1.
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
2.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
3.
Rectification (geometry)
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In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the facets of the original polytope. A rectification operator is denoted by the symbol r, for example, r is the rectified cube. Conway polyhedron notation uses ambo for this operator, in graph theory this operation creates a medial graph. Rectification is the point of a truncation process. The highest degree of rectification creates the dual polytope, a rectification truncates edges to points. A birectification truncates faces to points, a trirectification truncates cells to points, and so on. New vertices are placed at the center of the edges of the original polygon, each platonic solid and its dual have the same rectified polyhedron. The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual, the rectified octahedron, whose dual is the cube, is the cuboctahedron. The rectified icosahedron, whose dual is the dodecahedron, is the icosidodecahedron, a rectified square tiling is a square tiling. A rectified triangular tiling or hexagonal tiling is a trihexagonal tiling, examples If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. The resulting medial graph remains polyhedral, so by Steinitzs theorem it can be represented as a polyhedron, the Conway polyhedron notation equivalent to rectification is ambo, represented by a. Applying twice aa, is Conways expand operation, e, which is the same as Johnsons cantellation operation, t0,2 generated from regular polyhedral, each Convex regular 4-polytope has a rectified form as a uniform 4-polytope. Its rectification will have two types, a rectified polyhedron left from the original cells and polyhedron as new cells formed by each truncated vertex. A rectified is not the same as a rectified, however, a further truncation, called bitruncation, is symmetric between a 4-polytope and its dual. Examples A first rectification truncates edges down to points, If a polytope is regular, this form is represented by an extended Schläfli symbol notation t1 or r. A second rectification, or birectification, truncates faces down to points, If regular it has notation t2 or 2r. For polyhedra, a birectification creates a dual polyhedron, higher degree rectifications can be constructed for higher dimensional polytopes
4.
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
5.
Petrie polygon
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In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets. The Petrie polygon of a polygon is the regular polygon itself. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the interior to it. The plane in question is the Coxeter plane of the group of the polygon. These polygons and projected graphs are useful in visualizing symmetric structure of the regular polytopes. John Flinders Petrie was the son of Egyptologist Flinders Petrie. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability, in periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them. He first noted the importance of the skew polygons which appear on the surface of regular polyhedra. When my incredulity had begun to subside, he described them to me, one consisting of squares, six at each vertex, in 1938 Petrie collaborated with Coxeter, Patrick du Val, and H. T. Flather to produce The Fifty-Nine Icosahedra for publication, realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes. In 1972, a few months after his retirement, Petrie was killed by a car attempting to cross a motorway near his home in Surrey. The idea of Petrie polygons was later extended to semiregular polytopes, the Petrie polygon of the regular polyhedron has h sides, where h+2=24/. The regular duals, and, are contained within the same projected Petrie polygon, three of the Kepler–Poinsot polyhedra have hexagonal, and decagrammic, petrie polygons. The Petrie polygon projections are most useful for visualization of polytopes of dimension four and this table represents Petrie polygon projections of 3 regular families, and the exceptional Lie group En which generate semiregular and uniform polytopes for dimensions 4 to 8. Coxeter, H. S. M. Regular Polytopes, 3rd ed, Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons Ball, W. W. R. and H. S. M. Coxeter Mathematical Recreations and Essays, 13th ed. The Beauty of Geometry, Twelve Essays, Dover Publications LCCN 99-35678 Peter McMullen, Egon Schulte Abstract Regular Polytopes, ISBN 0-521-81496-0 Steinberg, Robert, ON THE NUMBER OF SIDES OF A PETRIE POLYGON Weisstein, Eric W. Petrie polygon. Weisstein, Eric W. Cross polytope graphs, Weisstein, Eric W. Gosset graph 3_21
6.
Dihedral group
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
7.
Coxeter element
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In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. Note that this assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple classes of Coxeter elements. There are many different ways to define the Coxeter number h of a root system. A Coxeter element is a product of all simple reflections, the product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. The Coxeter number is the number of roots divided by the rank, the number of reflections in the Coxeter group is half the number of roots. The Coxeter number is the order of any Coxeter element, if the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi The dimension of the corresponding Lie algebra is n, where n is the rank and h is the Coxeter number. The Coxeter number is the highest degree of an invariant of the Coxeter group acting on polynomials. Notice that if m is a degree of a fundamental invariant then so is h +2 − m, the eigenvalues of a Coxeter element are the numbers e2πi/h as m runs through the degrees of the fundamental invariants. Since this starts with m =2, these include the primitive hth root of unity, ζh = e2πi/h, an example, has h=30, so 64*30/g =12 -3 -6 -5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2= 960*15 =14400. Coxeter elements of A n −1 ≅ S n, considered as the group on n elements, are n-cycles, for simple reflections the adjacent transpositions, …. The dihedral group Dihm is generated by two reflections that form an angle of 2 π /2 m, and thus their product is a rotation by 2 π / m. For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π/h and this is called the Coxeter plane and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi/h. This plane was first systematically studied in, and subsequently used in to provide uniform proofs about properties of Coxeter elements, for polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the Platonic solids, in three dimensions, the symmetry of a regular polyhedron, with one directed petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry Sh, order h. Adding a mirror, the symmetry can be doubled to symmetry, Dhd. In orthogonal 2D projection, this becomes dihedral symmetry, Dihh, in four dimension, the symmetry of a regular polychoron, with one directed petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry +1/h, order h. In five dimension, the symmetry of a regular polyteron, with one directed petrie polygon marked, is represented by the composite of 5 reflections