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129-408: A convex polyhedron is a polyhedron that bounds a convex set . Every convex polyhedron can be constructed as the convex hull of its vertices, and for every finite set of points, not all on the same plane, the convex hull is a convex polyhedron. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a generalization of a 2-dimensional polygon and a 3-dimensional specialization of

258-416: A 2 3 ≈ 1.732 a 2 . {\displaystyle A=4\cdot \left({\frac {\sqrt {3}}{4}}a^{2}\right)=a^{2}{\sqrt {3}}\approx 1.732a^{2}.} The height of a regular tetrahedron is 6 3 a {\textstyle {\frac {\sqrt {6}}{3}}a} . The volume of a regular tetrahedron can be ascertained similarly as the other pyramids, one-third of

387-731: A 4 + d 1 4 + d 2 4 + d 3 4 + d 4 4 ) = ( a 2 + d 1 2 + d 2 2 + d 3 2 + d 4 2 ) 2 . {\displaystyle {\begin{aligned}{\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+{\frac {16R^{4}}{9}}&=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+{\frac {2R^{2}}{3}}\right)^{2},\\4\left(a^{4}+d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}\right)&=\left(a^{2}+d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}\right)^{2}.\end{aligned}}} With respect to

516-484: A , r = 1 3 R = a 24 , r M = r R = a 8 , r E = a 6 . {\displaystyle {\begin{aligned}R={\frac {\sqrt {6}}{4}}a,&\qquad r={\frac {1}{3}}R={\frac {a}{\sqrt {24}}},\\r_{\mathrm {M} }={\sqrt {rR}}={\frac {a}{\sqrt {8}}},&\qquad r_{\mathrm {E} }={\frac {a}{\sqrt {6}}}.\end{aligned}}} For

645-417: A stellated octahedron or stella octangula . Its interior is an octahedron , and correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e., rectifying the tetrahedron). The tetrahedron is yet related to another two solids: By truncation the tetrahedron becomes a truncated tetrahedron . The dual of this solid

774-425: A crescent shape, is not convex. The boundary of a convex set in the plane is always a convex curve . The intersection of all the convex sets that contain a given subset A of Euclidean space is called the convex hull of A . It is the smallest convex set containing A . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above

903-419: A cube can be grouped into two groups of four, each forming a regular tetrahedron, showing one of the two tetrahedra in the cube. The symmetries of a regular tetrahedron correspond to half of those of a cube: those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid not mapped to itself by point inversion . The regular tetrahedron has 24 isometries, forming

1032-515: A locally convex topological vector space such that rec ⁡ A ∩ rec ⁡ B {\displaystyle \operatorname {rec} A\cap \operatorname {rec} B} is a linear subspace. If A or B is locally compact then A  −  B is closed. The notion of convexity in the Euclidean space may be generalized by modifying the definition in some or other aspects. The common name "generalized convexity"

1161-469: A non-convex set . A polygon that is not a convex polygon is sometimes called a concave polygon , and some sources more generally use the term concave set to mean a non-convex set, but most authorities prohibit this usage. The complement of a convex set, such as the epigraph of a concave function , is sometimes called a reverse convex set , especially in the context of mathematical optimization . Given r points u 1 , ..., u r in

1290-534: A polytope , a more general concept in any number of dimensions . Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others, and there is no universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as

1419-634: A recession cone of a non-empty convex subset S , defined as: rec ⁡ S = { x ∈ X : x + S ⊆ S } , {\displaystyle \operatorname {rec} S=\left\{x\in X\,:\,x+S\subseteq S\right\},} where this set is a convex cone containing 0 ∈ X {\displaystyle 0\in X} and satisfying S + rec ⁡ S = S {\displaystyle S+\operatorname {rec} S=S} . Note that if S

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1548-463: A spherical tiling (of spherical triangles ), and projected onto the plane via a stereographic projection . This projection is conformal , preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. Regular tetrahedra can be stacked face-to-face in a chiral aperiodic chain called the Boerdijk–Coxeter helix . In four dimensions , all

1677-400: A convex hull extends naturally to geometries which are not Euclidean by defining a geodesically convex set to be one that contains the geodesics joining any two points in the set. Convexity can be extended for a totally ordered set X endowed with the order topology . Let Y ⊆ X . The subspace Y is a convex set if for each pair of points a , b in Y such that a ≤ b ,

1806-436: A convex polyhedron can be obtained by the process of polar reciprocation . 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. Abstract polyhedra also have duals, obtained by reversing the partial order defining the polyhedron to obtain its dual or opposite order . These have

1935-501: A convex polyhedron, or more generally any simply connected polyhedron with surface 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. All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even Euler characteristic may or may not be orientable. For example,

2064-445: A convex set S , and r nonnegative numbers λ 1 , ..., λ r such that λ 1 + ... + λ r = 1 , the affine combination ∑ k = 1 r λ k u k {\displaystyle \sum _{k=1}^{r}\lambda _{k}u_{k}} belongs to S . As the definition of a convex set is the case r = 2 , this property characterizes convex sets. Such an affine combination

2193-429: A convex set in Euclidean spaces can be generalized in several ways by modifying its definition, for instance by restricting the line segments that such a set is required to contain. Let S be a vector space or an affine space over the real numbers , or, more generally, over some ordered field (this includes Euclidean spaces, which are affine spaces). A subset C of S is convex if, for all x and y in C ,

2322-1229: A cube into three parts. Its dihedral angle —the angle between two planar—and its angle between lines from the center of a regular tetrahedron between two vertices is respectively: arccos ⁡ ( 1 3 ) = arctan ⁡ ( 2 2 ) ≈ 70.529 ∘ , arccos ⁡ ( − 1 3 ) = 2 arctan ⁡ ( 2 ) ≈ 109.471 ∘ . {\displaystyle {\begin{aligned}\arccos \left({\frac {1}{3}}\right)&=\arctan \left(2{\sqrt {2}}\right)\approx 70.529^{\circ },\\\arccos \left(-{\frac {1}{3}}\right)&=2\arctan \left({\sqrt {2}}\right)\approx 109.471^{\circ }.\end{aligned}}} The radii of its circumsphere R {\displaystyle R} , insphere r {\displaystyle r} , midsphere r M {\displaystyle r_{\mathrm {M} }} , and exsphere r E {\displaystyle r_{\mathrm {E} }} are: R = 6 4

2451-412: A face, and one centered on an edge. The first corresponds to the A 2 Coxeter plane . The two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a rectangle . When the intersecting plane is near one of the edges the rectangle is long and skinny. When halfway between the two edges

2580-418: A list that includes many of these formulas.) Volumes of more complicated polyhedra may not have simple formulas. Volumes of such polyhedra may be computed by subdividing the polyhedron into smaller pieces (for example, by triangulation ). For example, the volume of a regular polyhedron can be computed by dividing it into congruent pyramids , with each pyramid having a face of the polyhedron as its base and

2709-423: A polyhedron into a single number χ {\displaystyle \chi } defined by the formula 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

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2838-399: A polyhedron is called its symmetry group . All the elements that can be superimposed on each other by symmetries are said to form a symmetry orbit . For example, all the faces of a cube lie in one orbit, while all the edges lie in another. If all the elements of a given dimension, say all the faces, lie in the same orbit, the figure is said to be transitive on that orbit. For example, a cube

2967-410: A polyhedron that can be constructed by attaching more elementary polyhedrons. For example, triaugmented triangular prism is a composite polyhedron 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 elementary. A midsphere of a convex polyhedron is a sphere tangent to every edge of

3096-408: A polyhedron to create new faces—or facets—without creating any new vertices). A facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face . The stellation and faceting are inverse or reciprocal processes: the dual of some stellation is a faceting of the dual to the original polyhedron. Polyhedra may be classified and are often named according to

3225-422: A polyhedron, an intermediate sphere in radius between the insphere and circumsphere , for polyhedra for which all three of these spheres exist. Every convex polyhedron is combinatorially equivalent to a canonical polyhedron , a polyhedron that has a midsphere whose center coincides with the centroid of the polyhedron. The shape of the canonical polyhedron (but not its scale or position) is uniquely determined by

3354-768: A regular tetrahedron with side length a {\displaystyle a} , the radius of its circumscribed sphere R {\displaystyle R} , and distances d i {\displaystyle d_{i}} from an arbitrary point in 3-space to its four vertices, it is: d 1 4 + d 2 4 + d 3 4 + d 4 4 4 + 16 R 4 9 = ( d 1 2 + d 2 2 + d 3 2 + d 4 2 4 + 2 R 2 3 ) 2 , 4 (

3483-591: A right triangle with edges 4 3 {\displaystyle {\sqrt {\tfrac {4}{3}}}} , 3 2 {\displaystyle {\sqrt {\tfrac {3}{2}}}} , 1 6 {\displaystyle {\sqrt {\tfrac {1}{6}}}} . A space-filling tetrahedron packs with directly congruent or enantiomorphous ( mirror image ) copies of itself to tile space. The cube can be dissected into six 3-orthoschemes, three left-handed and three right-handed (one of each at each cube face), and cubes can fill space, so

3612-519: A set X , a convexity over X is a collection 𝒞 of subsets of X satisfying the following axioms: The elements of 𝒞 are called convex sets and the pair ( X , 𝒞 ) is called a convexity space . For the ordinary convexity, the first two axioms hold, and the third one is trivial. For an alternative definition of abstract convexity, more suited to discrete geometry , see the convex geometries associated with antimatroids . Convexity can be generalised as an abstract algebraic structure:

3741-431: 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

3870-533: A single point. (The Coxeter-Dynkin diagram of the generated polyhedron contains three nodes representing the three mirrors. The dihedral angle between each pair of mirrors is encoded in the diagram, as well as the location of a single generating point which is multiplied by mirror reflections into the vertices of the polyhedron.) Among the Goursat tetrahedra which generate 3-dimensional honeycombs we can recognize an orthoscheme (the characteristic tetrahedron of

3999-424: A single symmetry orbit: Some classes of polyhedra have only a single main axis of symmetry. These include the pyramids , bipyramids , trapezohedra , cupolae , as well as the semiregular prisms and antiprisms. Regular polyhedra are the most highly symmetrical. Altogether there are nine regular polyhedra: five convex and four star polyhedra. The five convex examples have been known since antiquity and are called

Polyhedron - Misplaced Pages Continue

4128-467: A solid, whether they describe it as a surface, or whether they describe it more abstractly based on its incidence geometry . 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

4257-410: A space is convex if it is possible to take convex combinations of points. Regular tetrahedron In geometry , a tetrahedron ( pl. : tetrahedra or tetrahedrons ), also known as a triangular pyramid , is a polyhedron composed of four triangular faces , six straight edges , and four vertices . The tetrahedron is the simplest of all the ordinary convex polyhedra . The tetrahedron

4386-399: A tetrahedron are perpendicular , then it is called an orthocentric tetrahedron . When only one pair of opposite edges are perpendicular, it is called a semi-orthocentric tetrahedron . In a trirectangular tetrahedron the three face angles at one vertex are right angles , as at the corner of a cube. An isodynamic tetrahedron is one in which the cevians that join the vertices to

4515-633: A tetrahedron with edge-length 2 2 {\displaystyle 2{\sqrt {2}}} , centered at the origin. For the other tetrahedron (which is dual to the first), reverse all the signs. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3- demicube , a polyhedron that is by alternating a cube. This form has Coxeter diagram [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] and Schläfli symbol h { 4 , 3 } {\displaystyle \mathrm {h} \{4,3\}} . The vertices of

4644-408: A topological vector space and C ⊆ X {\displaystyle C\subseteq X} be convex. Every subset A of the vector space is contained within a smallest convex set (called the convex hull of A ), namely the intersection of all convex sets containing A . The convex-hull operator Conv() has the characteristic properties of a hull operator : The convex-hull operation

4773-399: Is strictly convex if every point on the line segment connecting x and y other than the endpoints is inside the topological interior of C . A closed convex subset is strictly convex if and only if every one of its boundary points is an extreme point . A set C is absolutely convex if it is convex and balanced . The convex subsets of R (the set of real numbers) are

4902-410: Is orthogonal convexity . A set S in the Euclidean space is called orthogonally convex or ortho-convex , if any segment parallel to any of the coordinate axes connecting two points of S lies totally within S . It is easy to prove that an intersection of any collection of orthoconvex sets is orthoconvex. Some other properties of convex sets are valid as well. The definition of a convex set and

5031-690: Is a 60-90-30 triangle which is one-sixth of a tetrahedron face. The three faces interior to the tetrahedron are: a right triangle with edges 1 {\displaystyle 1} , 3 2 {\displaystyle {\sqrt {\tfrac {3}{2}}}} , 1 2 {\displaystyle {\sqrt {\tfrac {1}{2}}}} , a right triangle with edges 1 3 {\displaystyle {\sqrt {\tfrac {1}{3}}}} , 1 2 {\displaystyle {\sqrt {\tfrac {1}{2}}}} , 1 6 {\displaystyle {\sqrt {\tfrac {1}{6}}}} , and

5160-448: Is a polyhedron that forms a convex set as a solid. That being said, it is a three-dimensional solid whose every line segment connects two of its points lies its interior or on its boundary ; none of its faces are coplanar (they do not share the same plane) and none of its edges are collinear (they are not segments of the same line). A convex polyhedron can also be defined as a bounded intersection of finitely many half-spaces , or as

5289-456: Is a polyhedron with symmetries acting transitively on its faces. Their topology can be represented by a face configuration . All 5 Platonic solids and 13 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. Many of the symmetries or point groups in three dimensions are named after polyhedra having

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5418-466: Is also known as a "triangular pyramid". Like all convex polyhedra , a tetrahedron can be folded from a single sheet of paper. It has two such nets . For any tetrahedron there exists a sphere (called the circumsphere ) on which all four vertices lie, and another sphere (the insphere ) tangent to the tetrahedron's faces. A regular tetrahedron is a tetrahedron in which all four faces are equilateral triangles . In other words, all of its faces are

5547-433: Is an orientable manifold and whose Euler characteristic is 2 must be a topological sphere. 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. For every convex polyhedron, there exists a dual polyhedron having The dual of

5676-657: Is approximately 0.55129 steradians , 1809.8 square degrees , or 0.04387 spats . One way to construct a regular tetrahedron is by using the following Cartesian coordinates , defining the four vertices of a tetrahedron with edge length 2, centered at the origin, and two-level edges: ( ± 1 , 0 , − 1 2 ) and ( 0 , ± 1 , 1 2 ) {\displaystyle \left(\pm 1,0,-{\frac {1}{\sqrt {2}}}\right)\quad {\mbox{and}}\quad \left(0,\pm 1,{\frac {1}{\sqrt {2}}}\right)} Expressed symmetrically as 4 points on

5805-448: Is called a convex combination of u 1 , ..., u r . The collection of convex subsets of a vector space, an affine space, or a Euclidean space has the following properties: Closed convex sets are convex sets that contain all their limit points . They can be characterised as the intersections of closed half-spaces (sets of points in space that lie on and to one side of a hyperplane ). From what has just been said, it

5934-464: Is called a lattice polyhedron or integral polyhedron . The Ehrhart polynomial of a lattice 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 . There is a far-reaching equivalence between lattice polyhedra and certain algebraic varieties called toric varieties . This

