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

Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is one of the five Platonic solids, it has 6 faces, 12 edges, 8 vertices. The cube is a square parallelepiped, an equilateral cuboid and a right rhombohedron, it is a regular square prism in three orientations, a trigonal trapezohedron in four orientations. The cube is dual to the octahedron, it has octahedral symmetry. The cube is the only convex polyhedron; the cube has four special orthogonal projections, centered, on a vertex, edges and normal to its vertex figure. The first and third correspond to the B2 Coxeter planes; the cube can be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are while the interior consists of all points with −1 < xi < 1 for all i.

In analytic geometry, a cube's surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of edge length a: As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares and second powers. A cube has the largest volume among cuboids with a given surface area. A cube has the largest volume among cuboids with the same total linear size. For a cube whose circumscribing sphere has radius R, for a given point in its 3-dimensional space with distances di from the cube's eight vertices, we have: ∑ i = 1 8 d i 4 8 + 16 R 4 9 = 2. Doubling the cube, or the Delian problem, was the problem posed by ancient Greek mathematicians of using only a compass and straightedge to start with the length of the edge of a given cube and to construct the length of the edge of a cube with twice the volume of the original cube, they were unable to solve this problem, in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number.

The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123. The cube has three classes of symmetry, which can be represented by vertex-transitive coloring the faces; the highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color; the lowest symmetry D2h is a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol. A cube has eleven nets: that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the same color, one would need at least three colors; the cube is the cell of the only regular tiling of three-dimensional Euclidean space. It is unique among the Platonic solids in having faces with an number of sides and it is the only member of that group, a zonohedron; the cube can be cut into six identical square pyramids.

If these square pyramids are attached to the faces of a second cube, a rhombic dodecahedron is obtained. The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or hypercube. More properly, a hypercube is the analogue of the cube in n-dimensional Euclidean space and a tesseract is the order-4 hypercube. A hypercube is called a measure polytope. There are analogues of the cube in lower dimensions too: a point in dimension 0, a line segment in one dimension and a square in two dimensions; the quotient of the cube by the antipodal map yields the hemicube. If the original cube has edge length 1, its dual polyhedron has edge length 2 / 2; the cube is a special case in various classes of general polyhedra: The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. These two together form the stella octangula; the int

Star polyhedron

In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: Polyhedra. Concave polyhedra of a particular kind which alternate convex and concave or saddle vertices in a repetitive way. Mathematically these figures are examples of star domains. Mathematical studies of star polyhedra are concerned with regular, uniform polyhedra, or the duals of the uniform polyhedra. All these stars are of the self-intersecting kind; the regular star polyhedra are self-intersecting polyhedra. They may either have self-intersecting faces, or self-intersecting vertex figures. There are four regular star polyhedra, known as the Kepler–Poinsot polyhedra; the Schläfli symbol implies faces with p sides, vertex figures with q sides. Two of them have pentagrammic faces and two have pentagrammic vertex figures; these images show each form with a single face colored yellow to show the visible portion of that face.

There are many uniform star polyhedra including two infinite series, of prisms and of antiprisms, their duals. The uniform and dual uniform star polyhedra are self-intersecting polyhedra, they may either have self-intersecting vertex figures or both. The uniform star polyhedra have regular star polygon faces; the dual uniform star polyhedra have regular star polygon vertex figures. Beyond the forms above, there are unlimited classes of self-intersecting polyhedra. Two important classes are the stellations of convex polyhedra and their duals, the facettings of the dual polyhedra. For example, the complete stellation of the icosahedron can be interpreted as a self-intersecting polyhedron composed of 12 identical faces, each a wound polygon. Below is an illustration of this polyhedron with one face drawn in yellow. A self-intersecting polytope in any number of dimensions is called a star polytope. A regular polytope is a star polytope if either its vertex figure is a star polytope. In four dimensions, the 10 regular star polychora are called the Schläfli–Hess polychora.

Analogous to the regular star polyhedra, these 10 are all composed of facets which are either one of the five regular Platonic solids or one of the four regular star Kepler–Poinsot polyhedra. For example, the great grand stellated 120-cell, projected orthogonally into 3-space, looks like this: There are no regular star polytopes in dimensions higher than 4. A polyhedron which does not cross itself, such that all of the interior can be seen from one interior point, is an example of a star domain; the visible exterior portions of many self-intersecting star polyhedra form the boundaries of star domains, but despite their similar appearance, as abstract polyhedra these are different structures. For instance, the small stellated dodecahedron has 12 pentagram faces, but the corresponding star domain has 60 isosceles triangle faces, correspondingly different numbers of vertices and edges. Polyhedral star domains appear in various types of architecture religious in nature. For example they are seen on many baroque churches as symbols of the Pope who built the church, on Hungarian churches and on other religious buildings.

These stars can be used as decorations. Moravian stars can be constructed in various forms. Star polygon Stellation Polyhedral compound List of uniform polyhedra List of uniform polyhedra by Schwarz triangle Coxeter, H. S. M. M. S. Longuet-Higgins and J. C. P Miller, Uniform Polyhedra, Phil. Trans. 246 A pp. 401–450. Coxeter, H. S. M. Regular Polytopes, 3rd. Ed. Dover Publications, 1973. ISBN 0-486-61480-8. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Tarnai, T. Krähling, J. and Kabai, S.. Or Weisstein, Eric W. "Star Polyhedron". MathWorld

Dual polyhedron

In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are 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 many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.

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 defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. 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 poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.

R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; 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, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. 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 say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.

The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.

When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form

Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O itself is a subgroup of the Euclidean group E of all isometries. Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them; the symmetry group of an object is sometimes called full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO of the 3D space itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral; the point groups in three dimensions are used in chemistry to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, in this context they are called molecular point groups.

Finite Coxeter groups are a special set of point groups generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group is represented by a Coxeter -- Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. SO is a subgroup of E +, which consists of i.e. isometries preserving orientation. O is the direct product of SO and the group generated by inversion: O = SO × Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. There is a 1-to-1 correspondence between all groups of direct isometries H in O and all groups K of isometries in O that contain inversion: K = H × H = K ∩ SOFor instance, if H is C2 K is C2h, or if H is C3 K is S6. If a group of direct isometries H has a subgroup L of index 2 apart from the corresponding group containing inversion there is a corresponding group that contains indirect isometries but no inversion: M = L ∪ where isometry is identified with A.

An example would be C4 for H and S4 for M. Thus M is obtained from H by inverting the isometries in H ∖ L; this group M is as abstract group isomorphic with H. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries; this is clarifying when see below. In 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O and SO. Accordingly, in 3D, for every axis the cyclic group of k-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations is normal both in the group obtained by adding reflections in planes through the axis and in the group obtained by adding a reflection plane perpendicular to the axis; the isometries of R3 that leave the origin fixed, forming the group O, can be categorized as follows: SO: identity rotation about an axis through the origin by an angle not equal to 180° rotation about an axis through the origin by an angle of 180° the same with inversion, i.e. respectively: inversion rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis reflection in a plane through the originThe 4th and 5th in particular, in a wider sense the 6th are called improper rotations.

See the similar overview including translations. When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O. For example, two 3D objects have the same symmetry type: if both have mirror symmetry, but with respect to a different mirror plane if both have 3-fold rotational symmetry, but with respect to a different axis. In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type if and only if there is a rotation mapping the whole structure of the first symmetry group to that of the second; the conjugacy definition would allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. There are many infinite isometry groups.

We may create non-cyclical abelian groups by adding more rotations around the same axis. There are non-abelian groups generated by rotations around different axes; these are free groups. They will be infinite. All the infinite groups mentioned so far are not closed as topological subgroups of O. We now discuss

Dihedral group

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 finite groups, they play an important role in group theory and chemistry; the notation for the dihedral group differs in abstract algebra. In geometry, Dn or Dihn refers to the symmetries of a group of order 2n. In abstract algebra, D2n refers to this same dihedral group; 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. 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 there are n/2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices. In either case, there are 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.

The following picture shows the effect of the sixteen elements of D 8 on a stop sign: The first row shows the effect of the eight rotations, the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left. 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, this gives the symmetries of a polygon the algebraic structure of a finite group; the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity. 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, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative. In general, the group Dn has elements r0, …, rn−1 and s0, …, sn−1, with composition given by the following formulae: r i r j = r i + j, r i s j = s i + j, s i r j = s i − j, s i s j = r i − j.

In all cases and subtraction of subscripts are to be performed using modular arithmetic with modulus n. If we center the regular polygon at the origin elements of the dihedral group act as linear transformations of the plane; this lets us represent elements of Dn with composition being matrix multiplication. This is an example of a group representation. For example, the elements of the group D4 can be represented by the following eight matrices: r 0 =, r 1 =, r 2 =, r 3 =, s 0 =, s 1 =, s 2 =

Dihedral symmetry in three dimensions

In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn. There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations: Schönflies notation, Coxeter notation, orbifold notation. Chiral Dn, +, of order 2n – dihedral symmetry or para-n-gonal group Achiral Dnh, of order 4n – prismatic symmetry or full ortho-n-gonal group Dnd, of order 4n – antiprismatic symmetry or full gyro-n-gonal group For a given n, all three have n-fold rotational symmetry about one axis, 2-fold about a perpendicular axis, hence about n of those. For n = ∞ they correspond to three frieze groups. Schönflies notation is used, with Coxeter notation in brackets, orbifold notation in parentheses; the term horizontal is used with respect to a vertical axis of rotation. In 2D the symmetry group Dn includes reflections in lines; when the 2D plane is embedded horizontally in a 3D space, such a reflection can either be viewed as the restriction to that plane of a reflection in a vertical plane, or as the restriction to the plane of a rotation about the reflection line, by 180°.

In 3D the two operations are distinguished: the group Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order. With reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis we have Dnh. Dnd, has vertical mirror planes between the horizontal rotation axes, not through them; as a result the vertical axis is a 2n-fold rotoreflection axis. Dnh is the symmetry group for a regular n-sided prisms and for a regular n-sided bipyramid. Dnd is the symmetry group for a regular n-sided antiprism, for a regular n-sided trapezohedron. Dn is the symmetry group of a rotated prism. N = 1 is not included because the three symmetries are equal to other ones: D1 and C2: group of order 2 with a single 180° rotation D1h and C2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane D1d and C2h: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that planeFor n = 2 there is not one main axes and two additional axes, but there are three equivalent ones.

D2 +, of order 4 is one of the three symmetry group types with the Klein four-group as abstract group. It has three perpendicular 2-fold rotation axes, it is the symmetry group of a cuboid with an S written on two opposite faces, in the same orientation. D2h, of order 8 is the symmetry group of a cuboid D2d, of order 8 is the symmetry group of e.g.: a square cuboid with a diagonal drawn on one square face, a perpendicular diagonal on the other one a regular tetrahedron scaled in the direction of a line connecting the midpoints of two opposite edges. For Dnh, order 4n Cnh, order 2n Cnv, order 2n Dn, +, order 2nFor Dnd, order 4n S2n, order 2n Cnv, order 2n Dn, +, order 2nDnd is subgroup of D2nh. Dnh,: D5h,: D4d,: D5d,: D17d,: List of spherical symmetry groups Point groups in three dimensions Cyclic symmetry in three dimensions Coxeter, H. S. M. and Moser, W. O. J.. Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9. CS1 maint: Multiple names: authors list N.

W. Johnson: Geometries and Transformations, ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups Conway, John Horton. "The Orbifold Notation for Two-Dimensional Groups", Structural Chemistry, Springer Netherlands, 13: 247–257, doi:10.1023/A:1015851621002 Graphic overview of the 32 crystallographic point groups – form the first parts of the 7 infinite series and 5 of the 7 separate 3D point groups