Icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries; the set of orientation-preserving symmetries forms a group referred to as A5, the full symmetry group is the product A5 × Z2. The latter group is known as the Coxeter group H3, is represented by Coxeter notation, Coxeter diagram. Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries with the largest symmetry groups. Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups. Presentations corresponding to the above are: I: ⟨ s, t ∣ s 2, t 3, 5 ⟩ I h: ⟨ s, t ∣ s 3 − 2, t 5 − 2 ⟩; these correspond to the icosahedral groups being the triangle groups.
The first presentation was given by William Rowan Hamilton in 1856, in his paper on icosian calculus. Note that other presentations are possible, for instance as an alternating group; the icosahedral rotation group I is of order 60. The group I is isomorphic to A5, the alternating group of permutations of five objects; this isomorphism can be realized by I acting on various compounds, notably the compound of five cubes, the compound of five octahedra, or either of the two compounds of five tetrahedra. The group contains 5 versions of Th with 20 versions of D3, 6 versions of D5; the full icosahedral group Ih has order 120. It has I as normal subgroup of index 2; the group Ih is isomorphic to I × Z2, or A5 × Z2, with the inversion in the center corresponding to element, where Z2 is written multiplicatively. Ih acts on the compound of five cubes and the compound of five octahedra, but −1 acts as the identity, it acts on the compound of ten tetrahedra: I acts on the two chiral halves, −1 interchanges the two halves.
Notably, it does not act as S5, these groups are not isomorphic. The group contains 6 versions of D5d. I is isomorphic to PSL2, but Ih is not isomorphic to SL2; the following groups all have order 120, but are not isomorphic: S5, the symmetric group on 5 elements Ih, the full icosahedral group 2I, the binary icosahedral groupThey correspond to the following short exact sequences and product 1 → A 5 → S 5 → Z 2 → 1 I h = A 5 × Z 2 1 → Z 2 → 2 I → A 5 → 1 In words, A 5 is a normal subgroup of S 5 A 5 is a factor of I h, a direct product A 5 is a quotient group of 2 I Note that A 5 has an exceptional irreducible 3-dimensional representation, but S 5 does not have an irreducible 3-dimensional representation, corresponding to the full icosahedral group not being the symmetric group. These can be related to linear groups over the finite field with five elements, which exhibit the subgroups and covering groups directly. In computational contexts, the rotation icosahedral group I above can be explicitly represented by the following 60 rotatio
(2,3,7) triangle group
In the theory of Riemann surfaces and hyperbolic geometry, the triangle group is important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84, of its automorphism group. A note on terminology – the " triangle group" most refers, not to the full triangle group Δ, but rather to the ordinary triangle group D of orientation-preserving maps, index 2. Torsion-free normal subgroups of the triangle group are Fuchsian groups associated with Hurwitz surfaces, such as the Klein quartic, Macbeath surface and First Hurwitz triplet. To construct the triangle group, start with a hyperbolic triangle with angles π/2, π/3, π/7; this triangle, the smallest hyperbolic Schwarz triangle, tiles the plane by reflections in its sides. Consider the group generated by reflections in the sides of the triangle, a non-Euclidean crystallographic group with this triangle for fundamental domain; the triangle group is defined as the index 2 subgroup consisting of the orientation-preserving isometries, a Fuchsian group.
It has a presentation in terms of a pair of generators, g2, g3, modulo the following relations: g 2 2 = g 3 3 = 7 = 1. Geometrically, these correspond to rotations by 2 π 2, 2 π 3, 2 π 7 about the vertices of the Schwarz triangle; the triangle group admits a presentation in terms of the group of quaternions of norm 1 in a suitable order in a quaternion algebra. More the triangle group is the quotient of the group of quaternions by its center ±1. Let η = 2cos. From the identity 3 = 7 2. We see that Q is a real cubic extension of Q; the hyperbolic triangle group is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the pair of generators i,j and relations i2 = j2 = η, ij = −ji. One chooses a suitable Hurwitz quaternion order. Here the order Q H u r is generated by elements g 2 = 1 η i j g 3 = 1 2. In fact, the order is a free Z-module over the basis 1, g 2, g 3, g 2 g 3. Here the generators satisfy the relations g 2 2 = g 3 3 = 7 = − 1, which descend to the appropriate relations in the triangle group, after quotienting by the center.
Extending the scalars from Q to R, one obtains an isomorphism between the quaternion algebra and the algebra M of real 2 by 2 matrices. Choosing a concrete isomorphism allows one to exhibit the triangle group as a specific Fuchsian group in SL as a quotient of the modular group; this can be visualized by the associated tilings, as depicted at right: the tiling on the Poincaré disc is a quotient of the modular tiling on the upper half-plane. However, for many purposes, explicit isomorphisms are unnecessary. Thus, traces of group elements can be calculated by means of the reduced trace in the quaternion algebra, the formula tr = 2 cosh
Tessellation
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to a variety of geometries. A periodic tiling has a repeating pattern; some special kinds include regular tilings with regular polygonal tiles all of the same shape, semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons; such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings.
Tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles. Decorative mosaic tilings made of small squared blocks called tesserae were employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made an early documented study of tessellations, he wrote about semiregular tessellations in his Harmonices Mundi. Some two hundred years in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries.
Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Aleksei Shubnikov and Nikolai Belov, Heinrich Heesch and Otto Kienzle. In Latin, tessella is a small cubical piece of stone or glass used to make mosaics; the word "tessella" means "small square". It corresponds to the everyday term tiling, which refers to applications of tessellations made of glazed clay. Tessellation in two dimensions called planar tiling, is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules; these rules can be varied. Common ones are that there must be no gaps between tiles, that no corner of one tile can lie along the edge of another; the tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile.
There are only three shapes that can form such regular tessellations: the equilateral triangle and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps. Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner. Irregular tessellations can be made from other shapes such as pentagons, polyominoes and in fact any kind of geometric shape; the artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, these can be used to decorate physical surfaces such as church floors. More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries.
These tiles may be any other shapes. Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane; the Conway criterion is a sufficient but not necessary set of rules for deciding if a given shape tiles the plane periodically without reflections: some tiles fail the criterion but still tile the plane. No general rule has been found for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations. Mathematically, tessellations can be extended to spaces other than the Euclidean plane; the Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more dimensions, he further defined the Schläfli symbol notation to make it easy to describe polytopes.
For example, the Schläfli symbol for an equilateral triangle is. The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is. Other methods exist for describing polygonal tilings; when the tessellation
Uniform star polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are sometimes called nonconvex polyhedra to imply self-intersecting; each polyhedron can contain star polygon vertex figures or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra, 5 quasiregular ones, 48 semiregular ones. There are two infinite sets of uniform star prisms and uniform star antiprisms. Just as star polygons correspond to circular polygons with overlapping tiles, star polyhedra that do not pass through the center have polytope density greater than 1, correspond to spherical polyhedra with overlapping tiles; the remaining 10 nonprismatic uniform star polyhedra, those that pass through the center, are the hemipolyhedra as well as Miller's monster, do not have well-defined densities. The nonconvex forms are constructed from Schwarz triangles. All the uniform polyhedra are listed below by their symmetry groups and subgrouped by their vertex arrangements.
Regular polyhedra are labeled by their Schläfli symbol. Other nonregular uniform polyhedra are listed with their vertex configuration or their Uniform polyhedron index U. Note: For nonconvex forms below an additional descriptor Nonuniform is used when the convex hull vertex arrangement has same topology as one of these, but has nonregular faces. For example an nonuniform cantellated form may have rectangles created in place of the edges rather than squares. See Prismatic uniform polyhedron. There is the tetrahemihexahedron which has tetrahedral symmetry. There are two Schwarz triangles that generate unique nonconvex uniform polyhedra: one right triangle, one general triangle; the general triangle generates the octahemioctahedron, given further on with its full octahedral symmetry. There are 8 convex forms, 10 nonconvex forms with octahedral symmetry. There are four Schwarz triangles that generate nonconvex forms, two right triangles and two general triangles:. There are 8 convex forms and 46 nonconvex forms with icosahedral symmetry..
Some of the nonconvex snub forms have reflective vertex symmetry. Coxeter identified a number of degenerate star polyhedra by the Wythoff construction method, which contain overlapping edges or vertices; these degenerate forms include: Small complex icosidodecahedron Great complex icosidodecahedron Small complex rhombicosidodecahedron Great complex rhombicosidodecahedron Complex rhombidodecadodecahedron One further nonconvex degenerate polyhedron is the Great disnub dirhombidodecahedron known as Skilling's figure, vertex-uniform, but has pairs of edges which coincide in space such that four faces meet at some edges. It is counted as a degenerate uniform polyhedron rather than a uniform polyhedron because of its double edges, it has Ih symmetry. Star polygon List of uniform polyhedra List of uniform polyhedra by Schwarz triangle Coxeter, H. S. M.. "Uniform Polyhedra". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 246: 401–450. Doi:10.1098/rsta.1954.0003.
Wenninger, Magnus. Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. OCLC 1738087. Brückner, M. Vielecke und vielflache. Theorie und geschichte.. Leipzig, Germany: Teubner, 1900. Sopov, S. P. "A proof of the completeness on the list of elementary homogeneous polyhedra", Ukrainskiui Geometricheskiui Sbornik: 139–156, MR 0326550 Skilling, J. "The complete set of uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 278: 111–135, doi:10.1098/rsta.1975.0022, ISSN 0080-4614, JSTOR 74475, MR 0365333 Har'El, Z. Uniform Solution for Uniform Polyhedra. Geometriae Dedicata 47, 57-110, 1993. Zvi Har’El, Kaleido software, dual images Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993. Messer, Peter W. Closed-Form Expressions for Uniform Polyhedra and Their Duals. Discrete & Computational Geometry 27:353-375. Klitzing, Richard. "3D uniform polyhedra". Weisstein, Eric W. "Uniform Polyhedron". MathWorld
Equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, an equilateral triangle is equiangular, it is a regular polygon, so it is referred to as a regular triangle. Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: The area is A = 3 4 a 2 The perimeter is p = 3 a The radius of the circumscribed circle is R = a 3 The radius of the inscribed circle is r = 3 6 a or r = R 2 The geometric center of the triangle is the center of the circumscribed and inscribed circles The altitude from any side is h = 3 2 a Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that: The area of the triangle is A = 3 3 4 R 2 Many of these quantities have simple relationships to the altitude of each vertex from the opposite side: The area is A = h 2 3 The height of the center from each side, or apothem, is h 3 The radius of the circle circumscribing the three vertices is R = 2 h 3 The radius of the inscribed circle is r = h 3 In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, the medians to each side coincide.
A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii ra, rb, rc, where R and r are the radii of the circumcircle and incircle is equilateral if and only if any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, knowing that any one of them is true directly implies that we have an equilateral triangle. A = b = c 1 a + 1 b + 1 c = 25 R r − 2 r 2 4 R r s = 2 R + r s 2 = 3 r 2 + 12 R r s 2 = 3 3 T s = 3 3 r s = 3 3 2 R A = B = C = 60 ∘ cos A + cos B + cos C = 3 2 sin A 2 sin B 2 sin C 2 = 1 8 T = a 2 + b 2 + c 2 4 3 T = 3 4 2 3 R = 2 r 9 R 2 = a 2 + b 2 + c 2 r = r a +
Schwarz triangle function
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. Let πα, πβ, πγ be the interior angles at the vertices of the triangle. If any of α, β, γ are greater than zero the Schwarz triangle function can be given in terms of hypergeometric functions as: s = z α 2 F 1 2 F 1 where a = /2, b = /2, c = 1-α, a' = a - c + 1 = /2, b' = b - c + 1 = /2, c' = 2 - c = 1+α; this mapping has singular points at z=0, 1, ∞, corresponding to the vertices of the triangle with angles πα, πγ, πβ respectively. At these singular points, s = 0, s = Γ Γ Γ Γ Γ Γ, s = exp Γ Γ Γ Γ Γ Γ; this formula can be derived using the Schwarzian derivative. This function can be used to map the upper half-plane to a spherical triangle on the Riemann sphere if α + β + γ > 1, or a hyperbolic triangle on the Poincaré disk if α + β + γ < 1. When α + β + γ = 1 the triangle is a Euclidean triangle with straight edges: a=0, 2 F 1 = 1, the formula reduces to that given by the Schwarz–Christoffel transformation.
In the special case of ideal triangles, where all the angles are zero, the triangle function yields the modular lambda function. This function was introduced by H. A. Schwarz as the inverse function of the conformal mapping uniformizing a Schwarz triangle. Applying successive hyperbolic reflections in its sides, such a triangle generates a tessellation of the upper half plane; the conformal mapping of the upper half plane onto the interior of the geodesic triangle generalizes the Schwarz–Christoffel transformation. By the Schwarz reflection principle, the discrete group generated by hyperbolic reflections in the sides of the triangle induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two dimensional representation corresponds to the monodromy of the ordinary differential equation and induces a group of Möbius transformations on quotients of solutions. Since the triangle function is the inverse function of such a quotient, it is therefore an automorphic function for this discrete group of Möbius transformations.
This is a special case of a general method of Henri Poincaré that associates automorphic forms with ordinary differential equations with regular singular points. In this section two different models are given for hyperbolic geometry on the unit disk or equivalently the upper half plane; the group G = SU is formed of matrices g = with | α | 2 − | β | 2 = 1. It is a subgroup of Gc = SL, the group of complex 2 × 2 matrices with determinant 1; the group Gc acts by Möbius transformations on the extended complex plane. The subgroup G acts as automorphisms of the unit disk D and the subgroup G1 = SL acts as automorphisms of the upper half plane. If C =
Tetrahedral symmetry
A regular tetrahedron has 12 rotational symmetries, a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is one such symmetry for each permutation of the vertices of the tetrahedron; the set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4. Chiral and full are discrete point symmetries, they are among the crystallographic point groups of the cubic crystal system. Seen in stereographic projection 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 intersection of these circles meet at 3 gyration points. T, 332, +, or 23, of order 12 – chiral or rotational tetrahedral symmetry. There are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D2 or 222, with in addition four 3-fold axes, centered between the three orthogonal directions.
This group is isomorphic to the alternating group on 4 elements. The conjugacy classes of T are: identity 4 × rotation by 120° clockwise:, 4 × rotation by 120° counterclockwise 3 × rotation by 180°The rotations by 180°, together with the identity, form a normal subgroup of type Dih2, with quotient group of type Z3; the three elements of the latter are the identity, "clockwise rotation", "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. A4 is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of |G|, there does not exist a subgroup of G with order d: the group G = A4 has no subgroup of order 6. Although it is a property for the abstract group in general, it is clear from the isometry group of chiral tetrahedral symmetry: because of the chirality the subgroup would have to be C6 or D3, but neither applies. Td, *332, or 43m, of order 24 – achiral or full tetrahedral symmetry known as the triangle group.
This group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S4 axes. Td and O are isomorphic as abstract groups: they both correspond to S4, the symmetric group on 4 objects. Td is the set obtained by combining each element of O \ T with inversion. See the isometries of the regular tetrahedron; the conjugacy classes of Td are: identity 8 × rotation by 120° 3 × rotation by 180° 6 × reflection in a plane through two rotation axes 6 × rotoreflection by 90° Th, 3*2, or m3, of order 24 – pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions; the 3-fold axes are now S6 axes, there is a central inversion symmetry. Th is isomorphic to T × Z2: every element of Th is either an element of T, or one combined with inversion. Apart from these two normal subgroups, there is a normal subgroup D2h, of type Dih2 × Z2 = Z2 × Z2 × Z2, it is the direct product of the normal subgroup of T with Ci.
The quotient group is the same as above: of type Z3. The three elements of the latter are the identity, "clockwise rotation", "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation, it is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the permutations of the body diagonals and the same combined with inversion, it is the symmetry of a pyritohedron, similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side. It is a subgroup of the full icosahedral symmetry group, with 4 of the 10 3-fold axes; the conjugacy classes of Th include those of T, with the two classes of 4 combined, each with inversion: identity 8 × rotation by 120° 3 × rotation by 180° inversion 8 × rotoreflection by 60° 3 × reflection in a plane The Icosahedron colored as a snub tetrahedron has chiral symmetry.
Octahedral symmetry Icosahedral symmetry Binary tetrahedral group Symmetric group S4 Peter R. Cromwell, Polyhedra, p. 295 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 N. W. Johnson: Geometries and Transformations, ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups Weisstein, Eric W. "Tetrahedral group". MathWorld