributed to Jacopo de' Barbari, depicting Luca Pacioli and a student studying a glass rhombicuboctahedron half-filled with water.[104]

A skeletal polyhedron (specifically, a rhombicuboctahedron) drawn by Leonardo da Vinci to illustrate a book by Luca Pacioli

As with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the Italian Renaissance. Artists constructed skeletal polyhedra, depicting them from life as a part of their investigations into perspective.[105] Toroidal polyhedra, made of wood and used to support headgear, became a common exercise in perspective drawing, and were depicted in marquetry panels of the period as a symbol of geometry.[106] Piero della Francesca wrote about constructing perspective views of polyhedra, and rediscovered many of the Archimedean solids. Leonardo da Vinci illustrated skeletal models of several polyhedra for a book by Luca Pacioli,[107] with text largely plagiarized from della Francesca.[108] Polyhedral nets make an appearance in the work of Albrecht Dürer.[109]

Several works from this time investigate star polyhedra and other elaborations of the basic Platonic forms. A marble tarsia in the floor of St. Mark's Basilica, Venice, designed by Paolo Uccello, depicts a stellated dodecahedron.[110] As the Renaissance spread beyond Italy, later artists such as Wenzel Jamnitzer, Dürer and others also depicted polyhedra of increasing complexity, many of them novel, in imaginative etchings.[105] Johannes Kepler (1571–1630) used star polygons, typically pentagrams, to build star polyhedra. Some of these figures may have been discovered before Kepler's time, but he was the first to recognize that they could be considered "regular" if one removed the restriction that regular polyhedra must be convex.[111]

In the same period, Euler's polyhedral formula, a linear equation relating the numbers of vertices, edges, and faces of a polyhedron, was stated for the Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico.[112]

### 17th–19th centuries

René Descartes, in around 1630, wrote his book De solidorum elementis studying convex polyhedra as a general concept, not limited to the Platonic solids and their elaborations. The work was lost, and not rediscovered until the 19th century. One of its contributions was Descartes' theorem on total angular defect, which is closely related to Euler's polyhedral formula.[113] Leonhard Euler, for whom the formula is named, introduced it in 1758 for convex polyhedra more generally, albeit with an incorrect proof.[114] Euler's work (together with his earlier solution to the puzzle of the Seven Bridges of Königsberg) became the foundation of the new field of topology.[115] The core concepts of this field, including generalizations of the polyhedral formula, were developed in the late nineteenth century by Henri Poincaré, Enrico Betti, Bernhard Riemann, and others.[116]

In the early 19th century, Louis Poinsot extended Kepler's work, and discovered the remaining two regular star polyhedra. Soon after, Augustin-Louis Cauchy proved Poinsot's list complete, subject to an unstated assumption that the sequence of vertices and edges of each polygonal side cannot admit repetitions (an assumption that had been considered but rejected in the earlier work of A. F. L. Meister).[117] They became known as the Kepler–Poinsot polyhedra, and their usual names were given by Arthur Cayley.[118] Meanwhile, the discovery of higher dimensions in the early 19th century led Ludwig Schläfli by 1853 to the idea of higher-dimensional polytopes.[119] Additionally, in the late 19th century, Russian crystallographer Evgraf Fedorov completed the classification of parallelohedra, convex polyhedra that tile space by translations.[120]

### 20th–21st centuries

Mathematics in the 20th century dawned with Hilbert's problems, one of which, Hilbert's third problem, concerned polyhedra and their dissections. It was quickly solved by Hilbert's student Max Dehn, introducing the Dehn invariant of polyhedra.[121] Steinitz's theorem, published by Ernst Steinitz in 1992, characterized the graphs of convex polyhedra, bringing modern ideas from graph theory and combinatorics into the study of polyhedra.[122]

The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by H.S.M. Coxeter and others in 1938, with the now famous paper The Fifty-Nine Icosahedra.Cite error: Closing `</ref>` missing for `<ref>` tag

The radiolarian Circogonia icosahedra

Dymaxion map, created by the net of a regular icosahedron

Polyhedra have been discovered in many fields of science. The Platonic solids appeared in biological creatures, as in The Braarudosphaera bigelowii has a regular dodecahedral structure.[123] Ernst Haeckel described a number of species of radiolarians, some of whose shells are shaped like various regular polyhedra.[124] The outer protein shells of many viruses form regular polyhedra. For example, HIV is enclosed in a regular icosahedron, as is the head of a typical myovirus.[125][126] The regular icosahedron may also appeared in the applications of cartography when R. Buckminster Fuller used its net to his project known as Dymaxion map, frustatedly realized that the Greenland size is smaller than the South America.[127]

Polyhedra make a frequent appearance in modern computational geometry, computer graphics, and geometric design with topics including the reconstruction of polyhedral surfaces or surface meshes from scattered data points,[128] geodesics on polyhedral surfaces,[129] visibility and illumination in polyhedral scenes,[130] polycubes and other non-convex polyhedra with axis-parallel sides,[131] algorithmic forms of Steinitz's theorem,[132] and the still-unsolved problem of the existence of polyhedral nets for convex polyhedra.[133]

## See also

* Associahedron, a convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses
* Dihedron, a polyhedron with two faces
* Equiprojective polyhedra, orthogonal projection of the polygon onto a plane in a direction not parallel to a face of the polyhedron forms a polygon
* Extension complexity, the smallest number of facets among convex polytopes that has another one as a projection
* Holyhedron, each of the polyhedron's faces contains at least one polygon-shaped hole, and the boundaries share no point on the face's boundary
* List of books about polyhedra, listing polyhedral books regarding the modeling, studies, and history of polyhedra
* Monostatic polytope, a polytope standing on one face only
* Gauss–Bonnet theorem, a formula that links the curvature of a surface to its underlying topology
* Polyhedral number, a term for figurate numbers in different dimensions
* Polyhedral skeletal electron pair theory
* Polyhedral space, a certain metric space
* Polyhedral symbol, indicating the approximate geometry of the coordinating atoms around the central atom in coordination chemistry
* Polyhedral terrain, a polyhedral surface intersecting every line parallel to some particular line in a connected set or the empty set
* Polytope model, a mathematical framework for programs that perform large numbers of operations
* Rupert property, a property where a polyhedron of the same or larger size and the same size can pass through a hole
* Stella, a computer graphics software creating a polyhedral model

## Notes

1. ↑ The topological property of an isohedral polyhedra can be represented by a face configuration. All five Platonic solids and thirteen Catalan solids are isohedra, as well as the infinite families of trapezohedra and bipyramids. Some definitions of isohedra allow geometric variations including concave and self-intersecting forms.
2. ↑ This is one of several definitions of the term, depending on the author. Some definitions overlap with the quasi-regular class.
3. ↑ The Archimedean solids once had fourteenth solid known as the pseudorhombicuboctahedron, a mistaken construction of the rhombicuboctahedron. However, it was debarred for not having the vertex-transitive property, leading it to be instead classified as a Johnson solid.[53]

## References

1. ↑ "1.1 Definition of a Convex Polyhedron", Convex Polyhedra, Springer, 2005, pp. 8, 10 .
2. ↑ Bisztriczky, Tibor; McMullen, Peter; Schneider, Rolf et al., eds. (1994), "Polyhedra with hollow faces", Proceedings of the NATO Advanced Study Institute on Polytopes: Abstract, Convex and Computational, Dordrecht: Kluwer Acad. Publ., pp. 43–70, doi:10.1007/978-94-011-0924-6_3, ISBN 978-94-010-4398-4 ; for quote, see p. 43.
3. ↑ Proofs and Refutations: The logic of mathematical discovery, Cambridge Philosophy Classics, Cambridge: Cambridge University Press, 2015, p. 16, doi:10.1017/CBO9781316286425, ISBN 978-1-107-53405-6, "definitions are frequently proposed and argued about" .
4. ↑ Coxeter, H. S. M. (1939), "The regular sponges, or skew polyhedra", Scripta Mathematica 6: 240–244
5. ↑ Loeb, Arthur L. (2013), "Polyhedra: Surfaces or solids?", in Senechal, Marjorie, Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination (2nd ed.), Springer, pp. 65–75, doi:10.1007/978-0-387-92714-5_5, ISBN 978-0-387-92713-8
6. ↑ McCormack, Joseph P. (1931), Solid Geometry, D. Appleton-Century Company, p. 416 .
7. ↑ de Berg, M.; van Kreveld, M.; Overmars, M.; Schwarzkopf, O. (2000), Computational Geometry: Algorithms and Applications (2nd ed.), Springer, p. 64 .
8. ↑ Hazewinkel, Michiel, ed. (2001), "Polyhedron, abstract", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Polyhedron,_abstract&oldid=25452
9. ↑ Stewart, B. M. (1980), Adventures Among the Toroids: A study of orientable polyhedra with regular faces (2nd ed.), p. 6 .
10. ↑ 10.0 10.1 10.2 Grünbaum, Branko (1999), "Acoptic polyhedra", Advances in discrete and computational geometry (South Hadley, MA, 1996), Contemporary Mathematics, 223, Providence, Rhode Island: American Mathematical Society, pp. 163–199, doi:10.1090/conm/223/03137, ISBN 978-0-8218-0674-6, https://sites.math.washington.edu/~grunbaum/BG225.Acoptic polyhedra.pdf, retrieved 2022-07-01 .
11. ↑ Bokowski, J.; Guedes de Oliveira, A. (2000), "On the generation of oriented matroids", Discrete and Computational Geometry 24 (2–3): 197–208, doi:10.1007/s004540010027 .
12. ↑ 12.0 12.1 Burgiel, H.; Stanton, D. (2000), "Realizations of regular abstract polyhedra of types {3,6} and {6,3}", Discrete and Computational Geometry 24 (2–3): 241–255, doi:10.1007/s004540010030 .
13. ↑ 13.0 13.1 Grünbaum, Branko (2003), "Are your polyhedra the same as my polyhedra?", in Aronov, Boris; Basu, Saugata; Pach, János et al., Discrete and Computational Geometry: The Goodman–Pollack Festschrift, Algorithms and Combinatorics, 25, Berlin: Springer, pp. 461–488, doi:10.1007/978-3-642-55566-4_21, ISBN 978-3-642-62442-1, https://faculty.washington.edu/moishe/branko/BG249.Your polyh-my polyh.pdf
14. ↑ 14.0 14.1 14.2 Convex Polytopes, Graduate Texts in Mathematics, 221 (2nd ed.), New York: Springer-Verlag, 2003, p. 26, doi:10.1007/978-1-4613-0019-9, ISBN 978-0-387-00424-2 .
15. ↑ 15.0 15.1 15.2 Bruns, Winfried; Gubeladze, Joseph (2009), "Definition 1.1", Polytopes, Rings, and K-theory, Springer Monographs in Mathematics, Dordrecht: Springer, p. 5, doi:10.1007/b105283, ISBN 978-0-387-76355-2, https://books.google.com/books?id=pbgg1pFxW8YC&pg=PA5 .
16. ↑ Peirce, Charles S. (1976), Eisele, Carolyn, ed., The New Elements of Mathematics, Volume II: Algebra and Geometry, Mouton Publishers & Humanities Press, p. 297, ISBN 9783110818840, https://books.google.com/books?id=Z3ldDwAAQBAJ&pg=PA297
17. ↑ O'Keefe, Michael; Hyde, Bruce G. (2020), Crystal Structures: Patterns and Symmetry, Dover Publications, p. 134, ISBN 9780486836546, https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA134
18. ↑ 18.0 18.1 18.2 Ringel, Gerhard (1974), "Classification of surfaces", Map Color Theorem, Springer, pp. 34–53, doi:10.1007/978-3-642-65759-7_3, ISBN 978-3-642-65761-0
19. ↑ Euler's Gem: The polyhedron formula and the birth of topology, Princeton, NJ: Princeton University Press, 2008, ISBN 978-0-691-12677-7 , pp. 157, 180.
20. ↑ Stewart, B. M. (1980), Adventures Among the Toroids: A Study of Orientable Polyhedra with Regular Faces (2nd ed.), B. M. Stewart, ISBN 978-0-686-11936-4 .
21. ↑ Bokowski, Jürgen (2022). "Regular Leonardo polyhedra". The Art of Discrete and Applied Mathematics 5 (3). doi:10.26493/2590-9770.1535.8ad.
22. ↑ "3.2 Duality", Mathematical models (2nd ed.), Oxford: Clarendon Press, 1961, pp. 78–79 .
23. ↑ Grünbaum, B.; Shephard, G.C. (1969), "Convex polytopes", Bulletin of the London Mathematical Society 1 (3): 257–300, doi:10.1112/blms/1.3.257, http://www.wias-berlin.de/people/si/course/files/convex_polytopes-survey-Gruenbaum.pdf, retrieved 2017-02-21 . See in particular the bottom of page 260.
24. ↑ Cite error: Invalid `<ref>` tag; no text was provided for refs named `acoptic`
25. ↑ Coxeter, H. S. M. (1947), Regular Polytopes, Methuen, p. 16
26. ↑ Barnette, David (1973), "A proof of the lower bound conjecture for convex polytopes", Pacific Journal of Mathematics 46 (2): 349–354, doi:10.2140/pjm.1973.46.349, https://projecteuclid.org/euclid.pjm/1102946311
27. ↑ Luotoniemi, Taneli (2017), "Crooked houses: Visualizing the polychora with hyperbolic patchwork", in Swart, David; Séquin, Carlo H.; Fenyvesi, Kristóf, Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture, Phoenix, Arizona: Tessellations Publishing, pp. 17–24, ISBN 978-1-938664-22-9, https://archive.bridgesmathart.org/2017/bridges2017-17.html
28. ↑ "The polytopes with regular-prismatic vertex figures", Philosophical Transactions of the Royal Society of London, Series A (The Royal Society) 229 (670–680): 329–425, January 1930, doi:10.1098/rsta.1930.0009, Bibcode: 1930RSPTA.229..329C
29. ↑ "Example 44.2.3, the "punched-in icosahedron"", Geometry: Euclid and beyond, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2000, p. 442, doi:10.1007/978-0-387-22676-7, ISBN 0-387-98650-2
30. ↑ Posamentier, Alfred S.; Bannister, Robert L. (2014), Geometry, Its Elements and Structure: Second Edition (2nd ed.), Dover Publications, p. 543, ISBN 978-0-486-49267-4, https://books.google.com/books?id=XktMBAAAQBAJ&pg=PA543
31. ↑ Peterson, Dave (September 17, 2020). "Counting Diagonals of a Polyhedron". https://www.themathdoctors.org/counting-diagonals-of-a-polyhedron/.
32. ↑ "On indecomposable polyhedra", American Mathematical Monthly 55 (7): 411–413, 1948, doi:10.2307/2306130
33. ↑ Goldman, Ronald N. (1991), "Chapter IV.1: Area of planar polygons and volume of polyhedra", in Arvo, James, Graphic Gems Package: Graphics Gems II, Academic Press, pp. 170–171
34. ↑ Büeler, B.; Enge, A.; Fukuda, K. (2000), "Exact Volume Computation for Polytopes: A Practical Study", Polytopes — Combinatorics and Computation, pp. 131–154, doi:10.1007/978-3-0348-8438-9_6, ISBN 978-3-7643-6351-2
35. ↑ Sydler, J.-P. (1965), "Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions" (in fr), Comment. Math. Helv. 40: 43–80, doi:10.1007/bf02564364, https://eudml.org/doc/139296
36. ↑ Hazewinkel, Michiel, ed. (2001), "Dehn invariant", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Dehn_invariant&oldid=35803
37. ↑ Debrunner, Hans E. (1980), "Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln" (in de), Archiv der Mathematik 35 (6): 583–587, doi:10.1007/BF01235384 .
38. ↑ Alexandrov, Victor (2010), "The Dehn invariants of the Bricard octahedra", Journal of Geometry 99 (1–2): 1–13, doi:10.1007/s00022-011-0061-7 .
39. ↑ Polyhedron Models, Cambridge University Press, 1971
40. ↑ Mal'cev, A. V. (2012), Mathematics: Its Content, Methods and Meaning, Dover Publications, p. Volume III, 278, ISBN 978-0-486-15787-0, https://books.google.com/books?id=2cPDAgAAQBAJ&pg=RA2-PA278
41. ↑ McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette 74 (469): 243–256, doi:10.2307/3619822 See p. 247.
42. ↑ "Isogonal Prismatoids", Discrete & Computational Geometry 18 (1): 13-52, 1997, doi:10.1007/PL00009307
43. ↑ 43.0 43.1 Senechal, Marjorie (1989), "A Brief Introduction to Tilings", in Jarić, Marko, Introduction to the Mathematics of Quasicrystals, Academic Press, p. 12, https://books.google.com/books?id=OToVjZW9CKMC&pg=PA12
44. ↑ Powell, R. C. (2010), Symmetry, Group Theory, and the Physical Properties of Crystals, Lecture Notes in Physics, 824, Springer, p. 27, doi:10.1007/978-1-4419-7598-0, ISBN 978-1-441-97598-0, https://books.google.com/books?id=ojq5BQAAQBAJ&pg=PA27
45. ↑ Flusser, J.; Suk, T.; Zitofa, B. (2017), 2D and 3D Image Analysis by Moments, John Wiley & Sons, p. 127-128, ISBN 978-1-119-03935-8, https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA128
46. ↑ 46.0 46.1 46.2 46.3 Flusser, Suk & Zitofa (2017), p. 126.
47. ↑ Hergert, W.; Geilhufe, M. (2018), Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica, John Wiley & Sons, ISBN 978-3-527-41300-3, https://books.google.com/books?id=6mvpCgAAQBAJ&pg=PA56
48. ↑ Flusser, Suk & Zitofa (2017), p. 125.
49. ↑ Hergert & Geilhufe (2018), p. 57.
50. ↑ Buldygin, V. V.; Kharazishvili, A. B. (2000), Geometric Aspects of Probability Theory and Mathematical Statistics, Springer, p. 2, doi:10.1007/978-94-017-1687-1, ISBN 978-94-017-1687-1, https://books.google.com/books?id=mGD9CAAAQBAJ&pg=PA2
51. ↑ Kern, William F.; Bland, James R. (1938), Solid Mensuration with proofs, p. 75, https://books.google.com/books?id=Y6cAAAAAMAAJ .
52. ↑ Cromwell (1997), p. 51-52. harv error: no target: CITEREFCromwell1997 (help)
53. ↑ "An enduring error", Elemente der Mathematik 64 (3): 89–101, 2009, doi:10.4171/EM/120, https://digital.lib.washington.edu/dspace/bitstream/handle/1773/4592/An_enduring_error.pdf . Reprinted in Pitici, Mircea, ed. (2011), The Best Writing on Mathematics 2010, Princeton University Press, pp. 18–31 .
54. ↑ Diudea, M. V. (2018), Multi-shell Polyhedral Clusters, Carbon Materials: Chemistry and Physics, 10, Springer, p. 39, doi:10.1007/978-3-319-64123-2, ISBN 978-3-319-64123-2, https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 .
55. ↑ Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8 .
56. ↑ "Deltahedra", Mathematical Gazette 36 (318): 263–266, 1952, doi:10.2307/3608204 .
57. ↑ 57.0 57.1 Kaplan, Craig S.; Hart, George W. (27–29 July 2001), "Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons", Bridges: Mathematical Connections in Art, Music, and Science, Winfield, Kansas, United States of America, https://archive.bridgesmathart.org/2001/bridges2001-21.html#gsc.tab=0 .
58. ↑ Goldberg, Michael (1937), "A class of multi-symmetric polyhedra", Tohoku Mathematical Journal 43: 104–108, https://www.jstage.jst.go.jp/article/tmj1911/43/0/43_0_104/_article .
59. ↑ Schein, S.; Gayed, J. M. (2014), "Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses", Proceedings of the National Academy of Sciences 111 (8): 2920–2925, doi:10.1073/pnas.1310939111, ISSN 0027-8424, PMID 24516137, Bibcode: 2014PNAS..111.2920S .
60. ↑ Hartshorne (2000), p. 464. harv error: no target: CITEREFHartshorne2000 (help)
61. ↑ Timofeenko, A. V. (2010), "Junction of Non-composite Polyhedra", St. Petersburg Mathematical Journal 21 (3): 483–512, doi:10.1090/S1061-0022-10-01105-2, https://www.ams.org/journals/spmj/2010-21-03/S1061-0022-10-01105-2/S1061-0022-10-01105-2.pdf .
62. ↑ Schramm, Oded (1992-12-01), "How to cage an egg" (in en), Inventiones Mathematicae 107 (1): 543–560, doi:10.1007/BF01231901, ISSN 1432-1297, Bibcode: 1992InMat.107..543S .
63. ↑ Coxeter, H.S.M. (1985), "A special book review: M.C. Escher: His life and complete graphic work", The Mathematical Intelligencer 7 (1): 59–69, doi:10.1007/BF03023010 Coxeter's analysis of Stars is on pp. 61–62.
64. ↑ Grünbaum (2003), pp. 235–244.
65. ↑ 65.0 65.1 Bridge, N.J. (1974), "Facetting the dodecahedron", Acta Crystallographica A30 (4): 548–552, doi:10.1107/S0567739474001306, Bibcode: 1974AcCrA..30..548B .
66. ↑ Inchbald, G. (2006), "Facetting diagrams", The Mathematical Gazette 90 (518): 253–261, doi:10.1017/S0025557200179653 .
67. ↑ Malloy, Kaoime (2014), The Art of Theatrical Design: Elements of Visual Composition, Methods, and Practice, Taylor & Francis, p. 203, ISBN 978-1-317-69426-7, https://books.google.com/books?id=mW89BAAAQBAJ&pg=PT203 .
68. ↑ Si, Hang; Goerigk, Nadja (2016), "On Tetrahedralisations of Reduced Chazelle Polyhedra with Interior Steiner Points", Procedia Engineering 163: 33-45, doi:10.1016/j.proeng.2016.11.013
69. ↑ Unsolved Problems in Geometry: Unsolved Problems in Intuitive Mathematics, 1991, pp. 76, doi:10.1007/978-1-4612-0963-8, ISBN 978-1-4612-0963-8, https://books.google.com/books?id=rdDTBwAAQBAJ&pg=PA76
70. ↑ Hargittai, Istvan (November 1998), "Symmetry in crystallography", Acta Crystallographica, Section A 54 (6): 697–706, doi:10.1107/s0108767398006709, Bibcode: 1998AcCrA..54..697H, https://istvan.hargittai.com/files/10-Hargittai-SymmetryInCrystallography.pdf
71. ↑ Lekkerkerker, C. G.; Gruber, Pascale (1987), Geometry of Numbers, North-Holland, p. 166, ISBN 978-0-08-096023-4, https://books.google.com/books?id=eYeWNW8mzXMC&pg=PA166
72. ↑ "Polytopes that fill ℝn and scissors congruence", Discrete & Computational Geometry 13 (3–4): 573–583, 1995, doi:10.1007/BF02574064
73. ↑ Posamentier, Alfred S.; Maresch, Guenter; Thaller, Bernd; Spreitzer, Christian; Geretschlager, Robert; Stuhlpfarrer, David; Dorner, Christian (2022), Geometry In Our Three-dimensional World, World Scientific, pp. 232-234, ISBN 9789811237126, https://books.google.com/books?id=DGxYEAAAQBAJ&pg=PA232
74. ↑ "23.2 Flexible polyhedra", Geometric Folding Algorithms: Linkages, origami, polyhedra, Cambridge University Press, Cambridge, 2007, pp. 345–348, doi:10.1017/CBO9780511735172, ISBN 978-0-521-85757-4 .
75. ↑ Lijingjiao, I.; Tachi, T.; Guest, S. (2015), "Optimizing the Steffen flexible polyhedron", International Association for Shell and Spatial Structures, doi:10.17863/CAM.26518 .
76. ↑ Fuchs, Dmitry (2007), Mathematical Omnibus: Thirty lectures on classic mathematics, Providence, RI: American Mathematical Society, p. 347, doi:10.1090/mbk/046, ISBN 978-0-8218-4316-1, https://books.google.com/books?id=IiG9AwAAQBAJ&pg=PA347 .
77. ↑ Stachel, Hellmuth (2015), "Flexible Polyhedral Surfaces with Two Flat Poses", Symmetry 7 (2): 774–787, doi:10.3390/sym7020774, Bibcode: 2015Symm....7..774S .
78. ↑ Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997, p. 128, ISBN 0-691-08304-5, https://books.google.com/books?id=9kkuP3lsEFQC&pg=PA128
79. ↑ Enumerative Combinatorics, Volume I (1 ed.), Cambridge University Press, 1997, pp. 235–239, ISBN 978-0-521-66351-9
80. ↑ Kołodziejczyk, Krzysztof (1996), "An "odd" formula for the volume of three-dimensional lattice polyhedra", Geometriae Dedicata 61 (3): 271–278, doi:10.1007/BF00150027
81. ↑ Cox, David A. (2011), Toric varieties, John B. Little, Henry K. Schenck, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4819-7, OCLC 698027255
82. ↑ Stanley, Richard P. (1996), Combinatorics and commutative algebra (2nd ed.), Boston: Birkhäuser, ISBN 0-8176-3836-9, OCLC 33080168
83. ↑ "Zonohedra and generalized zonohedra", American Mathematical Monthly 99 (2): 108–111, 1992, doi:10.2307/2324178 .
84. ↑ Senechal, Marjorie, ed. (2013), "Dürer's Problem", Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, Springer, p. 86, doi:10.1007/978-0-387-92714-5, ISBN 978-0-387-92714-5, https://books.google.com/books?id=kZtCAAAAQBAJ&pg=PA86 .
85. ↑ "Orthogonal icosahedra", Nordisk Matematisk Tidskrift 15 (2): 90–96, 1967 .
86. ↑ "Unfolding orthogonal polyhedra", Surveys on discrete and computational geometry, Contemp. Math., 453, Amer. Math. Soc., Providence, RI, 2008, pp. 307–317, doi:10.1090/conm/453/08805, ISBN 978-0-8218-4239-3 .
87. ↑ "Mathematical Games: Is it possible to visualize a four-dimensional figure?", Scientific American 215 (5): 138–143, November 1966, doi:10.1038/scientificamerican1166-138
88. ↑ Regular Complex Polytopes, Cambridge: Cambridge University Press, 1974
89. ↑ Popko, Edward S. (2012), Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere, CRC Press, p. 463, ISBN 978-1-4665-0430-1, https://books.google.com/books?id=HjTSBQAAQBAJ&pg=PA463, ""A hosohedron is only possible on a sphere"" .
90. ↑ Kraynik, A.M.; Reinelt, D.A. (2007), "Foams, Microrheology of", in Mortensen, Andreas, Concise Encyclopedia of Composite Materials (2nd ed.), Elsevier, pp. 402–407 . See in particular p. 403: "foams consist of polyhedral gas bubbles ... each face on a polyhedron is a minimal surface with uniform mean curvature ... no face can be a flat polygon with straight edges".
91. ↑ Pearce, P. (1978), "14 Saddle polyhedra and continuous surfaces as environmental structures", Structure in nature is a strategy for design, MIT Press, p. 224, ISBN 978-0-262-66045-7, https://books.google.com/books?id=sfc2OEuE8oQC&pg=PA224 .
92. ↑ Grötschel, Martin; Lovász, László; Schrijver, Alexander (1993), Geometric algorithms and combinatorial optimization, Algorithms and Combinatorics, 2 (2nd ed.), Berlin: Springer-Verlag, p. 9, doi:10.1007/978-3-642-78240-4, ISBN 978-3-642-78242-8, https://books.google.com/books?id=hWvmCAAAQBAJ&pg=PA9
93. ↑ Kitchen, K. A. (October 1991), "The chronology of ancient Egypt", World Archaeology 23 (2): 201–208, doi:10.1080/00438243.1991.9980172
94. ↑ Gunn, Battiscombe; Peet, T. Eric (May 1929), "Four Geometrical Problems from the Moscow Mathematical Papyrus", The Journal of Egyptian Archaeology 15 (1): 167–185, doi:10.1177/030751332901500130
95. ↑ Friberg, Jöran (2000), "Mathematics at Ur in the Old Babylonian Period", Revue d'Assyriologie et d'archéologie orientale 94 (2): 97–188
96. ↑ Sparavigna, Amelia Carolina (2012), An Etruscan dodecahedron
97. ↑ Eves, Howard (January 1969), "A geometry capsule concerning the five platonic solids", The Mathematics Teacher 62 (1): 42–44, doi:10.5951/mt.62.1.0042
98. ↑ Field, J. V. (1997), "Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler", Archive for History of Exact Sciences 50 (3–4): 241–289, doi:10.1007/BF00374595
99. ↑ "Cracking bones and numbers: solving the enigma of numerical sequences on ancient Chinese artifacts", Archive for History of Exact Sciences 74 (4): 313–343, December 2019, doi:10.1007/s00407-019-00245-9
100. ↑ van der Waerden, B. L. (1983), "Chapter 7: Liu Hui and Aryabhata", Geometry and Algebra in Ancient Civilizations, Springer, pp. 192–217, doi:10.1007/978-3-642-61779-9_7
101. ↑ Knorr, Wilbur (1983), "On the transmission of geometry from Greek into Arabic", Historia Mathematica 10 (1): 71–78, doi:10.1016/0315-0860(83)90034-4
102. ↑ Rashed, Roshdi (2009), "Thābit ibn Qurra et l'art de la mesure", in Rashed, Roshdi (in fr), Thābit ibn Qurra: Science and Philosophy in Ninth-Century Baghdad, Scientia Graeco-Arabica, 4, Walter de Gruyter, pp. 173–175, ISBN 9783110220780, https://books.google.com/books?id=V5PTZi77YxwC&pg=PA173
103. ↑ Hisarligil, Hakan; Hisarligil, Beyhan Bolak (December 2017), "The geometry of cuboctahedra in medieval art in Anatolia", Nexus Network Journal 20 (1): 125–152, doi:10.1007/s00004-017-0363-7
104. ↑ Gamba, Enrico (2012), "The mathematical ideas of Luca Pacioli depicted by Iacopo de' Barbari in the Doppio ritratto", in Emmer, Michele, Imagine Math, Springer, pp. 267–271, doi:10.1007/978-88-470-2427-4_25, ISBN 978-88-470-2427-4
105. ↑ 105.0 105.1 Andrews, Noam (2022), The Polyhedrists: Art and Geometry in the Long Sixteenth Century, MIT Press, ISBN 9780262046640
106. ↑ Calvo-López, José; Alonso-Rodríguez, Miguel Ángel (February 2010), "Perspective versus stereotomy: From Quattrocento polyhedral rings to sixteenth-century Spanish torus vaults", Nexus Network Journal 12 (1): 75–111, doi:10.1007/s00004-010-0018-4
107. ↑ Field, J. V. (1997), "Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler", Archive for History of Exact Sciences 50 (3–4): 241–289, doi:10.1007/BF00374595
108. ↑ Montebelli, Vico (2015), "Luca Pacioli and perspective (part I)", Lettera Matematica 3 (3): 135–141, doi:10.1007/s40329-015-0090-4
109. ↑ Ghomi, Mohammad (2018), "Dürer's unfolding problem for convex polyhedra", Notices of the American Mathematical Society 65 (1): 25–27, doi:10.1090/noti1609, https://www.ams.org/publications/journals/notices/201801/rnoti-p25.pdf
110. ↑ Saffaro, Lucio (1992), "Cosmoids, fullerenes and continuous polygons", in Taliani, C.; Ruani, G.; Zamboni, R., Fullerenes: Status and Perspectives, Proceedings of the 1st Italian Workshop, Bologna, Italy, 6–7 February, Singapore: World Scientific, pp. 55–64, https://books.google.com/books?id=dOk7DwAAQBAJ&pg=PA55
111. ↑ Field, J. V. (1979), "Kepler's star polyhedra", Vistas in Astronomy 23 (2): 109–141, doi:10.1016/0083-6656(79)90001-1, Bibcode: 1979VA.....23..109F
112. ↑ Friedman, Michael (2018), A History of Folding in Mathematics: Mathematizing the Margins, Science Networks. Historical Studies, 59, Birkhäuser, p. 71, doi:10.1007/978-3-319-72487-4, ISBN 978-3-319-72486-7
113. ↑ Federico, Pasquale Joseph (1982), Descartes on Polyhedra: A Study of the "De solidorum elementis", Sources in the History of Mathematics and Physical Sciences, 4, Springer-Verlag, ISBN 0-387-90760-2
114. ↑ Francese, Christopher; Richeson, David (2007), "The flaw in Euler's proof of his polyhedral formula", The American Mathematical Monthly 114 (4): 286–296, doi:10.1080/00029890.2007.11920417
115. ↑ Alexanderson, Gerald L. (2006), "About the cover: Euler and Königsberg's bridges: a historical view", American Mathematical Society, New Series 43 (4): 567–573, doi:10.1090/S0273-0979-06-01130-X
116. ↑ Eckmann, Beno (2006), "The Euler characteristic – a few highlights in its long history", Mathematical Survey Lectures 1943–2004, Springer, pp. 177–188, doi:10.1007/978-3-540-33791-1_15, ISBN 978-3-540-33791-1
117. ↑ Grünbaum, Branko (1994), "Regular polyhedra", in Grattan-Guinness, I., Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, 2, Routledge, pp. 866–876, ISBN 0-415-03785-9, https://books.google.com/books?id=ZptYDwAAQBAJpg
118. ↑ Malkevitch, Joseph (2018), "Regular-faced polyhedra: remembering Norman Johnson", AMS Feature column (American Mathematical Society), https://www.ams.org/publicoutreach/feature-column/fc-2018-01, retrieved 2023-05-27
119. ↑ Coxeter (1947), pp. 141–143.
120. ↑ Austin, David (November 2013), "Fedorov's five parallelohedra", AMS Feature Column (American Mathematical Society), https://www.ams.org/samplings/feature-column/fc-2013-11
121. ↑ "On Hilbert's third problem", The Mathematical Gazette 86 (506): 241–247, July 2002, doi:10.2307/3621846
122. ↑ "Graphs of polyhedra; polyhedra as graphs", Discrete Mathematics 307 (3–5): 445–463, 2007, doi:10.1016/j.disc.2005.09.037
123. ↑ Hagino, K.; Onuma, R.; Kawachi, M.; Horiguchi, T. (2013), "Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-A in Braarudosphaera bigelowii (Prymnesiophyceae)", PLOS ONE 8 (12), doi:10.1371/journal.pone.0081749, PMID 24324722, Bibcode: 2013PLoSO...881749H
124. ↑ Haeckel, E (1904), Kunstformen der Natur . Available as Haeckel, E. Art forms in nature, Prestel USA (1998), ISBN 3-7913-1990-6. Online version at Kurt Stüber's Biolib (in German)
125. ↑ "Myoviridae", Virus Taxonomy, Elsevier, 2012, pp. 46–62, doi:10.1016/b978-0-12-384684-6.00002-1, ISBN 9780123846846
126. ↑ Strauss, James H.; Strauss, Ellen G. (2008), "The Structure of Viruses", Viruses and Human Disease, Elsevier, pp. 35–62, doi:10.1016/b978-0-12-373741-0.50005-2, ISBN 9780123737410
127. ↑ Cromwell 1997, p. 7. sfn error: no target: CITEREFCromwell1997 (help)
128. ↑ Lim, Seng Poh; Haron, Habibollah (March 2012), "Surface reconstruction techniques: a review", Artificial Intelligence Review 42 (1): 59–78, doi:10.1007/s10462-012-9329-z
129. ↑ "The discrete geodesic problem", SIAM Journal on Computing 16 (4): 647–668, 1987, doi:10.1137/0216045
130. ↑ "Global visibility algorithms for illumination computations", Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 1993, Anaheim, CA, USA, August 2–6, 1993, Association for Computing Machinery, 1993, pp. 239–246, doi:10.1145/166117.166148, ISBN 0-89791-601-8
131. ↑ Cherchi, Gianmarco (February 2019), Polycube Optimization and Applications: From the Digital World to Manufacturing, University of Cagliari
132. ↑ Rote, Günter (2011), "Realizing planar graphs as convex polytopes", in van Kreveld, Marc J.; Speckmann, Bettina, Graph Drawing – 19th International Symposium, GD 2011, Eindhoven, The Netherlands, September 21–23, 2011, Revised Selected Papers, Lecture Notes in Computer Science, 7034, Springer, pp. 238–241, doi:10.1007/978-3-642-25878-7_23, ISBN 978-3-642-25877-0
133. ↑ Demaine, Erik; O'Rourke, Joseph (2007), Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press

## External links

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### General theory

* Weisstein, Eric W.. "Polyhedron". http://mathworld.wolfram.com/Polyhedron.html.
* Polyhedra Pages
* Uniform Solution for Uniform Polyhedra by Dr. Zvi Har'El
* Symmetry, Crystals and Polyhedra

### Lists and databases of polyhedra

* Virtual Reality Polyhedra – The Encyclopedia of Polyhedra.
* Electronic Geometry Models – Contains a peer reviewed selection of polyhedra with unusual properties.
* Polyhedron Models – Virtual polyhedra.
* Paper Models of Uniform (and other) Polyhedra

### Free software

* A Plethora of Polyhedra – An interactive and free collection of polyhedra in Java. Features includes nets, planar sections, duals, truncations and stellations of more than 300 polyhedra.
* Hyperspace Star Polytope Slicer – Explorer java applet, includes a variety of 3d viewer options.
* openSCAD – Free cross-platform software for programmers. Polyhedra are just one of the things you can model. The openSCAD User Manual is also available.
* OpenVolumeMesh – An open source cross-platform C++ library for handling polyhedral meshes. Developed by the Aachen Computer Graphics Group, RWTH Aachen University.
* Polyhedronisme – Web-based tool for generating polyhedra models using Conway Polyhedron Notation. Models can be exported as 2D PNG images, or as 3D OBJ or VRML2 files.

### Resources for making physical models

* Paper Models of Polyhedra Free nets of polyhedra.
* Simple instructions for building over 30 paper polyhedra
* Polyhedra plaited with paper strips – Polyhedra models constructed without use of glue.

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Polyhedra

Listed by number of faces

1–10 faces|

* Monohedron
* Dihedron
* Trihedron
* Tetrahedron
* Pentahedron
* Hexahedron
* Heptahedron
* Octahedron
* Enneahedron
* Decahedron

11–20 faces|

* Hendecahedron
* Dodecahedron
* Tridecahedron
* Tetradecahedron
* Pentadecahedron
* Hexadecahedron
* Heptadecahedron
* Octadecahedron
* Enneadecahedron
* Icosahedron

Others|

* Triacontahedron (30)
* Hexecontahedron (60)
* Enneacontahedron (90)
* Hectotriadiohedron
* Apeirohedron

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Convex polyhedra

Platonic solids (regular)|

* tetrahedron
* cube
* octahedron
* dodecahedron
* icosahedron

Archimedean solids
(semiregular or uniform)

* truncated tetrahedron
* cuboctahedron
* truncated cube
* truncated octahedron
* rhombicuboctahedron
* truncated cuboctahedron
* snub cube
* icosidodecahedron
* truncated dodecahedron
* truncated icosahedron
* rhombicosidodecahedron
* truncated icosidodecahedron
* snub dodecahedron

Catalan solids
(duals of Archimedean)

* triakis tetrahedron
* rhombic dodecahedron
* triakis octahedron
* tetrakis hexahedron
* deltoidal icositetrahedron
* disdyakis dodecahedron
* pentagonal icositetrahedron
* rhombic triacontahedron
* triakis icosahedron
* pentakis dodecahedron
* deltoidal hexecontahedron
* disdyakis triacontahedron
* pentagonal hexecontahedron

Dihedral regular|

* dihedron
* hosohedron

Dihedral uniform|

* prisms
* antiprisms

duals:|

* bipyramids
* trapezohedra

Dihedral others|

* pyramids
* truncated trapezohedra
* gyroelongated bipyramid
* cupola
* bicupola
* pyramidal frusta
* bifrustum
* rotunda
* birotunda
* prismatoid
* scutoid

Degenerate polyhedra are in italics.

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