Spherical polyhedron
In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most conveniently derived in this way; the most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron; some "improper" polyhedra, such as the hosohedra and their duals the dihedra, exist as spherical polyhedra but have no flat-faced analogue. In the examples below, is a hosohedron and is the dual dihedron; the first known man-made polyhedra are spherical polyhedra carved in stone. Many have been found in Scotland, appear to date from the neolithic period. During the 10th Century, the Islamic scholar Abū al-Wafā' Būzjānī wrote the first serious study of spherical polyhedra. Two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra.
In the middle of the 20th Century, Coxeter used them to enumerate all but one of the uniform polyhedra, through the construction of kaleidoscopes. All the regular, semiregular polyhedra and their duals can be projected onto the sphere as tilings. Given by their Schläfli symbol or vertex figure a.b.c....: Spherical tilings allow cases that polyhedra do not, namely the hosohedra, regular figures as, dihedra, regular figures as. Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra – just as the sphere has a 2-to-1 covering map of the projective plane, projective polyhedra correspond under 2-fold cover to spherical polyhedra that are symmetric under reflection through the origin; the best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the centrally symmetric Platonic solids, as well as two infinite classes of dihedra and hosohedra: Hemi-cube, /2 Hemi-octahedron, /2 Hemi-dodecahedron, /2 Hemi-icosahedron, /2 Hemi-dihedron, /2, p>=1 Hemi-hosohedron, /2, p>=1 Spherical geometry Spherical trigonometry Polyhedron Projective polyhedron Toroidal polyhedron Conway polyhedron notation
Orbifold notation
In geometry, orbifold notation is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere, the frieze groups and wallpaper groups of the Euclidean plane, their analogues on the hyperbolic plane; the following types of Euclidean transformation can occur in a group described by orbifold notation: reflection through a line translation by a vector rotation of finite order around a point infinite rotation around a line in 3-space glide-reflection, i.e. reflection followed by translation. All translations which occur are assumed to form a discrete subgroup of the group symmetries being described.
Each group is denoted in orbifold notation by a finite string made up from the following symbols: positive integers 1, 2, 3, … the infinity symbol, ∞ the asterisk, * the symbol o, called a wonder and a handle because it topologically represents a torus closed surface. Patterns repeat by two translation; the symbol ×, called a miracle and represents a topological crosscap where a pattern repeats as a mirror image without crossing a mirror line. A string written in boldface represents a group of symmetries of Euclidean 3-space. A string not written in boldface represents a group of symmetries of the Euclidean plane, assumed to contain two independent translations; each symbol corresponds to a distinct transformation: an integer n to the left of an asterisk indicates a rotation of order n around a gyration point an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a kaleidoscopic point and reflects through a line an × indicates a glide reflection the symbol ∞ indicates infinite rotational symmetry around a line.
By abuse of language, we might say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way; the exceptional symbol o indicates that there are two linearly independent translations. An orbifold symbol is called good if it is not one of the following: p, pq, *p, *pq, for p,q>=2, p≠q. An object is chiral; the corresponding orbifold is non-orientable otherwise. The Euler characteristic of an orbifold can be read from its Conway symbol; each feature has a value: n without or before an asterisk counts as n − 1 n n after an asterisk counts as n − 1 2 n asterisk and × count as 1 o counts as 2. Subtracting the sum of these values from 2 gives the Euler characteristic. If the sum of the feature values is 2, the order is infinite, i.e. the notation represents a wallpaper group or a frieze group. Indeed, Conway's "Magic Theorem" indicates that the 17 wallpaper groups are those with the sum of the feature values equal to 2.
Otherwise, the order is 2 divided by the Euler characteristic. The following groups are isomorphic: 1* and *11 22 and 221 *22 and *221 2* and 2*1; this is. The symmetry of a 2D object without translational symmetry can be described by the 3D symmetry type by adding a third dimension to the object which does not add or spoil symmetry. For example, for a 2D image we can consider a piece of carton with that image displayed on one side, thus we have n• and *n•. The bullet is added on one- and two-dimensional groups to imply the existence of a fixed point. A 1D image can be drawn horizontally on a piece of carton, with a provision to avoid additional symmetry with respect to the line of the image, e.g. by drawing a horizontal bar under the image. Thus the discrete symmetry groups in one dimension are *•, *1•, ∞• and *∞•. Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the Cartesian product of the object and an asymmetric 2D or 1D object, respectively.
*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries §The diagram shows one fundamental domain in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, 2-fold gyration points as small green squares. A first few hyperbolic groups, ordered by their Euler characteristic are: Mutation of orbifolds Fibrifold notation - an extension of orbifold notation for 3d space groups John H. Conway, Olaf Delgado Friedrichs, Daniel H. Huson, W
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, 24 identical edges, each separating a triangle from a square; as such, it is a quasiregular polyhedron, i.e. an Archimedean solid, not only vertex-transitive but edge-transitive. It is the only radially equilateral convex polyhedron, its dual polyhedron is the rhombic dodecahedron. The cuboctahedron was known to Plato: Heron's Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name "Dymaxion" to this shape, used in an early version of the Dymaxion map, he called it the "Vector Equilibrium" because of its radial equilateral symmetry. He called a cuboctahedron consisting of rigid struts connected by flexible vertices a "jitterbug". With Oh symmetry, order 48, it is a rectified cube or rectified octahedron With Td symmetry, order 24, it is a cantellated tetrahedron or rhombitetratetrahedron.
With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are: A = a 2 ≈ 9.464 1016 a 2 V = 5 3 2 a 3 ≈ 2.357 0226 a 3. The cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, the two types of faces and square; the last two correspond to the B2 and A2 Coxeter planes. The skew projections show a hexagon passing through the center of the cuboctahedron; the cuboctahedron 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; the Cartesian coordinates for the vertices of a cuboctahedron centered at the origin are: An alternate set of coordinates can be made in 4-space, as 12 permutations of: This construction exists as one of 16 orthant facets of the cantellated 16-cell. The cuboctahedron's 12 vertices can represent the root vectors of the simple Lie group A3.
With the addition of 6 vertices of the octahedron, these vertices represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron. If these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created; the cuboctahedron can be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point. This dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra; the cuboctahedron is the unique convex polyhedron in which the long radius is the same as the edge length. This radial equilateral symmetry is a property of only a few polytopes, including the two-dimensional hexagon, the three-dimensional cuboctahedron, the four-dimensional 24-cell and 8-cell. Radially equilateral polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.
Therefore, all the interior elements which meet at the center of these polytopes have equilateral triangle inward faces, as in the dissection of the cuboctahedron into 6 square pyramids and 8 tetrahedra. Each of these radially equilateral polytopes occurs as cells of a characteristic space-filling tessellation: the tiling of regular hexagons, the rectified cubic honeycomb, the 24-cell honeycomb and the tesseractic honeycomb, respectively; each tessellation has a dual tessellation. The densest known regular sphere-packing in two and four dimensions uses the cell centers of one of these tessellations as sphere centers. A cuboctahedron has octahedral symmetry, its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either. A cuboctahedron can be obtained by taking an equatorial cross section of a four-dimensional 24-cell or 16-cell. A hexagon can be obtained by taking an equatorial cross section of a cuboctahedron.
The cuboctahedron is a rectified cube and a rectified octahedron. It is a cantellated tetrahedron. With this construction it is given the Wythoff symbol: 3 3 | 2. A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, six rectangles. While its edges are unequal, this solid remains vertex-uniform: the solid has the full tetrahedral symmet
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Hyperbolic plane geometry is the geometry of saddle surfaces and pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. A modern use of hyperbolic geometry is in the theory of special relativity Minkowski spacetime and gyrovector space; when geometers first realised they were working with something other than the standard Euclidean geometry they described their geometry under many different names. In the former Soviet Union, it is called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky; this page is about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. Hyperbolic geometry can be extended to three and more dimensions.
Hyperbolic geometry is more related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry and hyperbolic. All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines; this difference has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry. Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements. Single lines in hyperbolic geometry have the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, lines can be infinitely extended.
Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, adjacent angles of intersecting lines are supplementary; when we add a third line there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are infinitely many lines that do not intersect either of the given lines; these properties all are independent of the model used if the lines may look radically different. Non-intersecting lines in hyperbolic geometry have properties that differ from non-intersecting lines in Euclidean geometry: For any line R and any point P which does not lie on R, in the plane containing line R and point P there are at least two distinct lines through P that do not intersect R; this implies that there are through P an infinite number of coplanar lines that do not intersect R.
These non-intersecting lines are divided into two classes: Two of the lines are limiting parallels: there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never meeting it. All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, are called ultraparallel, diverging parallel or sometimes non-intersecting; some geometers use parallel lines instead of limiting parallel lines, with ultraparallel lines being just non-intersecting. These limiting parallels make an angle θ with PB. For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane, perpendicular to each pair of ultraparallel lines. In hyperbolic geometry, the circumference of a circle of radius r is greater than 2 π r. Let R = 1 − K, where K is the Gaussian curvature of the plane. In hyperbolic geometry, K is negative, so the square root is of a positive number.
The circumference of a circle of radius r is equal to: 2 π R sinh r R. And the area of the enclosed disk is: 4 π R 2 sinh 2 r 2 R = 2 π R 2. Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always greater than 2 π, though
Beltrami–Klein model
In geometry, the Beltrami–Klein model called the projective model, Klein disk model, the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere. The Beltrami–Klein model is named after the Italian geometer Eugenio Beltrami and the German Felix Klein while "Cayley" in Cayley–Klein model refers to the English geometer Arthur Cayley; the Beltrami–Klein model is analogous to the gnomonic projection of spherical geometry, in that geodesics are mapped to straight lines. This model is not conformal, meaning that angles and circles are distorted, whereas the Poincaré disk model preserves these. In this model and segments are straight Euclidean segments, whereas in the Poincaré disk model, lines are arcs that meet the boundary orthogonally; this model made its first appearance for hyperbolic geometry in two memoirs of Eugenio Beltrami published in 1868, first for dimension n = 2 and for general n, these essays proved the equiconsistency of hyperbolic geometry with ordinary Euclidean geometry.
The papers of Beltrami remained little noticed until and the model was named after Klein. This happened. In 1859 Arthur Cayley used the cross-ratio definition of angle due to Laguerre to show how Euclidean geometry could be defined using projective geometry, his definition of distance became known as the Cayley metric. In 1869, the young Felix Klein became acquainted with Cayley's work, he recalled that in 1870 he gave a talk on the work of Cayley at the seminar of Weierstrass and he wrote: "I finished with a question whether there might exist a connection between the ideas of Cayley and Lobachevsky. I was given the answer that these two systems were conceptually separated."Later, Felix Klein realized that Cayley's ideas give rise to a projective model of the non-Euclidean plane. As Klein puts it, "I allowed myself to be convinced by these objections and put aside this mature idea." However, in 1871, he returned to this idea, formulated it mathematically, published it. The distance function for the Beltrami–Klein model is a Cayley–Klein metric.
Given two distinct points p and q in the open unit ball, the unique straight line connecting them intersects the boundary at two ideal points, a and b, label them so that the points are, in order, a, p, q, b and |aq| > |ap| and |pb| > |qb|. The hyperbolic distance between p and q is then: d = 1 2 log | a q | | p b | | a p | | q b | The vertical bars indicate Euclidean distances between the points between them in the model, log is the natural logarithm and the factor of one half is needed to give the model the standard curvature of −1; when one of the points is the origin and Euclidean distance between the points is r the hyperbolic distance is: 1 2 ln = artanh r Where artanh is the inverse hyperbolic function or tangens hyperbolico. In two dimensions the Beltrami–Klein model is called the Klein disk model, it is a disk and the inside of the disk is a model of the entire hyperbolic plane. Lines in this model are represented by chords of the boundary circle; the points on the boundary circle are called ideal points.
Neither do points outside the disk. The model is not conformal, meaning that angles are distorted, circles on the hyperbolic plane are in general not circular in the model. Only circles that have their centre at the centre of the boundary circle are not distorted. All other circles are distorted, as are horocycles and hypercycles Chords that meet on the boundary circle are limiting parallel lines. Two chords are perpendicular if, when extended outside the disk, each goes through the pole of the other. Chords that go through the centre of the disk have their pole at infinity, orthogonal to the direction of the chord. Here is how one can use compass and straightedge constructions in the model to achieve the effect of the basic constructions in the hyperbolic plane; the pole of a line. While the pole is not a point in the hyperbolic plane most constructions will use the pole of a line in one or more ways. For a line: construct the tangents to the boundary circle through the ideal points of the line.
The point where these tangents intersect is the pole. For diameters of the disk: the pole is at infinity perpendicular to the diameter. To construct a perpendicular to a given line through a given point draw the ray from the pole of the line through the given point; the part of the ray, inside the disk is the perpendicular
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
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