Wythoff symbol
In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. A Wythoff symbol consists of a vertical bar, it represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 | 4 2 with Oh symmetry, 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D 2 h symmetry. With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space. In three dimensions, Wythoff's construction begins by choosing a generator point on the triangle. If the distance of this point from each of the sides is non-zero, the point must be chosen to be an equal distance from each edge.
A perpendicular line is dropped between the generator point and every face that it does not lie on. The three numbers in Wythoff's symbol, p, q and r, represent the corners of the Schwarz triangle used in the construction, which are π / p, π / q and π / r radians respectively; the triangle is represented with the same numbers, written. The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following: p | q r indicates that the generator lies on the corner p, p q | r indicates that the generator lies on the edge between p and q, p q r | indicates that the generator lies in the interior of the triangle. In this notation the mirrors are labeled by the reflection-order of the opposite vertex; the p, q, r values are listed before the bar. The one impossible symbol | p q r implies the generator point is on all mirrors, only possible if the triangle is degenerate, reduced to a point; this unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored.
The resulting figure has rotational symmetry only. The generator point can either be off each mirror, activated or not; this distinction creates 8 possible forms, neglecting one where the generator point is on all the mirrors. The Wythoff symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. A node is circled. There are seven generator points with each set of p, q, r: There are three special cases: p q | – This is a mixture of p q r | and p q s |, containing only the faces shared by both. | p q r – Snub forms are given by this otherwise unused symbol. | p q r s – A unique snub form for U75 that isn't Wythoff-constructible. There are 4 symmetry classes of reflection on the sphere, three in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are listed. Point groups: dihedral symmetry, p = 2, 3, 4 … tetrahedral symmetry octahedral symmetry icosahedral symmetry Euclidean groups: *442 symmetry: 45°-45°-90° triangle *632 symmetry: 30°-60°-90° triangle *333 symmetry: 60°-60°-60° triangleHyperbolic groups: *732 symmetry *832 symmetry *433 symmetry *443 symmetry *444 symmetry *542 symmetry *642 symmetry...
The above symmetry groups only include the integer solutions on the sphere. The list of Schwarz triangles includes rational numbers, determine the full set of solutions of nonconvex uniform polyhedra. In the tilings above, each triangle is a fundamental domain, colored by and odd reflections. Selected tilings created by the Wythoff con
Vertex (geometry)
In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices; the vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect, or any appropriate combination of rays and lines that result in two straight "sides" meeting at one place. A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called "convex" if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians. More a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, concave otherwise. Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope, in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices.
However, in graph theory, vertices may have fewer than two incident edges, not allowed for geometric vertices. There is a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, if a polygon is approximated by a smooth curve there will be a point of extreme curvature near each polygon vertex. However, a smooth curve approximation to a polygon will have additional vertices, at the points where its curvature is minimal. A vertex of a plane tiling or tessellation is a point. More a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x. There are two types of principal vertices: mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies in P. According to the two ears theorem, every simple polygon has at least two ears.
A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedron's surface has Euler characteristic V − E + F = 2, where V is the number of vertices, E is the number of edges, F is the number of faces; this equation is known as Euler's polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, a cube has 12 edges and 6 faces, hence 8 vertices. In computer graphics, objects are represented as triangulated polyhedra in which the object vertices are associated not only with three spatial coordinates but with other graphical information necessary to render the object such as colors, reflectance properties and surface normal. Weisstein, Eric W. "Polygon Vertex". MathWorld. Weisstein, Eric W. "Polyhedron Vertex". MathWorld. Weisstein, Eric W. "Principal Vertex". MathWorld
Hosohedron
In geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular n-gonal hosohedron has Schläfli symbol, with each spherical lune having internal angle 2π/n radians. For a regular polyhedron whose Schläfli symbol is, the number of polygonal faces may be found by: N 2 = 4 n 2 m + 2 n − m n The Platonic solids known to antiquity are the only integer solutions for m ≥ 3 and n ≥ 3; the restriction m ≥ 3 enforces. When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons can be represented as spherical lunes, having non-zero area. Allowing m = 2 admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron is represented as n abutting lunes, with interior angles of 2π/n. All these lunes share two common vertices; the digonal faces of a 2n-hosohedron, represents the fundamental domains of dihedral symmetry in three dimensions: Cnv, order 2n.
The reflection domains can be shown as alternately colored lunes as mirror images. Bisecting the lunes into two spherical triangles creates bipyramids and define dihedral symmetry Dnh, order 4n; the tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles. The dual of the n-gonal hosohedron is the n-gonal dihedron; the polyhedron is self-dual, is both a hosohedron and a dihedron. A hosohedron may be modified in the same manner as the other polyhedra to produce a truncated variation; the truncated n-gonal hosohedron is the n-gonal prism. In the limit the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation: Multidimensional analogues in general are called hosotopes. A regular hosotope with Schläfli symbol has each with a vertex figure; the two-dimensional hosotope, is a digon. The term “hosohedron” was coined by H. S. M. Coxeter, derives from the Greek ὅσος “as many”, the idea being that a hosohedron can have “as many faces as desired”.
Polyhedron Polytope McMullen, Peter. S. M. Dover Publications Inc. ISBN 0-486-61480-8 Weisstein, Eric W. "Hosohedron". MathWorld
Heptagon
In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" together with the Greek suffix "-agon" meaning angle. A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians, its Schläfli symbol is. The area of a regular heptagon of side length a is given by: A = 7 4 a 2 cot π 7 ≃ 3.634 a 2. This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with vertices at the center and at the heptagon's vertices, halving each triangle using the apothem as the common side; the apothem is half the cotangent of π / 7, the area of each of the 14 small triangles is one-fourth of the apothem. The exact algebraic expression, starting from the cubic polynomial x3 + x2 − 2x − 1 is given in complex numbers by: A = a 2 4 7 3, in which the imaginary parts offset each other leaving a real-valued expression; this expression cannot be algebraically rewritten without complex components, since the indicated cubic function is casus irreducibilis.
The area of a regular heptagon inscribed in a circle of radius R is 7 R 2 2 sin 2 π 7, while the area of the circle itself is π R 2. As 7 is a Pierpont prime but not a Fermat prime, the regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass; this type of construction is called a neusis construction. It is constructible with compass and angle trisector; the impossibility of straightedge and compass construction follows from the observation that 2 cos 2 π 7 ≈ 1.247 is a zero of the irreducible cubic x3 + x2 − 2x − 1. This polynomial is the minimal polynomial of 2cos, whereas the degree of the minimal polynomial for a constructible number must be a power of 2. An approximation for practical use with an error of about 0.2% is shown in the drawing. It is attributed to Albrecht Dürer. Let A lie on the circumference of the circumcircle. Draw arc BOC. B D = 1 2 B C gives an approximation for the edge of the heptagon; this approximation uses 3 2 ≈ 0.86603 for the side of the heptagon inscribed in the unit circle while the exact value is 2 sin π 7 ≈ 0.86777.
Example to illustrate the error: At a circumscribed circle radius r = 1 m, the absolute error of the 1st side would be -1.7 mm The regular heptagon belongs to the D7h point group, order 28. The symmetry elements are: a 7-fold proper rotation axis C7, a 7-fold improper rotation axis,S7, 7 vertical mirror planes, σv, 7 2-fold rotation axes, C2, in the plane of the heptagon and a horizontal mirror plane, σh in the heptagon's plane; the regular heptagon's side a, shorter diagonal b, longer diagonal c, with a<b<c, satisfy a 2 = c, b 2 = a, c 2 = b, 1 a = 1 b + 1 c and hence a b + a c
Hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of or t. English mathematician John Conway calls it a hextille; the internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane; the other two are the square tiling. The hexagonal tiling is the densest way to arrange circles in two dimensions; the Honeycomb conjecture states that the hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb was investigated by Lord Kelvin, who believed that the Kelvin structure is optimal. However, the less regular Weaire–Phelan structure is better; this structure exists in the form of graphite, where each sheet of graphene resembles chicken wire, with strong covalent carbon bonds. Tubular graphene sheets have been synthesised.
They have many potential applications, due to electrical properties. Silicene is similar. Chicken wire consists of a hexagonal lattice of wires; the hexagonal tiling appears in many crystals. In three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures, they are the densest known sphere packings in three dimensions, are believed to be optimal. Structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite, they differ in the way that the layers are staggered from each other, with the face-centered cubic being the more regular of the two. Pure copper, amongst other materials, forms a face-centered cubic lattice. There are three distinct uniform colorings of a hexagonal tiling, all generated from reflective symmetry of Wythoff constructions; the represent the periodic repeat of one colored tile, counting hexagonal distances as h first, k second. The same counting is used in the Goldberg polyhedra, with a notation h,k, can be applied to hyperbolic tilings for p>6.
The 3-color tiling is a tessellation generated by the order-3 permutohedrons. A chamferred hexagonal tiling replacing edges with new hexagons and transforms into another hexagonal tiling. In the limit, the original faces disappear, the new hexagons degenerate into rhombi, it becomes a rhombic tiling; the hexagons can be dissected into sets of 6 triangles. This process leads to two 2-uniform tilings, the triangular tiling: The hexagonal tiling can be considered an elongated rhombic tiling, where each vertex of the rhombic tiling is stretched into a new edge; this is similar to the relation of the rhombic dodecahedron and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions. It is possible to subdivide the prototiles of certain hexagonal tilings by two, four or nine equal pentagons: This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol, Coxeter diagram, progressing to infinity; this tiling is topologically related to regular polyhedra with vertex figure n3, as a part of sequence that continues into the hyperbolic plane.
It is related to the uniform truncated polyhedra with vertex figure n.6.6. This tiling is a part of a sequence of truncated rhombic polyhedra and tilings with Coxeter group symmetry; the cube can be seen as a rhombic hexahedron. The truncated forms have regular n-gons at the truncated vertices, nonregular hexagonal faces. Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, blue along the original edges, there are 8 forms, 7 which are topologically distinct. There are 3 types of monohedral convex hexagonal tilings, they are all isohedral. Each has parametric variations within a fixed symmetry. Type 2 contains glide reflections, is 2-isohedral keeping chiral pairs distinct. Hexagonal tilings can be made with the identical topology as the regular tiling. With isohedral faces, there are 13 variations. Symmetry given assumes all faces are the same color. Colors here represent the lattice positions.
Single-color lattices are parallelogon hexagons. Other isohedrally-tiled topological hexagonal tilings are seen as quadrilaterals and pentagons that are not edge-to-edge, but interpreted as colinear adjacent edges: The 2-uniform and 3-uniform tessellations have a rotational degree of freedom which distorts 2/3 of the hexagons, including a colinear case that can be seen as a non-edge-to-edge tiling of hexagons and larger triangles, it can be distorted into a chiral 4-colored tri-directional weaved pattern, distorting some hexagons into parallelograms. The weaved pattern with 2 colored faces have rotational 632 symmetry. A chevron pattern has pmg symmetry, lowered to p1 with 3 or 4 colored tiles; the hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing; the gap inside each hexagon allows for one circle, creating the densest packing from the triangular tiling, with each circle contact with the maximum of 6 circles.
There are 2 regular complex apeirogons, sharing the vertices of the
Tetrakis hexahedron
In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid, it can be called a disdyakis hexahedron or hexakis tetrahedron as the dual of an omnitruncated tetrahedron. The tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one mid-edge. Occurring formations of tetrahexahedra are observed in copper and fluorite systems. Polyhedral dice shaped like the tetrakis hexahedron are used by gamers. A 24-cell viewed under a vertex-first perspective projection has a surface topology of a tetrakis hexahedron and the geometric proportions of the rhombic dodecahedron, with the rhombic faces divided into two triangles; the tetrakis hexahedron appears as one of the simplest examples in building theory. Consider the Riemannian symmetric space associated to the group SL4, its Tits boundary has the structure of a spherical building whose apartments are 2-dimensional spheres. The partition of this sphere into spherical simplices can be obtained by taking the radial projection of a tetrakis hexahedron.
With Td, tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from 6 great circles on a sphere, it can be seen by a cube with its square faces triangulated by their vertices and face centers and a tetrahedron with its faces divided by vertices, mid-edges, a central point. The edges of the tetrakis hexahedron form 6 circles in the plane; each of these 6 circles represent a mirror line in tetrahedral symmetry. The 6 circles can be grouped into 3 sets of 2 pairs of orthogonal circles; these edges can be seen as a compound of 3 orthogonal square hosohedrons. If we denote the edge length of the base cube by a, the height of each pyramid summit above the cube is a/4; the inclination of each triangular face of the pyramid versus the cube face is arctan 26.565°. One edge of the isosceles triangles has length a, the other two have length 3a/4, which follows by applying the Pythagorean theorem to height and base length.
This yields an altitude of √5a/4 in the triangle. Its area is √5a/8, the internal angles are arccos and the complementary 180° − 2 arccos; the volume of the pyramid is a3/12. It can be seen as a cube with square pyramids covering each square face, it is similar to the 3D net for a 4D cubic pyramid, as the net for a square based is a square with triangles attached to each edge, the net for a cubic pyramid is a cube with square pyramids attached to each face. It is a polyhedra in a sequence defined by the face configuration V4.6.2n. This group is special for having all number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, continuing into the hyperbolic plane for any n ≥ 7. With an number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors; each face on these domains corresponds to the fundamental domain of a symmetry group with order 2,3,n mirrors at each triangle face vertex.
Disdyakis triacontahedron Disdyakis dodecahedron Kisrhombille tiling Compound of three octahedra Deltoidal icositetrahedron, another 24-face Catalan solid. Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. Wenninger, Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Tetrakis hexahedron at MathWorld. Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try: "dtO" or "kC" Tetrakis Hexahedron – Interactive Polyhedron model The Uniform Polyhedra
Geometry
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, 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 and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. 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, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study 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 means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
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, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp