Regular polyhedron

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used. A regular polyhedron is identified by its Schläfli symbol of the form, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra, four regular star polyhedra, making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. There are five convex regular polyhedra, known as the Platonic solids, four regular star polyhedra, the Kepler–Poinsot polyhedra, five regular compounds of regular polyhedra: The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition: The vertices of the polyhedron all lie on a sphere. All the dihedral angles of the polyhedron are equal All the vertex figures of the polyhedron are regular polygons.

All the solid angles of the polyhedron are congruent. A regular polyhedron has all of three related spheres which share its centre: An insphere, tangent to all faces. An intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices; the regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after them: Tetrahedral Octahedral Icosahedral Any shapes with icosahedral or octahedral symmetry will contain tetrahedral symmetry; the five Platonic solids have an Euler characteristic of 2. This reflects that the surface is a topological 2-sphere, so is true, for example,of any polyhedron, star-shaped with respect to some interior point; the sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not for tetrahedra. In a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, vice versa.

The regular polyhedra show this duality as follows: The tetrahedron is self-dual, i.e. it pairs with itself. The cube and octahedron are dual to each other; the icosahedron and dodecahedron are dual to each other. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron and great icosahedron are dual to each other. The Schläfli symbol of the dual is just the original written backwards. Stones carved in shapes resembling clusters of spheres or knobs have been found in Scotland and may be as much as 4,000 years old; some of these stones show not only the symmetries of the five Platonic solids, but some of the relations of duality amongst them. Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery. There is doubt regarding the mathematical interpretation of these objects, as many have non-platonic forms, only one has been found to be a true icosahedron, as opposed to a reinterpretation of the icosahedron dual, the dodecahedron.

It is possible that the Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua in the late 19th century of a dodecahedron made of soapstone, dating back more than 2,500 years. The earliest known written records of the regular convex solids originated from Classical Greece; when these solids were all discovered and by whom is not known, but Theaetetus, was the first to give a mathematical description of all five. H. S. M. Coxeter credits Plato with having made models of them, mentions that one of the earlier Pythagoreans, Timaeus of Locri, used all five in a correspondence between the polyhedra and the nature of the universe as it was perceived – this correspondence is recorded in Plato's dialogue Timaeus. Euclid's reference to Plato led to their common description as the Platonic solids. One might characterise the Greek definition as follows: A regular polygon is a planar figure with all edges equal and all corners equal A regular polyhedron is a solid figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.

This definition rules out, for example, the square pyramid, or the shape formed by joining two tetrahedra together. This concept of a regular polyhedron would remain unchallenged for 2000 years. Regular star polygons such as the pentagram were known to the ancient Greeks – the pentagram was used by the Pythagoreans as their secret sign, but they did not use them to construct polyhedra, it was not until the early 17th century that Johannes Kepler realised that pentagrams could be used as the faces of regular star polyhedra. Some of these star polyhedra may have been discovered

Polyhedron

In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from as poly - + - hedron. A convex polyhedron is the convex hull of finitely many points on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of the more general polytope 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, there is not universal agreement over which of these to choose; some of these definitions exclude shapes that have been counted as polyhedra or include shapes that are not considered as valid polyhedra. As Branko Grünbaum observed, "The Original Sin in the theory of polyhedra goes back to Euclid, through Kepler, Poinsot and many others... at each stage... the writers failed to define what are the polyhedra".

There is general agreement that a polyhedron is a solid or surface that can be described by its vertices, edges and sometimes by its 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. A common and somewhat naive definition of a polyhedron is that it is a solid whose boundary can be covered by finitely many planes or that it is a solid formed as the union of finitely many convex polyhedra. Natural refinements of this definition require the solid to be bounded, to have a connected interior, 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, 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, their faces may not form simple polygons, some edges may belong to more than two faces.

Definitions based on the idea of a bounding surface rather than a solid are common. For instance, O'Rourke defines a polyhedron as a union of convex polygons, arranged in space so that the intersection of any two polygons is a shared vertex or edge or the empty set and so that their union is a manifold. If a planar part of such a surface is not itself a convex polygon, O'Rourke requires it to be subdivided into smaller convex polygons, with flat dihedral angles between them. Somewhat more Grünbaum defines an acoptic polyhedron to be a collection of simple polygons that form an embedded manifold, with each vertex incident to at least three edges and each two faces intersecting only in shared vertices and edges of each. Cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra. Similar notions form the basis of topological definitions of polyhedra, as subdivisions of a topological manifold into topological disks whose pairwise intersections are required to be points, topological arcs, or the empty set.

However, there exist topological polyhedra. One modern approach is based on the theory of abstract polyhedra; these can be defined as ordered sets whose elements are the vertices and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face. Additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. If the sections of the partial order between elements three levels apart have the same structure as the abstract representation of a polygon these ordered sets carry the same information as a topological polyhedron. However, these requirements are relaxed, to instead require only that sections between elements two levels apart have the same structure as the abstract representation of a line segment. Geometric polyhedra, defined in other ways, can be described abstractly in this way, but it is possible to use abstract polyhedra as the basis of a definition of geometric polyhedra.

A realization of an abstract polyhedron is 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 be defined as a realization of an abstract polyhedron. Realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have been considered. Unlike the solid-based and surface-based definitions, this works well for star polyhedra. However, without additional restrictions, this definition allows degenerate or unfaithful polyhedra (for instance, by mapp

Tetrahedron

In geometry, a tetrahedron known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces; the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, may thus be called a 3-simplex. The tetrahedron is one kind of pyramid, 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, so a tetrahedron is 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 on which all four vertices lie, another sphere tangent to the tetrahedron's faces. A regular tetrahedron is one, it is one of the five regular Platonic solids. In a regular tetrahedron, all faces are the same size and shape and all edges are the same length.

Regular tetrahedra alone do not tessellate, but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, a tessellation. The regular tetrahedron is self-dual; the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, two level edges: and Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face level, the vertices are: v1 = v2 = v3 = v4 = with the edge length of sqrt. Still another set of coordinates are based on an alternated cube or demicube with edge length 2; this form has Coxeter diagram and Schläfli symbol h. The tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, the pair together form the stellated octahedron, whose vertices are those of the original cube. Tetrahedron:, Dual tetrahedron:, For a regular tetrahedron of edge length a: With respect to the base plane the slope of a face is twice that of an edge, 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 the fact that the medians of a triangle intersect at its centroid, this point divides each of them in two segments, one of, twice as long as the other. For a regular tetrahedron with side length a, radius R of its circumscribing sphere, distances di from an arbitrary point in 3-space to its four vertices, we have d 1 4 + d 2 4 + d 3 4 + d 4 4 4 + 16 R 4 9 = 2.

Square antiprism

In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is known as an anticube. If all its faces are regular, it is a semiregular uniform polyhedron; when eight points are distributed on the surface of a sphere with the aim of maximising the distance between them in some sense, the resulting shape corresponds to a square antiprism rather than a cube. Specific methods of distributing the points include, for example, the Thomson problem, maximising the distance of each point to the nearest point, or minimising the sum of all reciprocals of squares of distances between points. According to the VSEPR theory of molecular geometry in chemistry, based on the general principle of maximizing the distances between points, a square antiprism is the favoured geometry when eight pairs of electrons surround a central atom. One molecule with this geometry is the octafluoroxenate ion in the salt nitrosonium octafluoroxenate.

Few ions are cubical because such a shape would cause large repulsion between ligands. In addition, the element sulfur forms octatomic S8 molecules as its most stable allotrope; the S8 molecule has a structure based on the square antiprism, in which the eight atoms occupy the eight vertices of the antiprism, the eight triangle-triangle edges of the antiprism correspond to single covalent bonds between sulfur atoms. The main building block of the One World Trade Center has the shape of an tall tapering square antiprism, it is not a true antiprism because of its taper: the top square has half the area of the bottom one. A twisted prism can be made with the same vertex arrangement, it can be seen as the convex form with 4 tetrahedrons excavated around the sides. However, after this it can no longer be triangulated into tetrahedra without adding new vertices, it has half of the symmetry of the uniform solution: D4 order 4. A crossed square antiprism is a star polyhedron, topologically identical to the square antiprism with the same vertex arrangement, but it can't be made uniform.

Its vertex configuration is 3.3/2.3.4, with one triangle retrograde. It has d4d symmetry, order 8; the gyroelongated square pyramid is a Johnson solid constructed by augmenting one a square pyramid. The gyroelongated square bipyramid is a deltahedron constructed by replacing both squares of a square antiprism with a square pyramid; the snub disphenoid is another deltahedron, constructed by replacing the two squares of a square antiprism by pairs of equilateral triangles. The snub square antiprism can be seen as a square antiprism with a chain of equilateral triangles inserted around the middle; the sphenocorona and the sphenomegacorona are other Johnson solids that, like the square antiprism, consist of two squares and an number of equilateral triangles. The square antiprism can be truncated and alternated to form a snub antiprism: As an antiprism, the square antiprism belongs to a family of polyhedra that includes the octahedron, the pentagonal antiprism, the hexagonal antiprism, the octagonal antiprism.

The square antiprism is first in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n. Biscornu Compound of three square antiprisms Weisstein, Eric W. "Antiprism". MathWorld. Square Antiprism interactive model Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try: "A4"

Regular icosahedron

In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, the one with the most sides, it has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol, or sometimes by its vertex figure as 3.3.3.3.3 or 35. It is the dual of the dodecahedron, represented by, having three pentagonal faces around each vertex. A regular icosahedron is a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations; the name comes from Greek, Modern εἴκοσι, meaning'twenty', ἕδρα, meaning'seat'. The plural can be either "icosahedrons" or "icosahedra". If the edge length of a regular icosahedron is a, the radius of a circumscribed sphere is r u = a 2 ϕ 5 = a 4 10 + 2 5 = a sin 2 π 5 ≈ 0.951 056 5163 ⋅ a OEIS: A019881and the radius of an inscribed sphere is r i = ϕ 2 a 2 3 = 3 12 a ≈ 0.755 761 3141 ⋅ a OEIS: A179294while the midradius, which touches the middle of each edge, is r m = a ϕ 2 = 1 4 a = a cos π 5 ≈ 0.809 016 99 ⋅ a OEIS: A019863where ϕ is the golden ratio.

The surface area A and the volume V of a regular icosahedron of edge length a are: A = 5 3 a 2 ≈ 8.660 254 04 a 2 OEIS: A010527 V = 5 12 a 3 ≈ 2.181 694 99 a 3 OEIS: A102208The latter is F = 20 times the volume of a general tetrahedron with apex at the center of the inscribed sphere, where the volume of the tetrahedron is one third times the base area √3a2/4 times its height ri. The volume filling factor of the circumscribed sphere is: f = V 4 3 π r u 3 = 20 3 2 π ≈ 0.605 461 3829 The vertices of an icosahedron centered at the origin with an edge-length of 2 and a circumradius of ϕ + 2 ≈ 1.9 are described by circular permutations of: where ϕ = 1 + √5/2 is the golden ratio. Taking all permutations results in the Compound of two icosahedra. Note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings. If the original icosahedron has edge length 1, its dual dodecahedron has edge length √5 − 1/2 = 1/ϕ = ϕ − 1; the 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron.

This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle similarly subdividing each edge into the golden mean along the direction of its vector. The five octahedra defining any given icosahedron form a regular polyhedral compound, while the two icosahedra that can be defined in this way from any given octahedron form a uniform polyhedron compound; the locations of the vertices of a regular icosahedron can be described using spherical coordinates, for instance as latitude and longitude. If two vertices are taken to be at the north and south poles the other ten vertices are at latitude ±arctan ≈ ±26.57°. These ten vertices are at evenly spaced longitudes, alternating between south latitudes; this scheme takes advantage of the fact that the regular icosahedron is a pentagonal gyroelongated bipyramid, with D5d dihedral symmetry—that is, it is formed of two congruent pentagonal pyramids joined by a pentagonal antiprism. The icosahedron has three special orthogonal projections, centered on a face, an edge and a vertex: The

Prismatic uniform polyhedron

In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in the uniform prisms and the uniform antiprisms. All are therefore prismatoids; because they are isogonal, their vertex arrangement uniquely corresponds to a symmetry group. The difference between the prismatic and antiprismatic symmetry groups is that Dph has the vertices lined up in both planes, which gives it a reflection plane perpendicular to its p-fold axis; each has p reflection planes. The Dph symmetry group contains inversion if and only if p is while Dpd contains inversion symmetry if and only if p is odd. There are: prisms, for each rational number p/q > 2, with symmetry group Dph. If p/q is an integer, i.e. if q = 1, the prism or antiprism is convex. An antiprism with p/q < 2 is crossed or retrograde. If p/q ≤ 3/2 no uniform antiprism can exist, as its vertex figure would have to violate the triangle inequality. Note: The tetrahedron and octahedron are listed here with dihedral symmetry, although if uniformly colored, the tetrahedron has tetrahedral symmetry and the cube and octahedron have octahedral symmetry.

Uniform polyhedron Prism Antiprism Coxeter, Harold Scott MacDonald. P.. "Uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences; the Royal Society. 246: 401–450. Doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446. Cromwell, P.. 1997, ISBN 0-521-66432-2. Pbk. ISBN 0-521-66405-5. P.175 Skilling, John, "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554. Prisms and Antiprisms George W. Hart

Truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 equilateral triangle faces, 12 vertices and 18 edges, it can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length. A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called rectification; the rectification of a tetrahedron produces an octahedron. A truncated tetrahedron is the Goldberg polyhedron GIII, containing hexagonal faces. A truncated tetrahedron can be called a cantic cube, with Coxeter diagram, having half of the vertices of the cantellated cube. There are two dual positions of this construction, combining them creates the uniform compound of two truncated tetrahedra; the area A and the volume V of a truncated tetrahedron of edge length a are: A = 7 3 a 2 ≈ 12.124 355 65 a 2 V = 23 12 2 a 3 ≈ 2.710 575 995 a 3. The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = 207/208, as reported by two independent groups using Monte Carlo methods.

Although no mathematical proof exists that this is the best possible packing for the truncated tetrahedron, the high proximity to the unity and independency of the findings make it unlikely that an denser packing is to be found. In fact, if the truncation of the corners is smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to fill space. Cartesian coordinates for the 12 vertices of a truncated tetrahedron centered at the origin, with edge length √8, are all permutations of with an number of minus signs:, Another simple construction exists in 4-space as cells of the truncated 16-cell, with vertices as coordinate permutation of: The truncated tetrahedron can be represented as a spherical tiling, projected onto the plane via a stereographic projection; this projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. A lower symmetry version of the truncated tetrahedron is called a Friauf polyhedron in crystals such as complex metallic alloys.

This form fits 5 Friauf polyhedra around an axis, giving a 72-degree dihedral angle on a subset of 6-6 edges. It is named after J. B. Friauf and his 1927 paper "The crystal structure of the intermetallic compound MgCu2". Giant truncated tetrahedra were used for the "Man the Explorer" and "Man the Producer" theme pavilions in Expo 67, they were made of massive girders of steel bolted together in a geometric lattice. The truncated tetrahedra were interconnected with lattice steel platforms. All of these buildings were demolished after the end of Expo 67, as they had not been built to withstand the severity of the Montreal weather over the years, their only remnants are in the Montreal city archives, the Public Archives Of Canada and the photo collections of tourists of the times. The Tetraminx puzzle has a truncated tetrahedral shape; this puzzle shows a dissection of a truncated tetrahedron into 6 tetrahedra. It contains 4 central planes of rotations. In the mathematical field of graph theory, a truncated tetrahedral graph is a Archimedean graph, the graph of vertices and edges of the truncated tetrahedron, one of the Archimedean solids.

It has 18 edges. It is a connected cubic graph, connected cubic transitive graph, it is a part of a sequence of cantic polyhedra and tilings with vertex configuration 3.6.n.6. In this wythoff construction the edges between the hexagons represent degenerate digons; this polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations, Coxeter group symmetry. Quarter cubic honeycomb – Fills space using truncated tetrahedra and smaller tetrahedra Truncated 5-cell – Similar uniform polytope in 4-dimensions Truncated triakis tetrahedron Triakis truncated tetrahedron Octahedron – a rectified tetrahedron Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. Read, R. C.. An Atlas of Graphs, Oxford University Press Eric W. Weisstein, Truncated tetrahedron at MathWorld. Weisstein, Eric W. "Truncated tetrahedral graph". MathWorld. Klitzing, Richard. "3D convex uniform polyhedra x3x3o - tut".

Editable printable net of a truncated tetrahedron with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra