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Each n-orthoplex has 2n vertices. History of discoveryConvex polygons and polyhedraThe earliest surviving mathematical treatment of regular polygons and polyhedra comes to us from ancient Greek Ancient Greece The term Ancient Greece refers to the period of History of Greece lasting from the Greek Dark Ages ca. 1100 BC and the Dorian invasion, to 146 BC and the Roman Republic conquest of Greece after the Battle of Corinth .... mathematicians. The five Platonic solid Platonic solid In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex.... s were known to them. Pythagoras Pythagoras Pythagoras of Samos was an Ionians Ancient Greeks mathematician and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mysticism and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy.... knew of at least three of them and Theaetetus Theaetetus (mathematician) Theaetetus of Athens, son of Euphronius, of the Athenian deme Sunium, was a classical Greece mathematician. His principal contributions were on irrational number lengths, which was included in Book X of Euclid's Elements, and proving that there are precisely five Platonic solid.... (ca. 417 B.C. – 369 B.C.) described all five. Later, Euclid Euclid Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I .... wrote a systematic study of mathematics, publishing it under the title Elements Euclid's Elements Euclid's Elements is a mathematics and geometry treatise consisting of 13 books written by the Greek mathematics Euclid in Alexandria circa 300 BC.... , which built up a logical theory of geometry and number theory Number theory Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study.... . His work concluded with mathematical descriptions of the five Platonic solid Platonic solid In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex.... s.
Star polygons and polyhedraOur understanding remained static for many centuries after Euclid. The subsequent history of the regular polytopes can be characterised by a gradual broadening of the basic concept, allowing more and more objects to be considered among their number. Thomas Bredwardine (Bradwardinus) was the first to record a serious study of star polygons. Various star polyhedra appear in Renaissance art, but it was not until Johannes Kepler Johannes Kepler Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy.... studied the small stellated dodecahedron Small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex.... and the great stellated dodecahedron Great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra.It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex.... in 1619 that he realised these two were regular. Louis Poinsot Louis Poinsot Louis Poinsot was a France mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a couple .... discovered the great dodecahedron Great dodecahedron In geometry, the great dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces , with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path.... and great icosahedron Great icosahedron In geometry, the great icosahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 20 intersecting triangular faces, with five triangles meeting at each vertex in a pentagrammic sequence.... in 1809, and Augustin Cauchy proved the list complete in 1812. These polyhedra are known as collectively as the Kepler-Poinsot polyhedra.
Higher-dimensional polytopesIt was not until the 19th century that a Swiss mathematician, Ludwig Schläfli Ludwig Schläfli Ludwig Schl?fli was a Switzerland geometry and complex analysis who was one of the key figures in developing the notion of higher dimensional spaces.... , examined and characterised the regular polytopes in higher dimensions. His efforts were first published in full in (Schläfli, 1901), six years posthumously, although parts of it were published in 1855 and 1858 (Schläfli, 1855), (Schläfli, 1858). Interestingly, between 1880 and 1900, Schläfli's results were rediscovered independently by at least nine other mathematicians — see (Coxeter, 1948, pp143–144) for more details. Schläfli called such a figure a "polyschem" (in English, "polyscheme" or "polyschema"). The term "polytope" was introduced by Hoppe in 1882, and first used in English by Mrs. Stott Alicia Boole Stott Alicia Boole Stott was the third daughter of George Boole, born in Cork , Ireland. Before marrying Walter Stott, an actuary, in 1890, she was known as Alicia Boole.... some twenty years later. The term "polyhedroids" was also used in earlier literature (Hilbert, 1952). Coxeter (1948) is probably the most comprehensive printed treatment of Schläfli's and similar results to date. Schläfli showed that there are six regular convex polytopes in 4 dimensions Convex regular 4-polytope In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons .... , five of these correspond to the Platonic solids and the other one is the 24-cell 24-cell In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell .... . There are exactly three in each higher dimension, which correspond to the tetrahedron, cube and octahedron: these are the regular simplices Regular polytope In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n.... , measure polytopes Regular polytope In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n.... and cross polytopes Regular polytope In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n.... . Descriptions of these may be found in the List of regular polytopes List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.... . Also of interest are the nonconvex regular 4-polytopes Schläfli-Hess polychoron In four dimensional geometry, Schl?fli-Hess polychora are the complete set of 10 Regular polytope self-intersecting Star polytope . They are named in honor of their discoverers: Ludwig Schl?fli and Edmund Hess.... , partially discovered by Schläfli. By the end of the 19th century, mathematicians such as Arthur Cayley Arthur Cayley Arthur Cayley was a British mathematician. He helped found the modern British school of pure mathematics.As a child, Cayley enjoyed solving complex maths problems for amusement.... and Ludwig Schläfli Ludwig Schläfli Ludwig Schl?fli was a Switzerland geometry and complex analysis who was one of the key figures in developing the notion of higher dimensional spaces.... had developed the theory of regular polytopes in four and higher dimensions, such as the tesseract Tesseract In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square .... and the 24-cell 24-cell In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell .... . The latter are hard to visualise, but still retain the aesthetically pleasing symmetry of their lower dimensional cousins. Harder still to imagine are the more modern abstract regular polytopes Abstract polytope In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorics properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths etc.... such as the 57-cell 57-cell In mathematics, the 57-cell is a duality abstract polytope . Its 57 Cell s are hemi-dodecahedron. It also has 57 vertices, 171 edges and 171 faces.... or the 11-cell 11-cell In mathematics, the 11-cell is a duality abstract polytope . Its 11 cells are hemi-icosahedron. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has... . From the mathematical point of view, however, these objects have the same aesthetic qualities as their more familiar two and three-dimensional relatives. At the start of the 20th century, the definition of a regular polytope was as follows.
This is a "recursive" definition. It defines regularity of higher dimensional figures in terms of regular figures of a lower dimension. There is an equivalent (non-recursive) definition, which states that a polytope is regular if it has a sufficient degree of symmetry.
So for example, the cube is regular because if we choose a vertex of the cube, and one of the three edges it is on, and one of the two faces containing the edge, then this triplet, or flag Flag (geometry) In geometry, a flag is a sequence of faces of a Abstract polytope, each contained in the next, with just one face from each dimension.More formally, a flag ψ of an n-polytope is a set such that Fi ≤ Fi+1 and there is precisely one Fi in ψ for each i, .... , (vertex, edge, face) can be mapped to any other such flag by a suitable symmetry of the cube. Thus we can define a regular polytope very succinctly:
In the 20th century, some important developments were made. The symmetry Symmetry Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection.... group Group (mathematics) In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element.... s of the classical regular polytopes were generalised into what are now called Coxeter group Coxeter group In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries.... s. Coxeter groups also include the symmetry groups of regular tessellation Tessellation A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces.... s of space or of the plane. For example, the symmetry group of an infinite chessboard Chessboard A chessboard is the type of checkerboard used in the game of chess, and consists of 64 squares arranged in two alternating colors . The colors are called "black" and "white" , although the actual colors are usually dark green and buff for boards used in competition, and often natural shades of light and dark woods for home boards.... would be the Coxeter group [4,4]. Apeirotopes — infinite polytopesIn the first part of the 20th century, Coxeter and Petrie discovered three infinite structures , and . They called them regular skew polyhedra, because they seemed to satisfy the definition of a regular polyhedron — all the vertices, edges and faces are alike, all the angles are the same, and the figure has no free edges (because they can never be reached). Nowadays we call them infinite polyhedra or apeirohedra. The regular tilings of the plane , and can also be regarded as infinite polyhedra. In the 1960s Branko Grünbaum Branko Grünbaum Branko Gr?nbaum is a Croatian-born mathematician and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel.... issued a call to the geometric community to consider more abstract types of regular polytopes that he called polystromata. He developed the theory of polystromata, showing examples of new objects he called regular apeirotopes Apeirogon An apeirogon is a Degeneracy polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.... , that is, regular polytopes with infinitely Infinity Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology.... many faces. A simple example of an apeirogon Apeirogon An apeirogon is a Degeneracy polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.... would be a zig-zag. It seems to satisfy the definition of a regular polygon — all the edges are the same length, all the angles are the same, and the figure has no loose ends (because they can never be reached). More importantly, perhaps, there are symmetries of the zig-zag that can map any pair of a vertex and attached edge to any other. Since then, other regular apeirogons and higher apeirotopes have continued to be discovered. Regular complex polytopesA complex number Complex number In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:... has a real part, which is the bit we are all familiar with, and an imaginary part, which is a multiple of the square root of minus one. A complex Hilbert space Hilbert space The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces.... has its x, y, z, etc. coordinates as complex numbers. This effectively doubles the number of dimensions. A polytope constructed in such a unitary space is called a complex polytope Complex polytope A complex polytope is a generalization of a polytope which exists in a Complex number Hilbert space, where each real dimension is accompanied by an imaginary one.... . Abstract polytopesGrünbaum also discovered the 11-cell 11-cell In mathematics, the 11-cell is a duality abstract polytope . Its 11 cells are hemi-icosahedron. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has... , a four-dimensional self-dual Dual polyhedron In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron.... object whose facets are not icosahedra, but are "hemi-icosahedra" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the same face (Grünbaum, 1977). The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12. This concept may be easier for the reader to grasp if one considers the relationship of the cube and the hemicube. An ordinary cube has 8 corners, they could be labeled A to H, with A opposite H, B opposite G, and so on. In a hemicube, A and H would be treated as the same corner. So would B and G, and so on. The edge AB would become the same edge as GH, and the face ABEF would become the same face as CDGH. The new shape has only three faces, 6 edges and 4 corners. The 11-cell cannot be formed with regular geometry in flat (Euclidean) hyperspace, but only in positively-curved (elliptic) hyperspace. A few years after Grünbaum's discovery of the 11-cell 11-cell In mathematics, the 11-cell is a duality abstract polytope . Its 11 cells are hemi-icosahedron. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has... , H. S. M. Coxeter independently discovered the same shape. He had earlier discovered a similar polytope, the 57-cell 57-cell In mathematics, the 57-cell is a duality abstract polytope . Its 57 Cell s are hemi-dodecahedron. It also has 57 vertices, 171 edges and 171 faces.... (Coxeter 1982, 1984). By 1994 Grünbaum had refined his ideas and called them abstract polytope Abstract polytope In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorics properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths etc.... s. An abstract polytope is defined as a partially ordered set (poset), whose elements are the polytope's faces (vertices, edges, faces etc.) ordered by containment. Certain restrictions are imposed on the set that are similar to properties satisfied by the classical regular polytopes (including the Platonic solids). The restrictions, however, are loose enough that regular tessellations, hemicubes, and even objects as strange as the 11-cell or stranger, are all examples of regular polytopes. A geometric polytope is understood to be a realization of the abstract polytope, such that there is a one-to-one mapping from the abstract elements to the geometric. Thus, any geometric polytope may be described by the appropriate abstract poset, though not all abstract polytopes have proper geometric realizations. The theory has since been further developed, largely by Egon Schulte and Peter McMullen (McMullen, 2002), but other researchers have also made contributions. Regularity of Abstract PolytopesRegularity has a related, though different meaning for abstract polytopeAbstract polytope In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorics properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths etc.... s, since angles and lengths of edges have no meaning. The definition of regularity in terms of the transitivity of flags as given in the introduction applies to abstract polytopes. Any classical regular polytope has an abstract equivalent which is regular, obtained by taking the set of faces. But non-regular classical polytopes can have regular abstract equivalents, since abstract polytopes don't care about angles and edge lengths, for example. And a regular abstract polytope may not be realisable as a classical polytope. All polygons are regular in the abstract world, for example, whereas only those having equal angles and edges of equal length are regular in the classical world. Vertex figure of abstract polytopesThe concept of vertex figure is also defined differently for an abstract polytope Abstract polytope In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorics properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths etc.... . The vertex figure of a given abstract n-polytope at a given vertex V is the set of all abstract faces which contain V, including V itself. More formally, it is the abstract section
where Fn is the maximal face, i.e. the notional n-face which contains all other faces. Note that each i-face, i ≥ 0 of the original polytope becomes an (i−1)-face of the vertex figure. Unlike the case for Euclidean polytopes, an abstract polytope with regular facets and vertex figures may or may not be regular itself - for example, the square pyramid, all of whose facets and vertex figures are regular abstract polygons. The classical vertex figure will, however, be a realisation of the abstract one. ConstructionsPolygonsThe traditional way to construct a regular polygon, or indeed any other figure on the plane, is by compass and straightedge Compass and straightedge Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or angles using only an Idealization ruler and Compass .... . Constructing some regular polygons in this way is very simple (the easiest is perhaps the equilateral triangle), some are more complex, and some are impossible ("not constructible"). The simplest few regular polygons that are impossible to construct are the n-sided polygons with n equal to 7, 9, 11, 13, 14, 18, 19, 21,... Constructibility Constructible polygon In mathematics, a constructible polygon is a regular polygon that can be Compass and straightedge constructions. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not.... in this sense refers only to ideal constructions with ideal tools. Of course reasonably accurate approximations can be constructed by a range of methods; while theoretically possible constructions may be impractical. PolyhedraEuclid's Elements gave what amount to ruler-and-compass constructions for the five Platonic solids. (See, for example, .) However, the merely practical question of how one might draw a straight line in space, even with a ruler, might lead one to question what exactly it means to "construct" a regular polyhedron. (One could ask the same question about the polygons, of course.) The English word "construct" has the connotation of systematically building the thing constructed. The most common way presented to construct a regular polyhedron is via a fold-out net Net (polyhedron) In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron.... . To obtain a fold-out net of a polyhedron, one takes the surface of the polyhedron and cuts it along just enough edges so that the surface may be laid out flat. This gives a plan for the net of the unfolded polyhedron. Since the Platonic solids have only triangles, squares and pentagons for faces, and these are all constructible with a ruler and compass, there exist ruler-and-compass methods for drawing these fold-out nets. The same applies to star polyhedra, although here we must be careful to make the net for only the visible outer surface. If this net is drawn on cardboard, or similar foldable material (for example, sheet metal), the net may be cut out, folded along the uncut edges, joined along the appropriate cut edges, and so forming the polyhedron for which the net was designed. For a given polyhedron there may be many fold-out nets. For example, there are 11 for the cube, and over 900000 for the dodecahedron. Some interesting fold-out nets of the cube, octahedron, dodecahedron and icosahedron are available . Numerous children's toys, generally aimed at the teen or pre-teen age bracket, allow experimentation with regular polygons and polyhedra. For example, klikko provides sets of plastic triangles, squares, pentagons and hexagons that can be joined edge-to-edge in a large number of different ways. A child playing with such a toy could re-discover the Platonic solids (or the Archimedean solid Archimedean solid In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex .... s), especially if given a little guidance from a knowledgeable adult. In theory, almost any material may be used to construct regular polyhedra. Instructions for building origami Origami is the traditional Japanese art of paper folding. The goal of this art is to create a representation of an object using geometric folds and crease patterns preferably without the use of gluing or cutting the paper, and using only one piece of paper.... models may be found , for example. They may be carved out of wood, modeled out of wire, formed from stained glass. The imagination is the limit. Higher dimensions  The first approach is to embed the higher-dimensional objects in three-dimensional space, using methods analogous to the ways in which three-dimensional objects are drawn on the plane. For example, the fold out nets mentioned in the previous section have higher-dimensional equivalents. Some of these may be viewed at . One might even imagine building a model of this fold-out net, as one draws a polyhedron's fold-out net on a piece of paper. Sadly, we could never do the necessary folding of the 3-dimensional structure to obtain the 4-dimensional polytope, or polychoron Polychoron In geometry, a four-dimensional polytope is sometimes called a polychoron , from the Greek language root poly, meaning "many", and choros meaning "room" or "space".... , because of the constraints of the physical universe. Another way to "draw" the higher-dimensional shapes in 3 dimensions is via some kind of projection, for example, the analogue of either orthographic Orthographic projection Orthographic projection is a means of representing a Three-dimensional space object in 2D.It is a form of parallel projection, where the view direction is orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface.... or perspective Perspective (graphical) File:Staircase perspective.jpgPerspective in the graphic arts, such as drawing, is an approximate representation, on a flat surface , of an image as it is perceived by the eye.... projection. Coxeter's famous book on polytopes (Coxeter, 1948) has some examples of such orthographic projections. Other examples may be found on the web (see for example ). Note that immersing even 4-dimensional polychora directly into two dimensions is quite confusing. Easier to understand are 3-d models of the projections. Such models are occasionally found in science museums or mathematics departments of universities (such as that of the Université Libre de Bruxelles Université Libre de Bruxelles The Universit? Libre de Bruxelles is a French language-speaking university in Brussels, Belgium. It has about 20,000 students.... ). The intersection of a four (or higher) dimensional regular polytope with a three-dimensional hyperplane will be a polytope (not necessarily regular). If the hyperplane is moved through the shape, the three-dimensional slices can be combined, animated Animation Animation is the rapid display of a sequence of images of 2-D or 3-D artwork or model positions in order to create an illusion of movement. It is an optical illusion of Motion due to the phenomenon of persistence of vision, and can be created and demonstrated in a number of ways.... into a kind of four dimensional object, where the fourth dimension is taken to be time. In this way, we can see (if not fully grasp) the full four-dimensional structure of the four-dimensional regular polytopes, via such cutaway cross sections. This is analogous to the way a CAT scan reassembles two-dimensional images to form a 3-dimensional representation of the organs being scanned. The ideal would be an animated hologram of some sort, however, even a simple animation such as the one shown can already give some limited insight into the structure of the polytope. Another way a three-dimensional viewer can comprehend the structure of a four-dimensional polychoron is through being "immersed" in the object, perhaps via some form of virtual reality Virtual reality Virtual reality is a technology which allows a user to interact with a computer-simulated environment, whether that environment is a simulation of the real world or an imaginary world.... technology. To understand how this might work, imagine what one would see if space were filled with cubes. The viewer would be inside one of the cubes, and would be able to see cubes in front of, behind, above, below, to the left and right of himself. If one could travel in these directions, one could explore the array of cubes, and gain an understanding of its geometrical structure. An infinite array of cubes Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs.... is not a polytope in the traditional sense. In fact, it is a tessellation of 3-dimensional (Euclidean Euclidean space Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space.... ) space. However, a 4-dimensional polychoron can be considered a tessellation of a 3-dimensional non-Euclidean space, namely, a tessellation of the surface of a four-dimensional sphere Sphere A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface.... .  University of Illinois at Urbana-Champaign The University of Illinois at Urbana-Champaign is a public university research university in the state of Illinois, United States. It is the oldest and largest campus in the University of Illinois system.... has a number of pictures of what one would see if embedded in a tessellation Tessellation A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces.... of hyperbolic space Hyperbolic space In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1.... with dodecahedra. Such a tessellation forms an example of an infinite abstract regular polytope. Normally, for abstract regular polytopes, a mathematician considers that the object is "constructed" if the structure of its symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned.... is known. This is because of an important theorem in the study of abstract regular polytopes, providing a technique that allows the abstract regular polytope to be constructed from its symmetry group in a standard and straightforward manner. Regular polytopes in natureFor examples of polygons in nature, see: Each of the Platonic solids occurs naturally in one form or another: Higher polytopes can obviously not exist in a three-dimensional world. However this might not rule them out altogether. In cosmology Cosmology Cosmology is study of the Universe in its totality, and by extension, humanity's place in it. Though the word cosmology is recent , study of the Universe has a long history involving science, philosophy, esotericism, and religion.... and in string theory String theory String theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum gravity. The String s of string theory are one-dimensional oscillating lines, but they are no longer considered fundamental to the theory, which can be formulated in terms of points or surfaces too.... , physicists commonly model the Universe as having many more dimensions E8 (mathematics) In mathematics, E8 is the name given to a family of closely related structures. In particular, it is the name of four exceptional simple Lie algebra Lie algebras as well as that of the six associated simple Lie group Lie groups.... . It is possible that the Universe itself has the form of some higher polytope, regular or otherwise. Astronomers have even searched Homology sphere In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n = 1. That is,... the sky in the last few years, for tell-tale signs of a few regular candidates, so far without definite results. See also
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