1.
Schlegel diagram
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In geometry, a Schlegel diagram is a projection of a polytope from R d into R d −1 through a point beyond one of its facets or faces. The resulting entity is a subdivision of the facet in R d −1 that is combinatorially equivalent to the original polytope. Named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes, in dimensions 3 and 4, a Schlegel diagram is a projection of a polyhedron into a plane figure and a projection of a 4-polytope to 3-space, respectively. As such, Schlegel diagrams are used as a means of visualizing four-dimensional polytopes. The most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows, if it is projected from any external point, since each ray cuts it twice, it will be represented by a polygonal area divided twice over into polygons. It is always possible by suitable choice of the centre of projection to make the projection of one face completely contain the projections of all the other faces and this is called a Schlegel diagram of the polyhedron. The Schlegel diagram completely represents the morphology of the polyhedron, Sommerville also considers the case of a simplex in four dimensions, The Schlegel diagram of simplex in S4 is a tetrahedron divided into four tetrahedra. More generally, a polytope in n-dimensions has a Schegel diagram constructed by a perspective projection viewed from a point outside of the polytope, all vertices and edges of the polytope are projected onto a hyperplane of that facet. If the polytope is convex, a point near the facet will exist which maps the facet outside, and all other facets inside, so no edges need to cross in the projection. Net – A different approach for visualization by lowering the dimension of a polytope is to build a net, disconnecting facets and this maintains the geometric scale and shape, but makes the topological connections harder to see. Victor Schlegel Theorie der homogen zusammengesetzten Raumgebilde, Nova Acta, Ksl, deutsche Akademie der Naturforscher, Band XLIV, Nr. 4, Druck von E. Blochmann & Sohn in Dresden, Victor Schlegel Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper, Waren. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 Grünbaum, Branko, Kaibel, Volker, Klee, Victor, convex polytopes, New York & London, Springer-Verlag, ISBN 0-387-00424-6. George W. Hart, 4D Polytope Projection Models by 3D Printing Nrich maths – for the teenager
2.
Duoprism
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In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an -polytope, the lowest-dimensional duoprisms exist in 4-dimensional space as 4-polytopes being the Cartesian product of two polygons in 2-dimensional Euclidean space. More precisely, it is the set of points, P1 × P2 = where P1, such a duoprism is convex if both bases are convex, and is bounded by prismatic cells. Four-dimensional duoprisms are considered to be prismatic 4-polytopes, a duoprism constructed from two regular polygons of the same edge length is a uniform duoprism. An alternative, more concise way of specifying a particular duoprism is by prefixing with numbers denoting the base polygons, for example,3, other alternative names, q-gonal-p-gonal prism q-gonal-p-gonal double prism q-gonal-p-gonal hyperprism The term duoprism is coined by George Olshevsky, shortened from double prism. John Horton Conway proposed a similar name proprism for product prism, the duoprisms are proprisms formed from exactly two polytopes. A 4-dimensional uniform duoprism is created by the product of a regular n-sided polygon and it is bounded by n m-gonal prisms and m n-gonal prisms. For example, the Cartesian product of a triangle and a hexagon is a duoprism bounded by 6 triangular prisms and 3 hexagonal prisms, when m and n are identical, the resulting duoprism is bounded by 2n identical n-gonal prisms. For example, the Cartesian product of two triangles is a duoprism bounded by 6 triangular prisms, when m and n are identically 4, the resulting duoprism is bounded by 8 square prisms, and is identical to the tesseract. The m-gonal prisms are attached to each other via their m-gonal faces, similarly, the n-gonal prisms are attached to each other via their n-gonal faces, and form a second loop perpendicular to the first. These two loops are attached to each other via their square faces, and are mutually perpendicular, as m and n approach infinity, the corresponding duoprisms approach the duocylinder. As such, duoprisms are useful as non-quadric approximations of the duocylinder, a cell-centered perspective projection makes a duoprism look like a torus, with two sets of orthogonal cells, p-gonal and q-gonal prisms. The p-q duoprisms are identical to the q-p duoprisms, but look different in these projections because they are projected in the center of different cells, vertex-centered orthogonal projections of p-p duoprisms project into symmetry for odd degrees, and for even degrees. There are n vertices projected into the center, for 4,4, it represents the A3 Coxeter plane of the tesseract. The 5,5 projection is identical to the 3D rhombic triacontahedron, the regular skew polyhedron, exists in 4-space as the n2 square faces of a n-n duoprism, using all 2n2 edges and n2 vertices. The 2n n-gonal faces can be seen as removed, like the antiprisms as alternated prisms, there is a set of 4-dimensional duoantiprisms, 4-polytopes that can be created by an alternation operation applied to a duoprism. The alternated vertices create nonregular tetrahedral cells, except for the special case, the 16-cell is the only convex uniform duoantiprism. The duoprisms, t0,1,2,3, can be alternated into, ht0,1,2,3, the duoantiprisms, which cannot be made uniform in general
3.
Coxeter diagram
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In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction, each node represents a mirror. An unlabeled branch implicitly represents order-3, each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams correspond to and are used to root systems. Branches of a Coxeter–Dynkin diagram are labeled with a number p. When p =2 the angle is 90° and the mirrors have no interaction, if a branch is unlabeled, it is assumed to have p =3, representing an angle of 60°. Two parallel mirrors have a branch marked with ∞, in principle, n mirrors can be represented by a complete graph in which all n /2 branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, diagrams can be labeled by their graph structure. The first forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes, plagioschemes are simplices represented by branching graphs, and cycloschemes are simplices represented by cyclic graphs. Every Coxeter diagram has a corresponding Schläfli matrix with matrix elements ai, j = aj, as a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. It is related closely to the Cartan matrix, used in the similar but directed graph Dynkin diagrams in the cases of p =2,3,4, and 6. The determinant of the Schläfli matrix, called the Schläflian, and its sign determines whether the group is finite, affine and this rule is called Schläflis Criterion. The eigenvalues of the Schläfli matrix determines whether a Coxeter group is of type, affine type. The indefinite type is further subdivided, e. g. into hyperbolic. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups and we use the following definition, A Coxeter group with connected diagram is hyperbolic if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter group is compact if all subgroups are finite, Finite and affine groups are also called elliptical and parabolic respectively. Hyperbolic groups are also called Lannér, after F. Lannér who enumerated the compact groups in 1950
4.
Decagonal prism
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In geometry, the decagonal prism is the eighth in the infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra, the decagonal prism has 12 faces,30 edges, and 20 vertices. If faces are all regular, it is a semiregular or prismatic uniform polyhedron, the decagonal prism exists as cells in two four-dimensional uniform 4-polytopes, Weisstein, Eric W. Prism. 3-d model of a Decagonal Prism
5.
Square (geometry)
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
6.
Vertex figure
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In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Take some vertex of a polyhedron, mark a point somewhere along each connected edge. Draw lines across the faces, joining adjacent points. When done, these form a complete circuit, i. e. a polygon. This polygon is the vertex figure, more precise formal definitions can vary quite widely, according to circumstance. For example Coxeter varies his definition as convenient for the current area of discussion, most of the following definitions of a vertex figure apply equally well to infinite tilings, or space-filling tessellation with polytope cells. Make a slice through the corner of the polyhedron, cutting all the edges connected to the vertex. The cut surface is the vertex figure and this is perhaps the most common approach, and the most easily understood. Different authors make the slice in different places, Wenninger cuts each edge a unit distance from the vertex, as does Coxeter. For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint, other authors make the cut through the vertex at the other end of each edge. For irregular polyhedra, these approaches may produce a figure that does not lie in a plane. A more general approach, valid for convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices. Cromwell makes a cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere, many combinatorial and computational approaches treat a vertex figure as the ordered set of points of all the neighboring vertices to the given vertex. In the theory of polytopes, the vertex figure at a given vertex V comprises all the elements which are incident on the vertex, edges, faces. More formally it is the -section Fn/V, where Fn is the greatest face and this set of elements is elsewhere known as a vertex star. A vertex figure for an n-polytope is an -polytope, for example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a 4-polytope is a polyhedron. Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two vertices from an original face
7.
Tetragonal disphenoid
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In geometry, a disphenoid is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which two edges that are opposite each other have equal lengths. Other names for the shape are sphenoid, bisphenoid, isosceles tetrahedron, equifacial tetrahedron, almost regular tetrahedron. All the solid angles and vertex figures of a disphenoid are the same, however, a disphenoid is not a regular polyhedron, because, in general, its faces are not regular polygons, and its edges have three different lengths. If the faces of a disphenoid are equilateral triangles, it is a tetrahedron with Td tetrahedral symmetry. When the faces of a disphenoid are isosceles triangles, it is called a tetragonal disphenoid, in this case it has D2d dihedral symmetry. A sphenoid with scalene triangles as its faces is called a rhombic disphenoid, unlike the tetragonal disphenoid, the rhombic disphenoid has no reflection symmetry, so it is chiral. Both tetragonal disphenoids and rhombic disphenoids are isohedra, as well as being congruent to each other and it is not possible to construct a disphenoid with right triangle or obtuse triangle faces. When right triangles are glued together in the pattern of a disphenoid, two more types of tetrahedron generalize the disphenoid and have similar names. The digonal disphenoid has faces with two different shapes, both triangles, with two faces of each shape. The phyllic disphenoid similarly has faces with two shapes of scalene triangles, disphenoids can also be seen as digonal antiprisms or as alternated quadrilateral prisms. A tetrahedron is a if and only if its circumscribed parallelepiped is right-angled. We also have that a tetrahedron is a if and only if the center in the circumscribed sphere. The disphenoids are the polyhedra having infinitely many non-self-intersecting closed geodesics. On a disphenoid, all closed geodesics are non-self-intersecting and they are the polyhedra having a net in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints. The volume of a disphenoid with opposite edges of length l, m and n is given by V =72. There is also the following interesting relation connecting the volume and the circumradius,16 T2 R2 = l 2 m 2 n 2 +9 V2, the squares of the lengths of the bimedians are 12,12,12. If the four faces of a tetrahedron have the same perimeter, if the four faces of a tetrahedron have the same area, then it is a disphenoid
8.
Coxeter notation
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The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. For Coxeter groups defined by pure reflections, there is a correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors, the Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the An group is represented by, to imply n nodes connected by n-1 order-3 branches, example A2 = = or represents diagrams or. Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like, Coxeter allowed for zeros as special cases to fit the An family, like A3 = = = =, like = =. Coxeter groups formed by cyclic diagrams are represented by parenthesese inside of brackets, if the branch orders are equal, they can be grouped as an exponent as the length the cycle in brackets, like =, representing Coxeter diagram or. More complicated looping diagrams can also be expressed with care, the paracompact complete graph diagram or, is represented as with the superscript as the symmetry of its regular tetrahedron coxeter diagram. The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs, so the Coxeter diagram = A2×A2 = 2A2 can be represented by × =2 =. For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, Coxeters notation represents rotational/translational symmetry by adding a + superscript operator outside the brackets which cuts the order of the group in half. This is called a direct subgroup because what remains are only direct isometries without reflective symmetry, + operators can also be applied inside of the brackets, and creates semidirect subgroups that include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches next to it, the subgroup index is 2n for n + operators. So the snub cube, has symmetry +, and the tetrahedron, has symmetry. Johnson extends the + operator to work with a placeholder 1 nodes, in general this operation only applies to mirrors bounded by all even-order branches. The 1 represents a mirror so can be seen as, or, like diagram or, the effect of a mirror removal is to duplicate connecting nodes, which can be seen in the Coxeter diagrams, =, or in bracket notation, = =. Each of these mirrors can be removed so h = = = and this can be shown in a Coxeter diagram by adding a + symbol above the node, = =. If both mirrors are removed, a subgroup is generated, with the branch order becoming a gyration point of half the order, q = = +. For example, = = = ×, order 4. = +, the opposite to halving is doubling which adds a mirror, bisecting a fundamental domain, and doubling the group order
9.
Dual polytope
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron, duality preserves the symmetries of a polyhedron. Therefore, for classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. There are many kinds of duality, the kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality. The duality of polyhedra is often defined in terms of polar reciprocation about a concentric sphere. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2. The vertices of the dual are the reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required plane at infinity. Some theorists prefer to stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged
10.
10-10 duopyramid
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In geometry of 4 dimensions, a 10-10 duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two decagons. It has 100 vertices,200 edges,120 faces, in 20 decagonal prism cells and it has Coxeter diagram, and symmetry, order 800. The uniform 10-10 duoprism can be constructed from × or × symmetry, order 400 or 100, the regular complex polytope 102, in C2 has a real representation as a 10-10 duoprism in 4-dimensional space. 102 has 100 vertices, and 20 10-edges and its symmetry is 102, order 200. It also has a lower construction, or 10×10, with symmetry 1010. This is the if the red and blue 10-edges are considered distinct. The dual of a 10-10 duoprism is called a 10-10 duopyramid and it has 100 tetragonal disphenoid cells,200 triangular faces,120 edges, and 20 vertices. Orthogonal projection The regular complex polygon 210 has 20 vertices in C2 with a representation in R4 matching the same vertex arrangement of the 10-10 duopyramid. It has 100 2-edges corresponding to the edges of the 10-10 duopyramid. The vertices and edges makes a complete graph with each vertex from one decagon is connected to every vertex on the other. 3-3 duoprism 3-4 duoprism 5-5 duoprism Tesseract Convex regular 4-polytope Duocylinder Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc. Coxeter, The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 978-0-486-40919-1 Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 20010, ISBN 978-1-56881-220-5 Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 Olshevsky, George, archived from the original on 4 February 2007. Catalogue of Convex Polychora, section 6, George Olshevsky
11.
Convex polytope
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A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms polytope and convex polyhedron interchangeably. In addition, some require a polytope to be a bounded set. The terms bounded/unbounded convex polytope will be used whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or -manifold, Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. A comprehensive and influential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum, in 2003 the 2nd edition of the book was published, with significant additional material contributed by new writers. In Grünbaums book, and in other texts in discrete geometry. Grünbaum points out that this is solely to avoid the repetition of the word convex. A polytope is called if it is an n-dimensional object in Rn. Many examples of bounded convex polytopes can be found in the article polyhedron, a convex polytope may be defined in a number of ways, depending on what is more suitable for the problem at hand. Grünbaums definition is in terms of a set of points in space. Other important definitions are, as the intersection of half-spaces and as the hull of a set of points. This is equivalent to defining a bounded convex polytope as the hull of a finite set of points. Such a definition is called a vertex representation, for a compact convex polytope, the minimal V-description is unique and it is given by the set of the vertices of the polytope. A convex polytope may be defined as an intersection of a number of half-spaces. Such definition is called a half-space representation, there exist infinitely many H-descriptions of a convex polytope. However, for a convex polytope, the minimal H-description is in fact unique and is given by the set of the facet-defining halfspaces. A closed half-space can be written as an inequality, a 1 x 1 + a 2 x 2 + ⋯ + a n x n ≤ b where n is the dimension of the space containing the polytope under consideration
12.
Vertex-uniform
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
13.
Facet-transitive
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In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, in other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex polyhedra are the shapes that will make fair dice. They can be described by their face configuration, a polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral and they are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, a polyhedron which is isohedral and isogonal is said to be noble. A polyhedron is if it contains k faces within its symmetry fundamental domain. Similarly a k-isohedral tiling has k separate symmetry orbits, a monohedral polyhedron or monohedral tiling has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces, a facet-transitive or isotopic figure is a n-dimensional polytopes or honeycomb, with its facets congruent and transitive. The dual of an isotope is an isogonal polytope, by definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, an isotopic 3-dimensional figure is isohedral. An isotopic 4-dimensional figure is isochoric, edge-transitive Anisohedral tiling Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.367 Transitivity Olshevsky, George. Archived from the original on 4 February 2007
14.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
15.
4-polytope
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In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements, vertices, edges, faces, each face is shared by exactly two cells. The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron, topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space, similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space, a 4-polytope is a closed four-dimensional figure. It comprises vertices, edges, faces and cells, a cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are also 4-polytopes, i. e. it is not a compound, the most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube. 4-polytopes cannot be seen in space due to their extra dimension. Several techniques are used to help visualise them, Orthogonal projection Orthogonal projections can be used to show various symmetry orientations of a 4-polytope. They can be drawn in 2D as vertex-edge graphs, and can be shown in 3D with solid faces as visible projective envelopes. Perspective projection Just as a 3D shape can be projected onto a flat sheet, sectioning Just as a slice through a polyhedron reveals a cut surface, so a slice through a 4-polytope reveals a cut hypersurface in three dimensions. A sequence of sections can be used to build up an understanding of the overall shape. The extra dimension can be equated with time to produce an animation of these cross sections. The topology of any given 4-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 4-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, like all polytopes, 4-polytopes may be classified based on properties like convexity and symmetry. Self-intersecting 4-polytopes are also known as star 4-polytopes, from analogy with the shapes of the non-convex star polygons. A 4-polytope is regular if it is transitive on its flags and this means that its cells are all congruent regular polyhedra, and similarly its vertex figures are congruent and of another kind of regular polyhedron
16.
Cartesian product
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In Set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B, products can be specified using set-builder notation, e. g. A table can be created by taking the Cartesian product of a set of rows, If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More generally, a Cartesian product of n sets, also known as an n-fold Cartesian product, can be represented by an array of n dimensions, an ordered pair is a 2-tuple or couple. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, an illustrative example is the standard 52-card deck. The standard playing card ranks form a 13-element set, the card suits form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form, both sets are distinct, even disjoint. The main historical example is the Cartesian plane in analytic geometry, usually, such a pairs first and second components are called its x and y coordinates, respectively, cf. picture. The set of all such pairs is thus assigned to the set of all points in the plane, a formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, the Kuratowski definition, is =, note that, under this definition, X × Y ⊆ P, where P represents the power set. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, let A, B, C, and D be sets. × C ≠ A × If for example A =, then × A = ≠ = A ×, the Cartesian product behaves nicely with respect to intersections, cf. left picture. × = ∩ In most cases the above statement is not true if we replace intersection with union, cf. middle picture. Other properties related with subsets are, if A ⊆ B then A × C ⊆ B × C, the cardinality of a set is the number of elements of the set. For example, defining two sets, A = and B =, both set A and set B consist of two elements each. Their Cartesian product, written as A × B, results in a new set which has the following elements, each element of A is paired with each element of B. Each pair makes up one element of the output set, the number of values in each element of the resulting set is equal to the number of sets whose cartesian product is being taken,2 in this case
17.
Square
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
18.
Decagon
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In geometry, a decagon is a ten-sided polygon or 10-gon. A regular decagon has all sides of length and each internal angle will always be equal to 144°. Its Schläfli symbol is and can also be constructed as a pentagon, t. By simple trigonometry, d =2 a, and it can be written algebraically as d = a 5 +25. The side of a regular decagon inscribed in a circle is −1 +52 =1 ϕ. As 10 =2 ×5, a power of two times a Fermat prime, it follows that a regular decagon is constructible using compass and straightedge, or by an edge-bisection of a regular pentagon. An alternative method is as follows, Construct a pentagon in a circle by one of the shown in constructing a pentagon. Extend a line from each vertex of the pentagon through the center of the circle to the side of that same circle. Where each line cuts the circle is a vertex of the decagon, the five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon, both in the construction with given circumcircle as well as with given side length is the golden ratio dividing a line segment by exterior division the determining construction element. In the construction with given circumcircle the circular arc around G with radius GE3 produces the segment AH, a M ¯ M H ¯ = A H ¯ A M ¯ =1 +52 = Φ ≈1.618. In the construction with side length the circular arc around D with radius DA produces the segment E10F. E1 E10 ¯ E1 F ¯ = E10 F ¯ E1 E10 ¯ = R a =1 +52 = Φ ≈1.618, the regular decagon has Dih10 symmetry, order 20. There are 3 subgroup dihedral symmetries, Dih5, Dih2, and Dih1, and 4 cyclic group symmetries, Z10, Z5, Z2, and Z1. These 8 symmetries can be seen in 10 distinct symmetries on the decagon, john Conway labels these by a letter and group order. Full symmetry of the form is r20 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g10 subgroup has no degrees of freedom but can seen as directed edges
19.
Regular complex polytope
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In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a collection of points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes. Precise definitions exist only for the regular polytopes, which are configurations. The regular complex polytopes have been characterized, and can be described using a symbolic notation developed by Coxeter. Some complex polytopes which are not fully regular have also been described, the complex line C1 has one dimension with real coordinates and another with imaginary coordinates. Applying real coordinates to both dimensions is said to give it two dimensions over the real numbers, a real plane, with the imaginary axis labelled as such, is called an Argand diagram. Because of this it is called the complex plane. Complex 2-space is thus a four-dimensional space over the reals, a complex n-polytope in complex n-space is the analogue of a real n-polytope in real n-space. There is no natural complex analogue of the ordering of points on a real line, because of this a complex polytope cannot be seen as a contiguous surface and it does not bound an interior in the way that a real polytope does. In the case of polytopes, a precise definition can be made by using the notion of symmetry. For any regular polytope the symmetry group acts transitively on the flags, thus, by definition, regular complex polytopes are configurations in complex unitary space. The regular complex polytopes were discovered by Shephard, and the theory was developed by Coxeter. A complex polytope exists in the space of equivalent dimension. For example, the vertices of a polygon are points in the complex plane C2. Thus, an edge can be given a system consisting of a single complex number. In a regular polytope the vertices incident on the edge are arranged symmetrically about their centroid. So we may assume that the vertices on the edge satisfy the equation x p −1 =0 where p is the number of incident vertices. Thus, in the Argand diagram of the edge, the points lie at the vertices of a regular polygon centered on the origin
20.
Duopyramid
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In geometry of 4 dimensions or higher, a duopyramid is a dual polytope of a duoprism. As a dual uniform polychoron, it is called a p-q duopyramid with a composite Schläfli symbol +, the regular 16-cell can be seen as a 4-4 duopyramid, symmetry, order 128. A p-q duopyramid has Coxeter group symmetry, order 4pq, when p and q are identical, the symmetry is doubled as, order 8p2. Edges exist on all pairs of vertices between the p-gon and q-gon, the 1-skeleton of a p-q duopyramid represents edges of each p and q polygon and pq complete bipartite graph between them. A p-q duopyramid can be seen as two regular polygons of p and q sides with the same center and orthogonal orientations in 4 dimensions. Along with the p and q edges of the two polygons, all permutations of vertices in one polygon to vertices in the other form edges, All faces are triangular, with one edge of one polygon connected to one vertex of the other polygon. The p and q sided polygons are hollow, passing through the polytope center, cells are tetrahedra constructed as all permutations of edge pairs between each polygon. It can be understood by analogy to the relation of the 3D prisms and their dual bipyramids with Schläfli symbol +, the symmetry will be the product of the symmetry of the two polygons. So a rectangle-rectangle duopyramid would be identical to the uniform 4-4 duopyramid. The coordinates of a p-q duopyramid can be given as, i=1. p, the 2n vertices of a n-n duopyramid can be orthogonally projected into two regular n-gons with edges between all vertices of each n-gon. The regular 16-cell can be seen as a 4-4 duopyramid, being dual to the 4-4 duoprism, as a 4-4 duopyramid, the 16-cells symmetry is, order 64, and doubled to, order 128 with the 2 central squares interchangeable. The regular 16-cell has a symmetry, order 384. Archived from the original on 4 February 2007, polygloss - glossary of higher-dimensional terms
21.
Isosceles triangle
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In geometry, an isosceles triangle is a triangle that has two sides of equal length. By the isosceles triangle theorem, the two angles opposite the sides are themselves equal, while if the third side is different then the third angle is different. By the Steiner–Lehmus theorem, every triangle with two angle bisectors of equal length is isosceles, in an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle, the vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base, if needed, and the term leg is not generally used. A triangle with two equal sides has exactly one axis of symmetry, which goes through the vertex angle. Thus the axis of symmetry coincides with the bisector of the vertex angle, the median drawn to the base, the altitude drawn from the vertex angle. Whether the isosceles triangle is acute, right or obtuse depends on the vertex angle, in Euclidean geometry, the base angles cannot be obtuse or right because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. The Euler line of any triangle goes through the orthocenter, its centroid. In an isosceles triangle with two equal sides, the Euler line coincides with the axis of symmetry. This can be seen as follows, if the vertex angle is acute, then the orthocenter, the centroid, and the circumcenter all fall inside the triangle. In an isosceles triangle the incenter lies on the Euler line, the Steiner inellipse of any triangle is the unique ellipse that is internally tangent to the triangles three sides at their midpoints. For any isosceles triangle with area T and perimeter p, we have 2 p b 3 − p 2 b 2 +16 T2 =0. By substituting the height, the formula for the area of a triangle can be derived from the general formula one-half the base times the height. This is what Herons formula reduces to in the isosceles case, if the apex angle and leg lengths of an isosceles triangle are known, then the area of that triangle is, T =2 = a 2 sin cos . This is derived by drawing a line from the base of the triangle. The bases of two right triangles are both equal to the hypotenuse times the sine of the bisected angle by definition of the term sine. For the same reason, the heights of these triangles are equal to the times the cosine of the bisected angle
22.
Vertex arrangement
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In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes, for example, a square vertex arrangement is understood to mean four points in a plane, equal distance and angles from a center point. Two polytopes share the same vertex arrangement if they share the same 0-skeleton, the same set of vertices can be connected by edges in different ways. For example, the pentagon and pentagram have the same vertex arrangement, a vertex arrangement is often described by the convex hull polytope which contains it. For example, the regular pentagram can be said to have a vertex arrangement. Infinite tilings can also share common vertex arrangements, for example, this triangular lattice of points can be connected to form either isosceles triangles or rhombic faces. Polyhedra can also share an edge arrangement while differing in their faces, for example, of the ten nonconvex regular Schläfli-Hess polychora, there are only 7 unique face arrangements. Synonyms for special cases include company for a 2-regiment and army for a 0-regiment, n-skeleton - a set of elements of dimension n and lower in a higher polytope. Vertex figure - A local arrangement of faces in a polyhedron around a single vertex, archived from the original on 4 February 2007. Archived from the original on 4 February 2007, archived from the original on 4 February 2007
23.
Complete bipartite graph
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Graph theory itself is typically dated as beginning with Leonhard Eulers 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, Llull himself had made similar drawings of complete graphs three centuries earlier. That is, it is a graph such that for every two vertices v1 ∈ V1 and v2 ∈ V2, v1v2 is an edge in E. A complete bipartite graph with partitions of size |V1|=m and |V2|=n, is denoted Km, n, for any k, K1, k is called a star. All complete bipartite graphs which are trees are stars, the graph K1,3 is called a claw, and is used to define the claw-free graphs. The graph K3,3 is called the utility graph and this usage comes from a standard mathematical puzzle in which three utilities must each be connected to three buildings, it is impossible to solve without crossings due to the nonplanarity of K3,3. Given a bipartite graph, testing whether it contains a complete bipartite subgraph Ki, a planar graph cannot contain K3,3 as a minor, an outerplanar graph cannot contain K3,2 as a minor. Conversely, every nonplanar graph contains either K3,3 or the complete graph K5 as a minor, Kn, n is a Moore graph and a -cage. The complete bipartite graphs Kn, n and Kn, n+1 have the possible number of edges among all triangle-free graphs with the same number of vertices. The complete bipartite graph Km, n has a vertex covering number of min, the complete bipartite graph Km, n has a maximum independent set of size max. The adjacency matrix of a bipartite graph Km, n has eigenvalues √, −√ and 0, with multiplicity 1,1. The Laplacian matrix of a bipartite graph Km, n has eigenvalues n+m, n, m. A complete bipartite graph Km, n has mn−1 nm−1 spanning trees, a complete bipartite graph Km, n has a maximum matching of size min. A complete bipartite graph Kn, n has a proper n-edge-coloring corresponding to a Latin square, every complete bipartite graph is a modular graph, every triple of vertices has a median that belongs to shortest paths between each pair of vertices
24.
3-3 duoprism
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In geometry of 4 dimensions, a 3-3 duoprism, the smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of two triangles. It has 9 vertices,18 edges,15 faces, in 6 triangular prism cells and it has Coxeter diagram, and symmetry, order 72. There are three constructions for the honeycomb with two lower symmetries, the regular complex polytope 32, in C2 has a real representation as a 3-3 duoprism in 4-dimensional space. 32 has 9 vertices, and 6 3-edges and its symmetry is 32, order 18. It also has a lower construction, or 3×3, with symmetry 33. This is the if the red and blue 3-edges are considered distinct. The dual of a 3-3 duoprism is called a 3-3 duopyramid and it has 9 tetragonal disphenoid cells,18 triangular faces,15 edges, and 6 vertices. It can be seen in orthogonal projection as a 6-gon circle of vertices, orthogonal projection The regular complex polygon 23 has 6 vertices in C2 with a real represention in R4 matching the same vertex arrangement of the 3-3 duopyramid. It has 9 2-edges corresponding to the edges of the 3-3 duopyramid. It can be seen in a projection with 3 sets of colored edges. This arrangement of vertices and edges makes a complete graph with each vertex from one triangle is connected to every vertex on the other. It is also called a Thomsen graph or 4-cage, 3-4 duoprism Tesseract 5-5 duoprism Convex regular 4-polytope Duocylinder Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc. Coxeter, The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 0-486-40919-8 Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 Olshevsky, George, archived from the original on 4 February 2007. Catalogue of Convex Polychora, section 6, George Olshevsky
25.
3-4 duoprism
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In geometry of 4 dimensions, a 3-4 duoprism, the second smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of a triangle and a square. The 3-4 duoprism exists in some of the uniform 5-polytopes in the B5 family, the quasiregular complex polytope 3×4, in C2 has a real representation as a 3-4 duoprism in 4-dimensional space. It has 12 vertices, and 4 3-edges and 3 4-edges and its symmetry is 34, order 12. The birectified 5-cube, has a uniform 3-4 duoprism vertex figure and it has 12 tetragonal disphenoid cells,24 isosceles triangular faces,12 edges, and 7 vertices. Polytope and polychoron Convex regular polychoron Duocylinder Tesseract Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc. Coxeter, The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 0-486-40919-8 Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 Olshevsky, George, archived from the original on 4 February 2007. Catalogue of Convex Polychora, section 6, George Olshevsky
26.
5-5 duoprism
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In geometry of 4 dimensions, a 5-5 duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two pentagons. It has 25 vertices,50 edges,35 faces, in 10 pentagonal prism cells and it has Coxeter diagram, and symmetry, order 200. Seen in a skew 2D orthogonal projection,20 of the vertices are in two rings, while 5 project into the center. The 5-5 duoprism here has an identical 2D projective appearance to the 3D rhombic triacontahedron, in this projection, the square faces project into wide and narrow rhombi seen in penrose tiling. The regular complex polytope 52, in C2 has a representation as a 5-5 duoprism in 4-dimensional space. 52 has 25 vertices, and 10 5-edges and its symmetry is 52, order 50. It also has a lower construction, or 5×5, with symmetry 55. This is the if the red and blue 5-edges are considered distinct. The birectified order-5 120-cell, constructed by all rectified 600-cells, a 5-5 duoprism vertex figure, the dual of a 5-5 duoprism is called a 5-5 duopyramid. It has 25 tetragonal disphenoid cells,50 triangular faces,35 edges and it has 25 2-edges corresponding to the connecting edges of the 5-5 duopyramid, while the 10 edges connecting the two pentagons are not included. The vertices and edges makes a complete graph with each vertex from one pentagon is connected to every vertex on the other. 3-3 duoprism 3-4 duoprism Tesseract Convex regular 4-polytope Duocylinder Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc. Coxeter, The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 0-486-40919-8 Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 Olshevsky, George, archived from the original on 4 February 2007. Catalogue of Convex Polychora, section 6, George Olshevsky
27.
Tesseract
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In geometry, the tesseract is the four-dimensional analog of the cube, the tesseract is to the cube as the cube is to the square. Just as the surface of the consists of six square faces. The tesseract is one of the six convex regular 4-polytopes, the tesseract is also called an 8-cell, C8, octachoron, octahedroid, cubic prism, and tetracube. It is the four-dimensional hypercube, or 4-cube as a part of the family of hypercubes or measure polytopes. In this publication, as well as some of Hintons later work, the tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol with hyperoctahedral symmetry of order 384, constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol ×, with symmetry order 96. As a 4-4 duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol ×, with symmetry order 64, as an orthotope it can be represented by composite Schläfli symbol × × × or 4, with symmetry order 16. Since each vertex of a tesseract is adjacent to four edges, the dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol. The standard tesseract in Euclidean 4-space is given as the hull of the points. That is, it consists of the points, A tesseract is bounded by eight hyperplanes, each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge, there are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes,24 squares,32 edges, the construction of a hypercube can be imagined the following way, 1-dimensional, Two points A and B can be connected to a line, giving a new line segment AB. 2-dimensional, Two parallel line segments AB and CD can be connected to become a square, 3-dimensional, Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. 4-dimensional, Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube and it is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a two-dimensional space. Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices, the scheme is similar to the construction of a cube from two squares, juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length, the regular complex polytope 42, in C2 has a real representation as a tesseract or 4-4 duoprism in 4-dimensional space. 42 has 16 vertices, and 8 4-edges and its symmetry is 42, order 32. It also has a lower construction, or 4×4, with symmetry 44
28.
Convex regular 4-polytope
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In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the analogs of the regular polyhedra in three dimensions and the regular polygons in two dimensions. Regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century, There are six convex and ten star regular 4-polytopes, giving a total of sixteen. The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century, Schläfli discovered that there are precisely six such figures. Schläfli also found four of the regular star 4-polytopes and he skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures. That excludes cells and vertex figures as, and, the six convex and ten star polytopes described are the only solutions to these constraints. There are four nonconvex Schläfli symbols that have cells and vertex figures, and pass the dihedral test. The regular convex 4-polytopes are the analogs of the Platonic solids in three dimensions and the convex regular polygons in two dimensions. Five of them may be thought of as close analogs of the Platonic solids, There is one additional figure, the 24-cell, which has no close three-dimensional equivalent. Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and these are fitted together along their respective faces in a regular fashion. The following tables lists some properties of the six convex regular 4-polytopes, the symmetry groups of these 4-polytopes are all Coxeter groups and given in the notation described in that article. The number following the name of the group is the order of the group, John Conway advocates the names simplex, orthoplex, tesseract, octaplex or polyoctahedron, dodecaplex or polydodecahedron, and tetraplex or polytetrahedron. The Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analog of Eulers polyhedral formula, the topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients. The following table shows some 2-dimensional projections of these 4-polytopes, various other visualizations can be found in the external links below. The Coxeter-Dynkin diagram graphs are given below the Schläfli symbol. The Schläfli–Hess 4-polytopes are the set of 10 regular self-intersecting star polychora. They are named in honor of their discoverers, Ludwig Schläfli, each is represented by a Schläfli symbol in which one of the numbers is 5/2. They are thus analogous to the regular nonconvex Kepler–Poinsot polyhedra and their names given here were given by John Conway, extending Cayleys names for the Kepler–Poinsot polyhedra, along with stellated and great, he adds a grand modifier
29.
Harold Scott MacDonald Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
30.
Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
31.
John Horton Conway
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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey, Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at an early age, his mother has recalled that he could recite the powers of two when he was four years old. By the age of eleven his ambition was to become a mathematician, after leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a terribly introverted adolescent in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person and he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the problem posed by Davenport on writing numbers as the sums of fifth powers. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos and he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University, Conway is especially known for the invention of the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics, there is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, at times Conway has said he hates the game of life–largely because it has come to overshadow some of the other deeper and more important things he has done. Nevertheless, the game did help launch a new branch of mathematics, the Game of Life is now known to be Turing complete. Conways career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner, when Gardner featured Conways Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, for instance, he discussed Conways game of Sprouts, Hackenbush, and his angel and devil problem. In the September 1976 column he reviewed Conways book On Numbers and Games, Conway is widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays and he also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, as well as philosophers football and he developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conways soldiers
32.
Norman Johnson (mathematician)
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Norman Woodason Johnson is a mathematician, previously at Wheaton College, Norton, Massachusetts. He earned his Ph. D. from the University of Toronto in 1966 with a title of The Theory of Uniform Polytopes. In 1966 he enumerated 92 convex non-uniform polyhedra with regular faces, victor Zalgaller later proved that Johnsons list was complete, and the set is now known as the Johnson solids. The theory of polytopes and honeycombs, Ph. D. Dissertation,1966 Hyperbolic Coxeter Groups, paper, convex polyhedra with regular faces, paper containing the original enumeration of the 92 Johnson solids and the conjecture that there are no others. Norman W. Johnson at the Mathematics Genealogy Project Norman W. Johnson Endowed Fund in Mathematics and Computer Science at Wheaton College