6063-415: Is clear that such intersections are convex, and they will also be closed sets. To prove the converse, i.e., every closed convex set may be represented as such intersection, one needs the supporting hyperplane theorem in the form that for a given closed convex set C and point P outside it, there is a closed half-space H that contains C and not P . The supporting hyperplane theorem is a special case of

6192-566: Is closed and convex then rec ⁡ S {\displaystyle \operatorname {rec} S} is closed and for all s 0 ∈ S {\displaystyle s_{0}\in S} , rec ⁡ S = ⋂ t > 0 t ( S − s 0 ) . {\displaystyle \operatorname {rec} S=\bigcap _{t>0}t(S-s_{0}).} Theorem (Dieudonné). Let A and B be non-empty, closed, and convex subsets of

6321-459: Is common instead to slice the polyhedron by a small sphere centered at the vertex. 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. The surface area of a polyhedron is the sum of areas of its faces, for definitions of polyhedra for which

6450-413: Is face-transitive, while a truncated cube has two symmetry orbits of faces. The same abstract structure may support more or less symmetric geometric polyhedra. But where a polyhedral name is given, such as icosidodecahedron , the most symmetrical geometry is often implied. There are several types of highly symmetric polyhedron, classified by which kind of element – faces, edges, or vertices – belong to

6579-416: 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

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6708-810: Is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice . In a real vector-space, the Minkowski sum of two (non-empty) sets, S 1 and S 2 , is defined to be the set S 1  +  S 2 formed by the addition of vectors element-wise from the summand-sets S 1 + S 2 = { x 1 + x 2 : x 1 ∈ S 1 , x 2 ∈ S 2 } . {\displaystyle S_{1}+S_{2}=\{x_{1}+x_{2}:x_{1}\in S_{1},x_{2}\in S_{2}\}.} More generally,

6837-482: Is known as the bellows theorem. 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 . An orthogonal polyhedron is one all of whose edges are parallel to axes of a Cartesian coordinate system. This implies that all faces meet at right angles , but this condition

6966-695: Is needed for the set of convex sets to form a lattice , in which the " join " operation is the convex hull of the union of two convex sets Conv ⁡ ( S ) ∨ Conv ⁡ ( T ) = Conv ⁡ ( S ∪ T ) = Conv ⁡ ( Conv ⁡ ( S ) ∪ Conv ⁡ ( T ) ) . {\displaystyle \operatorname {Conv} (S)\vee \operatorname {Conv} (T)=\operatorname {Conv} (S\cup T)=\operatorname {Conv} {\bigl (}\operatorname {Conv} (S)\cup \operatorname {Conv} (T){\bigr )}.} The intersection of any collection of convex sets

7095-412: Is not scissors-congruent to any other polyhedra which can fill the space, see Hilbert's third problem ). The tetrahedral-octahedral honeycomb fills space with alternating regular tetrahedron cells and regular octahedron cells in a ratio of 2:1. An irregular tetrahedron which is the fundamental domain of a symmetry group is an example of a Goursat tetrahedron . The Goursat tetrahedra generate all

7224-436: Is orientable or non-orientable by considering a topological cell complex with the same incidences between its vertices, edges, and faces. A more subtle distinction between polyhedron surfaces is given by their Euler characteristic , which combines the numbers of vertices V {\displaystyle V} , edges E {\displaystyle E} , and faces F {\displaystyle F} of

7353-407: Is the three-dimensional case of the more general concept of a Euclidean simplex , and may thus also be called a 3-simplex . The tetrahedron is one kind of pyramid , which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a triangle (any of the four faces can be considered the base), so a tetrahedron

7482-405: Is the triakis tetrahedron , a regular tetrahedron with four triangular pyramids attached to each of its faces. i.e., its kleetope . Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb , which is a tessellation. Some tetrahedra that are not regular, including

7611-402: Is the identity, and the symmetry group is the trivial group . An irregular tetrahedron has Schläfli symbol ( )∨( )∨( )∨( ). It has 8 isometries. If edges (1,2) and (3,4) are of different length to the other 4 then the 8 isometries are the identity 1, reflections (12) and (34), and 180° rotations (12)(34), (13)(24), (14)(23) and improper 90° rotations (1234) and (1432) forming

7740-427: Is used, because the resulting objects retain certain properties of convex sets. Let C be a set in a real or complex vector space. C is star convex (star-shaped) if there exists an x 0 in C such that the line segment from x 0 to any point y in C is contained in C . Hence a non-empty convex set is always star-convex but a star-convex set is not always convex. An example of generalized convexity

7869-456: Is weaker: Jessen's icosahedron has faces meeting at right angles, but does not have axis-parallel edges. Aside from the rectangular cuboids , orthogonal polyhedra are nonconvex. They are the 3D analogs of 2D orthogonal polygons, also known as rectilinear polygons . Orthogonal polyhedra are used in computational geometry , where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding

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7998-841: The Hahn–Banach theorem of functional analysis . Let C be a convex body in the plane (a convex set whose interior is non-empty). We can inscribe a rectangle r in C such that a homothetic copy R of r is circumscribed about C . The positive homothety ratio is at most 2 and: 1 2 ⋅ Area ⁡ ( R ) ≤ Area ⁡ ( C ) ≤ 2 ⋅ Area ⁡ ( r ) {\displaystyle {\tfrac {1}{2}}\cdot \operatorname {Area} (R)\leq \operatorname {Area} (C)\leq 2\cdot \operatorname {Area} (r)} The set K 2 {\displaystyle {\mathcal {K}}^{2}} of all planar convex bodies can be parameterized in terms of

8127-541: The Minkowski sum of a finite family of (non-empty) sets S n is the set formed by element-wise addition of vectors ∑ n S n = { ∑ n x n : x n ∈ S n } . {\displaystyle \sum _{n}S_{n}=\left\{\sum _{n}x_{n}:x_{n}\in S_{n}\right\}.} For Minkowski addition,

8256-454: The Minkowski sums of line segments, and include several important space-filling polyhedra. A space-filling polyhedron packs with copies of itself to fill space. Such a close-packing or space-filling is often called a tessellation of space or a honeycomb. Space-filling polyhedra must have a Dehn invariant equal to zero. Some honeycombs involve more than one kind of polyhedron. A convex polyhedron in which all vertices have integer coordinates

8385-514: The Platonic solids . These are the triangular pyramid or tetrahedron , cube , octahedron , dodecahedron and icosahedron : There are also four regular star polyhedra, known as the Kepler–Poinsot polyhedra after their discoverers. The dual of a regular polyhedron is also regular. Uniform polyhedra are vertex-transitive and every face is a regular polygon . They may be subdivided into

8514-514: The Schläfli orthoscheme and the Hill tetrahedron , can tessellate. Given that the regular tetrahedron with edge length a {\displaystyle a} . The surface area of a regular tetrahedron A {\displaystyle A} is four times the area of an equilateral triangle: A = 4 ⋅ ( 3 4 a 2 ) =

8643-438: The characteristic angles 𝟀, 𝝉, 𝟁), plus 3 2 {\displaystyle {\sqrt {\tfrac {3}{2}}}} , 1 2 {\displaystyle {\sqrt {\tfrac {1}{2}}}} , 1 6 {\displaystyle {\sqrt {\tfrac {1}{6}}}} (edges that are the characteristic radii of the regular tetrahedron). The 3-edge path along orthogonal edges of

8772-592: The convex hull of a tree in which all edges are mutually perpendicular. In a 3-dimensional orthoscheme, the tree consists of three perpendicular edges connecting all four vertices in a linear path that makes two right-angled turns. The 3-orthoscheme is a tetrahedron having two right angles at each of two vertices, so another name for it is birectangular tetrahedron . It is also called a quadrirectangular tetrahedron because it contains four right angles. Coxeter also calls quadrirectangular tetrahedra "characteristic tetrahedra", because of their integral relationship to

8901-720: The convex hull of finitely many points, in either case, restricted to intersections or hulls that have nonzero volume. Important classes of convex polyhedra include the family of prismatoid , the Platonic solids , the Archimedean solids and their duals the Catalan solids , and the regular polygonal faces polyhedron. The prismatoids are the polyhedron whose vertices lie on two parallel planes and their faces are likely to be trapezoids and triangles. Examples of prismatoids are pyramids , wedges , parallelipipeds , prisms , antiprisms , cupolas , and frustums . The Platonic solids are

9030-446: The deltahedron (whose faces are all equilateral triangles and Johnson solids (whose faces are arbitrary regular polygons). The convex polyhedron can be categorized into elementary polyhedron or composite polyhedron. An elementary polyhedron is a convex regular-faced polyhedron that cannot be produced into two or more polyhedrons by slicing it with a plane. Quite opposite to a composite polyhedron, it can be alternatively defined as

9159-528: The graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis . Spaces in which convex sets are defined include the Euclidean spaces , the affine spaces over the real numbers , and certain non-Euclidean geometries . The notion of

9288-423: The incenters of the opposite faces are concurrent . An isogonic tetrahedron has concurrent cevians that join the vertices to the points of contact of the opposite faces with the inscribed sphere of the tetrahedron. A disphenoid is a tetrahedron with four congruent triangles as faces; the triangles necessarily have all angles acute. The regular tetrahedron is a special case of a disphenoid. Other names for

9417-427: The line segment connecting x and y is included in C . This means that the affine combination (1 − t ) x + ty belongs to C for all x,y in C and t in the interval [0, 1] . This implies that convexity is invariant under affine transformations . Further, it implies that a convex set in a real or complex topological vector space is path-connected (and therefore also connected ). A set C

9546-402: The operations of Minkowski summation and of forming convex hulls are commuting operations. The Minkowski sum of two compact convex sets is compact. The sum of a compact convex set and a closed convex set is closed. The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed. It uses the concept of

9675-575: The regular , quasi-regular , or semi-regular , and may be convex or starry. The duals of the uniform polyhedra have irregular faces but are face-transitive , and every vertex figure is a regular polygon. A uniform polyhedron has the same symmetry orbits as its dual, with the faces and vertices simply swapped over. The duals of the convex Archimedean polyhedra are sometimes called the Catalan solids . The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are convex or not. An isohedron

9804-400: The self-crossing polyhedra ) or include shapes that are often not considered as valid polyhedra (such as solids whose boundaries are not manifolds ). 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". Nevertheless, there

9933-424: The symmetry group known as full tetrahedral symmetry T d {\displaystyle \mathrm {T} _{\mathrm {d} }} . This symmetry group is isomorphic to the symmetric group S 4 {\displaystyle S_{4}} . They can be categorized as follows: The regular tetrahedron has two special orthogonal projections , one centered on a vertex or equivalently on

10062-406: 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

10191-884: The unit sphere , centroid at the origin, with lower face parallel to the x y {\displaystyle xy} plane, the vertices are: ( 8 9 , 0 , − 1 3 ) , ( − 2 9 , 2 3 , − 1 3 ) , ( − 2 9 , − 2 3 , − 1 3 ) , ( 0 , 0 , 1 ) {\displaystyle {\begin{aligned}\left({\sqrt {\frac {8}{9}}},0,-{\frac {1}{3}}\right),&\quad \left(-{\sqrt {\frac {2}{9}}},{\sqrt {\frac {2}{3}}},-{\frac {1}{3}}\right),\\\left(-{\sqrt {\frac {2}{9}}},-{\sqrt {\frac {2}{3}}},-{\frac {1}{3}}\right),&\quad (0,0,1)\end{aligned}}} with

10320-412: The zero set   {0} containing only the zero vector   0 has special importance : For every non-empty subset S of a vector space S + { 0 } = S ; {\displaystyle S+\{0\}=S;} in algebraic terminology, {0} is the identity element of Minkowski addition (on the collection of non-empty sets). Minkowski addition behaves well with respect to

10449-525: The Platonic solid is named after the Greek philosopher Plato , who associated those four solids with nature. The regular tetrahedron was considered as the classical element of fire , because of his interpretation of its sharpest corner being most penetrating. The regular tetrahedron is self-dual, meaning its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form

10578-399: 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

10707-416: The associated symmetry. These include: Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. Examples include the snub cuboctahedron and snub icosidodecahedron . 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

10836-530: The base and its height. Because the base is an equilateral, it is: V = 1 3 ⋅ ( 3 4 a 2 ) ⋅ 6 3 a = a 3 6 2 ≈ 0.118 a 3 . {\displaystyle V={\frac {1}{3}}\cdot \left({\frac {\sqrt {3}}{4}}a^{2}\right)\cdot {\frac {\sqrt {6}}{3}}a={\frac {a^{3}}{6{\sqrt {2}}}}\approx 0.118a^{3}.} Its volume can also be obtained by dissecting

10965-425: The base plane the slope of a face (2 √ 2 ) is twice that of an edge ( √ 2 ), corresponding to the fact that the horizontal distance covered from the base to the apex along an edge is twice that along the median of a face. In other words, if C is the centroid of the base, the distance from C to a vertex of the base is twice that from C to the midpoint of an edge of the base. This follows from

11094-439: 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 1 3 | ∑ F ( Q F ⋅ N F ) area ⁡ ( F ) | , {\displaystyle {\frac {1}{3}}\left|\sum _{F}(Q_{F}\cdot N_{F})\operatorname {area} (F)\right|,} where

11223-475: The characteristic 3-orthoscheme of the cube is a space-filling tetrahedron in this sense. (The characteristic orthoscheme of the cube is one of the Hill tetrahedra , a family of space-filling tetrahedra. All space-filling tetrahedra are scissors-congruent to a cube.) A disphenoid can be a space-filling tetrahedron in the directly congruent sense, as in the disphenoid tetrahedral honeycomb . Regular tetrahedra, however, cannot fill space by themselves (moreover, it

11352-457: The column for Greek cardinal numbers. The names of tetrahedra, hexahedra, octahedra (8-sided polyhedra), dodecahedra (12-sided polyhedra), and icosahedra (20-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. Some polyhedra have two distinct sides to their surface. For example,

11481-410: The combinatorial structure of the given polyhedron. Some polyhedrons do not have the property of convexity, and they are called non-convex polyhedrons . Such polyhedrons are star polyhedrons and Kepler–Poinsot polyhedrons , which constructed by either stellation (process of extending the faces—within their planes—so that they meet) or faceting (whose process of removing parts of

11610-479: The commonly used subdivision methods is the Longest Edge Bisection (LEB) , which identifies the longest edge of the tetrahedron and bisects it at its midpoint, generating two new, smaller tetrahedra. When this process is repeated multiple times, bisecting all the tetrahedra generated in each previous iteration, the process is called iterative LEB. A similarity class is the set of tetrahedra with

11739-409: The convex regular 4-polytopes with tetrahedral cells (the 5-cell , 16-cell and 600-cell ) can be constructed as tilings of the 3-sphere by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface. Tetrahedra which do not have four equilateral faces are categorized and named by the symmetries they do possess. If all three pairs of opposite edges of

11868-860: The convex body diameter D , its inradius r (the biggest circle contained in the convex body) and its circumradius R (the smallest circle containing the convex body). In fact, this set can be described by the set of inequalities given by 2 r ≤ D ≤ 2 R {\displaystyle 2r\leq D\leq 2R} R ≤ 3 3 D {\displaystyle R\leq {\frac {\sqrt {3}}{3}}D} r + R ≤ D {\displaystyle r+R\leq D} D 2 4 R 2 − D 2 ≤ 2 R ( 2 R + 4 R 2 − D 2 ) {\displaystyle D^{2}{\sqrt {4R^{2}-D^{2}}}\leq 2R(2R+{\sqrt {4R^{2}-D^{2}}})} and can be visualized as

11997-456: The cube . The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a 3-dimensional point group is formed. Two other isometries (C 3 , [3] ), and (S 4 , [2 ,4 ]) can exist if the face or edge marking are included. Tetrahedral diagrams are included for each type below, with edges colored by isometric equivalence, and are gray colored for unique edges. Its only isometry

12126-563: The cube is an example of a Heronian tetrahedron . Every regular polytope, including the regular tetrahedron, has its characteristic orthoscheme . There is a 3-orthoscheme, which is the "characteristic tetrahedron of the regular tetrahedron". The regular tetrahedron [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] is subdivided into 24 instances of its characteristic tetrahedron [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] by its planes of symmetry. The 24 characteristic tetrahedra of

12255-412: The cube), a double orthoscheme (the characteristic tetrahedron of the cube face-bonded to its mirror image), and the space-filling disphenoid illustrated above . The disphenoid is the double orthoscheme face-bonded to its mirror image (a quadruple orthoscheme). Thus all three of these Goursat tetrahedra, and all the polyhedra they generate by reflections, can be dissected into characteristic tetrahedra of

12384-501: The cube. The cube [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] can be dissected into six such 3-orthoschemes [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] four different ways, with all six surrounding the same √ 3 cube diagonal. The cube can also be dissected into 48 smaller instances of this same characteristic 3-orthoscheme (just one way, by all of its symmetry planes at once). The characteristic tetrahedron of

12513-840: The edge length of 2 6 3 {\textstyle {\frac {2{\sqrt {6}}}{3}}} . A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. For one such embedding, the Cartesian coordinates of the vertices are ( 1 , 1 , 1 ) , ( 1 , − 1 , − 1 ) , ( − 1 , 1 , − 1 ) , ( − 1 , − 1 , 1 ) . {\displaystyle {\begin{aligned}(1,1,1),&\quad (1,-1,-1),\\(-1,1,-1),&\quad (-1,-1,1).\end{aligned}}} This yields

12642-742: The fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see proof ). Its solid angle at a vertex subtended by a face is arccos ⁡ ( 23 27 ) = π 2 − 3 arcsin ⁡ ( 1 3 ) = 3 arccos ⁡ ( 1 3 ) − π {\displaystyle {\begin{aligned}\arccos \left({\frac {23}{27}}\right)&={\frac {\pi }{2}}-3\arcsin \left({\frac {1}{3}}\right)\\&=3\arccos \left({\frac {1}{3}}\right)-\pi \end{aligned}}} This

12771-453: The five ancientness polyhedrons— tetrahedron , octahedron , icosahedron , cube , and dodecahedron —classified by Plato in his Timaeus whose connecting four classical elements of nature. The Archimedean solids are the class of thirteen polyhedrons whose faces are all regular polygons and whose vertices are symmetric to each other; their dual polyhedrons are Catalan solids . The class of regular polygonal faces polyhedron are

12900-471: The formation of highly irregular elements that could compromise simulation results. The iterative LEB of the regular tetrahedron has been shown to produce only 8 similarity classes. Furthermore, in the case of nearly equilateral tetrahedra where their two longest edges are not connected to each other, and the ratio between their longest and their shortest edge is less than or equal to 3 / 2 {\displaystyle {\sqrt {3/2}}} ,

13029-446: The image of the function g that maps a convex body to the R point given by ( r / R , D /2 R ). The image of this function is known a ( r , D , R ) Blachke-Santaló diagram. Alternatively, the set K 2 {\displaystyle {\mathcal {K}}^{2}} can also be parametrized by its width (the smallest distance between any two different parallel support hyperplanes), perimeter and area. Let X be

13158-552: 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

13287-444: The intersection is a square . The aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly. If the tetrahedron is bisected on this plane, both halves become wedges . This property also applies for tetragonal disphenoids when applied to the two special edge pairs. The tetrahedron can also be represented as

13416-465: The interval [ a , b ] = { x ∈ X | a ≤ x ≤ b } is contained in Y . That is, Y is convex if and only if for all a , b in Y , a ≤ b implies [ a , b ] ⊆ Y . A convex set is not connected in general: a counter-example is given by the subspace {1,2,3} in Z , which is both convex and not connected. The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axioms . Given

13545-406: The intervals and the points of R . Some examples of convex subsets of the Euclidean plane are solid regular polygons , solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids . The Kepler-Poinsot polyhedra are examples of non-convex sets. A set that is not convex is called

13674-438: 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. The remainder of this article considers only three-dimensional polyhedra. A convex polyhedron

13803-563: 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. 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, 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

13932-522: 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. For a complete list of the Greek numeral prefixes see Numeral prefix § Table of number prefixes in English , in

14061-484: The one-holed toroid and the Klein bottle both have χ = 0 {\displaystyle \chi =0} , with the first being orientable and the other not. 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

14190-983: The operation of taking convex hulls, as shown by the following proposition: Let S 1 , S 2 be subsets of a real vector-space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls Conv ⁡ ( S 1 + S 2 ) = Conv ⁡ ( S 1 ) + Conv ⁡ ( S 2 ) . {\displaystyle \operatorname {Conv} (S_{1}+S_{2})=\operatorname {Conv} (S_{1})+\operatorname {Conv} (S_{2}).} This result holds more generally for each finite collection of non-empty sets: Conv ( ∑ n S n ) = ∑ n Conv ( S n ) . {\displaystyle {\text{Conv}}\left(\sum _{n}S_{n}\right)=\sum _{n}{\text{Conv}}\left(S_{n}\right).} In mathematical terminology,

14319-485: The orthoscheme is 1 {\displaystyle 1} , 1 3 {\displaystyle {\sqrt {\tfrac {1}{3}}}} , 1 6 {\displaystyle {\sqrt {\tfrac {1}{6}}}} , first from a tetrahedron vertex to an tetrahedron edge center, then turning 90° to an tetrahedron face center, then turning 90° to the tetrahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face

14448-402: The regular polyhedra (and many other uniform polyhedra) by mirror reflections, a process referred to as Wythoff's kaleidoscopic construction . For polyhedra, Wythoff's construction arranges three mirrors at angles to each other, as in a kaleidoscope . Unlike a cylindrical kaleidoscope, Wythoff's mirrors are located at three faces of a Goursat tetrahedron such that all three mirrors intersect at

14577-428: The regular polytopes and their symmetry groups. For example, the special case of a 3-orthoscheme with equal-length perpendicular edges is characteristic of the cube , which means that the cube can be subdivided into instances of this orthoscheme. If its three perpendicular edges are of unit length, its remaining edges are two of length √ 2 and one of length √ 3 , so all its edges are edges or diagonals of

14706-472: The regular tetrahedron occur in two mirror-image forms, 12 of each. If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths 4 3 {\displaystyle {\sqrt {\tfrac {4}{3}}}} , 1 {\displaystyle 1} , 1 3 {\displaystyle {\sqrt {\tfrac {1}{3}}}} around its exterior right-triangle face (the edges opposite

14835-419: The rotation (12)(34), giving the group C 2 isomorphic to the cyclic group , Z 2 . Tetrahedra subdivision is a process used in computational geometry and 3D modeling to divide a tetrahedron into several smaller tetrahedra. This process enhances the complexity and detail of tetrahedral meshes, which is particularly beneficial in numerical simulations, finite element analysis, and computer graphics. One of

14964-429: 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. For every vertex one can define a vertex figure , which describes

15093-451: The same geometric shape, regardless of their specific position, orientation, and scale. So, any two tetrahedra belonging to the same similarity class may be transformed to each other by an affine transformation. The outcome of having a limited number of similarity classes in iterative subdivision methods is significant for computational modeling and simulation. It reduces the variability in the shapes and sizes of generated tetrahedra, preventing

15222-400: The same shape include bisphenoid, isosceles tetrahedron and equifacial tetrahedron. A 3-orthoscheme is a tetrahedron where all four faces are right triangles . A 3-orthoscheme is not a disphenoid, because its opposite edges are not of equal length. It is not possible to construct a disphenoid with right triangle or obtuse triangle faces. An orthoscheme is an irregular simplex that is

15351-417: The same size and shape (congruent) and all edges are the same length. A convex polyhedron in which all of its faces are equilateral triangles is the deltahedron . There are eight convex deltahedra, one of which is the regular tetrahedron. The regular tetrahedron is also one of the five regular Platonic solids , a set of polyhedrons in which all of their faces are regular polygons . Known since antiquity,

15480-441: The same surface distances as each other, or the same as certain convex polyhedra. 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

15609-469: 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

15738-491: The same volumes and Dehn invariants can be cut up and reassembled into each other. 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. Another of Hilbert's problems, Hilbert's 18th problem , concerns (among other things) polyhedra that tile space . Every such polyhedron must have Dehn invariant zero. The Dehn invariant has also been connected to flexible polyhedra by

15867-494: The strong bellows theorem, which states that the Dehn invariant of any flexible polyhedron remains invariant as it flexes. Many of the most studied polyhedra are highly symmetrical , that is, their appearance is unchanged by some reflection or rotation of space. Each such symmetry may change the location of a given vertex, face, or edge, but the set of all vertices (likewise faces, edges) is unchanged. The collection of symmetries of

15996-449: The sum is over faces F of the polyhedron, Q F is an arbitrary point on face F , N F is the unit vector perpendicular to F pointing outside the solid, and the multiplication dot is the dot product . 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

16125-431: The surface of a polyhedron to a polygonal net . Convex set In geometry , a set of points is convex if it contains every line segment between two points in the set. Equivalently, a convex set or a convex region is a set that intersects every line in a line segment , single point, or the empty set . For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example,

16254-655: The symmetry group D 2d . A tetragonal disphenoid has Coxeter diagram [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] and Schläfli symbol s{2,4}. It has 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the Klein four-group V 4 or Z 2 , present as the point group D 2 . A rhombic disphenoid has Coxeter diagram [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] and Schläfli symbol sr{2,2}. This has two pairs of equal edges (1,3), (2,4) and (1,4), (2,3) but otherwise no edges equal. The only two isometries are 1 and

16383-418: The vertex from the other vertices. 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; 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

16512-474: The volume in these cases. 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

16641-519: Was used by Stanley to prove the Dehn–Sommerville equations for simplicial polytopes . It is possible for some polyhedra to 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

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