In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees; the triangular tiling has Schläfli symbol of. Conway calls it a deltille, named from the triangular shape of the Greek letter delta; the triangular tiling can be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille. It is one of three regular tilings of the plane; the other two are the hexagonal tiling. There are 9 distinct uniform colorings of a triangular tiling. Three of them can be derived from others by repeating colors: 111212 and 111112 from 121213 by combining 1 and 3, while 111213 is reduced from 121314. There is one class of Archimedean colorings, 111112, not 1-uniform, containing alternate rows of triangles where every third is colored; the example shown is 2-uniform, but there are infinitely many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows.
The vertex arrangement of the triangular tiling is called an A2 lattice. It is the 2-dimensional case of a simplectic honeycomb; the A*2 lattice can be constructed by the union of all three A2 lattices, equivalent to the A2 lattice. + + = dual of = The vertices of the triangular tiling are the centers of the densest possible circle packing. Every circle is in contact with 6 other circles in the packing; the packing density is π⁄√12 or 90.69%. The voronoi cell of a triangular tiling is a hexagon, so the voronoi tessellation, the hexagonal tiling, has a direct correspondence to the circle packings. Triangular tilings can be made with the equivalent topology as the regular tiling. With identical faces and vertex-transitivity, there are 5 variations. Symmetry given assumes all faces are the same color; the planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a pyramid; these can be expanded to Platonic solids: five and three triangles on a vertex define an icosahedron and tetrahedron respectively.
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbols, continuing into the hyperbolic plane. It is topologically related as a part of sequence of Catalan solids with face configuration Vn.6.6, continuing into the hyperbolic plane. 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 4 regular complex apeirogons. Regular complex apeirogons have edges, where edges can contain 2 or more vertices. Regular apeirogons pr are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, vertex figures are r-gonal; the first is made of 2-edges, next two are triangular edges, the last has overlapping hexagonal edges. There are three Laves tilings made of single type of triangles: Triangular tiling honeycomb Simplectic honeycomb Tilings of regular polygons List of uniform tilings Isogrid Coxeter, H.
S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs Grünbaum, Branko. C.. Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. CS1 maint: Multiple names: authors list Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. P35 John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Weisstein, Eric W. "Triangular Grid". MathWorld. Weisstein, Eric W. "Regular tessellation". MathWorld. Weisstein, Eric W. "Uniform tessellation". MathWorld. Klitzing, Richard. "2D Euclidean tilings x3o6o - trat - O2"
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur in architecture and decorative art in textiles and tiles as well as wallpaper. A proof that there were only 17 distinct groups of possible patterns was first carried out by Evgraf Fedorov in 1891 and derived independently by George Pólya in 1924; the proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done. The seventeen possible wallpaper groups are listed below in § The seventeen groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups. Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are different in style, scale or orientation may belong to the same group. Consider the following examples: Examples A and B have the same wallpaper group.
Example C has a different wallpaper group, called p4g or 4*2. The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it looks the same after the transformation. For example, translational symmetry is present when the pattern can be translated some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe; the pattern is unchanged. Speaking, a true symmetry only exists in patterns that repeat and continue indefinitely. A set of only, five stripes does not have translational symmetry—when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, classification is applied to finite patterns, small imperfections may be ignored. Sometimes two categorizations are meaningful, one based on shapes alone and one including colors.
When colors are ignored there may be more symmetry. In black and white there are 17 wallpaper groups; the types of transformations that are relevant here are called Euclidean plane isometries. For example: If we shift example B one unit to the right, so that each square covers the square, adjacent to it the resulting pattern is the same as the pattern we started with; this type of symmetry is called a translation. Examples A and C are similar. If we turn example B clockwise by 90°, around the centre of one of the squares, again we obtain the same pattern; this is called a rotation. Examples A and C have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C. We can flip example B across a horizontal axis that runs across the middle of the image; this is called a reflection. Example B has reflections across a vertical axis, across two diagonal axes; the same can be said for A. However, example C is different, it only has reflections in vertical directions, not across diagonal axes.
If we flip across a diagonal line, we do not get the same pattern back. This is part of the reason that the wallpaper group of A and B is different from the wallpaper group of C. Another transformation is "Glide", a combination of reflection and translation parallel to the line of reflection. Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations. Two such isometry groups are of the same type if they are the same up to an affine transformation of the plane, thus e.g. a translation of the plane does not affect the wallpaper group. The same applies for a change of angle between translation vectors, provided that it does not add or remove any symmetry. Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those that preserve orientation, it follows from the Bieberbach theorem that all wallpaper groups are different as abstract groups.
2D patterns with double translational symmetry can be categorized according to their symmetry group type. Isometries of the Euclidean plane fall into four categories. Translations, denoted by Tv, where v is a vector in R2; this has the effect of shifting the plane applying displacement vector v. Rotations, denoted by Rc,θ, where c is a point in the plane, θ is the angle of rotation. Reflections, or mirror isometries, denoted by FL, where L is a line in R2.. This has the effect of reflecting the plane in the line L, called the reflection axis or the associated mirror. Glide reflections, denoted by GL,d, where L is a line in R2 and d is a distance; this is a combination of a reflection in the line L and a translation along L by a distance d. The condition
Hexagonal crystal family
In crystallography, the hexagonal crystal family is one of the 6 crystal families, which includes 2 crystal systems and 2 lattice systems. The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, is the union of the hexagonal crystal system and the trigonal crystal system. There are 52 space groups associated with it, which are those whose Bravais lattice is either hexagonal or rhombohedral; the hexagonal crystal family consists of two lattice systems: rhombohedral. Each lattice system consists of one Bravais lattice. In the hexagonal family, the crystal is conventionally described by a right rhombic prism unit cell with two equal axes, an included angle of 120° and a height perpendicular to the two base axes; the hexagonal unit cell for the rhombohedral Bravais lattice is the R-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell with coordinates and.
Hence, there are 3 lattice points per unit cell in total and the lattice is non-primitive. The Bravais lattices in the hexagonal crystal family can be described by rhombohedral axes; the unit cell is a rhombohedron. This is a unit cell with parameters a = b = c. In practice, the hexagonal description is more used because it is easier to deal with a coordinate system with two 90° angles. However, the rhombohedral axes are shown in textbooks because this cell reveals 3m symmetry of crystal lattice; the rhombohedral unit cell for the hexagonal Bravais lattice is the D-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell with coordinates and. However, such a description is used; the hexagonal crystal family consists of two crystal systems: hexagonal. A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system; the trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis.
These 5 point groups have 7 corresponding space groups assigned to the rhombohedral lattice system and 18 corresponding space groups assigned to the hexagonal lattice system. The hexagonal crystal system consists of the 7 point groups that have a single six-fold rotation axis; these 7 point groups have 27 space groups, all of which are assigned to the hexagonal lattice system. Graphite is an example of a crystal; the trigonal crystal system is the only crystal system whose point groups have more than one lattice system associated with their space groups: the hexagonal and rhombohedral lattices both appear. The 5 point groups in this crystal system are listed below, with their international number and notation, their space groups in name and example crystals; the point groups in this crystal system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation, mineral examples, if they exist. Hexagonal close packed is one of the two simple types of atomic packing with the highest density, the other being the face centered cubic.
However, unlike the fcc, it is not a Bravais lattice as there are two nonequivalent sets of lattice points. Instead, it can be constructed from the hexagonal Bravais lattice by using a two atom motif associated with each lattice point. Quartz is a crystal that belongs to the hexagonal lattice system but exists in two polymorphs that are in two different crystal systems; the crystal structures of α-quartz are described by two of the 18 space groups associated with the trigonal crystal system, while the crystal structures of β-quartz are described by two of the 27 space groups associated with the hexagonal crystal system. The lattice angles and the lengths of the lattice vectors are all the same for both the cubic and rhombohedral lattice systems; the lattice angles for simple cubic, face-centered cubic, body-centered cubic lattices are π/2 radians, π/3 radians, arccos radians, respectively. A rhombohedral lattice will result from lattice angles other than these. Crystal structure Close-packing Wurtzite Hahn, Theo, ed..
International Tables for Crystallography, Volume A: Space Group Symmetry. A. Berlin, New York: Springer-Verlag. Doi:10.1107/97809553602060000100. ISBN 978-0-7923-6590-7. Media related to Trigonal lattices at Wikimedia Commons Mineralogy database
In mathematics, Eisenstein integers also known as Eulerian integers, are complex numbers of the form z = a + b ω, where a and b are integers and ω = − 1 + i 3 2 = e 2 π i 3 is a primitive cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane; the Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field Q — the third cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each z = a + bω is a root of the monic polynomial z 2 − z +. In particular, ω satisfies the equation ω 2 + ω + 1 = 0; the product of two Eisenstein integers a + bω and c + dω is given explicitly by ⋅ = + ω. The norm of an Eisenstein integer is just the square of its modulus, is given by | a + b ω | 2 = a 2 − a b + b 2 = 1 4, a positive ordinary integer; the conjugate of ω satisfies ω ¯ = ω 2. The group of units in this ring is the cyclic group formed by the sixth roots of unity in the complex plane:, the Eisenstein integers of norm 1.
If x and y are Eisenstein integers, we say that x divides y if there is some Eisenstein integer z such that y = zx. A non-unit Eisenstein integer x is said to be an Eisenstein prime if its only non-unit divisors are of the form ux, where u is any of the six units. There are two types of Eisenstein primes. First, an ordinary prime number, congruent to 2 mod 3 is an Eisenstein prime. Second, 3 and any rational prime congruent to 1 mod 3 is equal to the norm x2 − xy + y2 of an Eisentein integer x + ωy. Thus, such a prime may be factored as, these factors are Eisenstein primes: they are the Eisenstein integers whose norm is a rational prime; the ring of Eisenstein integers forms a Euclidean domain whose norm N is given by the square modulus, as above: N = a 2 − a b + b 2. A division algorithm, applied to any dividend α and divisor β ≠ 0, gives a quotient κ and a remainder ρ smaller than the divisor, satisfying: α = κ β + ρ with N < N. Here α, β, κ, ρ are all Eisenstein integers; this algorithm implies the Euclidean algorithm, which proves Euclid's Lemma and the unique factorization of Eisenstein integers into Eisenstein primes.
One division algorithm is. First perform the division in the field of complex numbers, write the quotient in terms of ω: α β = 1 | β | 2 α β ¯ = a + b i = a + 1 3 b + 2 3 b ω, for rational a, b ∈
Close-packing of equal spheres
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement. Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is π 3 2 ≃ 0.74048. The same packing density can be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction; the Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven by T. C. Hales. Highest density is known only in case of 2, 3, 8 and 24 dimensions. Many crystal structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them; the cubic and hexagonal arrangements are close to one another in energy, it may be difficult to predict which form will be preferred from first principles.
There are two simple regular lattices. They are called face-centered hexagonal close-packed, based on their symmetry. Both are based upon sheets of spheres arranged at the vertices of a triangular tiling; the fcc lattice is known to mathematicians as that generated by the A3 root system. The problem of close-packing of spheres was first mathematically analyzed by Thomas Harriot around 1587, after a question on piling cannonballs on ships was posed to him by Sir Walter Raleigh on their expedition to America. Cannonballs were piled in a rectangular or triangular wooden frame, forming a three-sided or four-sided pyramid. Both arrangements produce a face-centered cubic lattice – with different orientation to the ground. Hexagonal close-packing would result in a six-sided pyramid with a hexagonal base; the cannonball problem asks which flat square arrangements of cannonballs can be stacked into a square pyramid. Édouard Lucas formulated the problem as the Diophantine equation ∑ n = 1 N n 2 = M 2 or 1 6 N = M 2 and conjectured that the only solutions are N = 1, M = 1, N = 24, M = 70.
Here N is the number of layers in the pyramidal stacking arrangement and M is the number of cannonballs along an edge in the flat square arrangement. In both the fcc and hcp arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres and two smaller gaps surrounded by four spheres; the distances to the centers of these gaps from the centers of the surrounding spheres is √3⁄2 for the tetrahedral, √2 for the octahedral, when the sphere radius is 1. Relative to a reference layer with positioning A, two more positionings B and C are possible; every sequence of A, B, C without immediate repetition of the same one is possible and gives an dense packing for spheres of a given radius. The most regular ones are fcc = ABC ABC ABC... hcp = AB AB AB AB.... There is an uncountably infinite number of disordered arrangements of planes that are sometimes collectively referred to as "Barlow packings", after crystallographer William BarlowIn close-packing, the center-to-center spacing of spheres in the xy plane is a simple honeycomb-like tessellation with a pitch of one sphere diameter.
The distance between sphere centers, projected on the z axis, is: pitch Z = 6 ⋅ d 3 ≈ 0.816 496 58 d, where d is the diameter of a sphere. The coordination number of hcp and fcc is 12 and their atomic packing factors are equal to the number mentioned above, 0.74. When forming any sphere-packing lattice, the first fact to notice is that whenever two spheres touch a straight line may be drawn from the center of one sphere to the center of the other intersecting the point of contact; the distance between the centers along the shortest path namely that straight line will therefore be r1 + r2 where r1 is the radius of the first sphere and r2 is the radius of the second. In close packing all of the spheres share a common radius, r; therefore two centers would have a distance 2r. To form an A-B-A-B-... hexagonal close packing of spheres, the coordinate points of the lattice will be the spheres' centers. Suppose, the goal is to fill a box with spheres according to hcp; the box would be placed on the x-y-z coordinate space.
First form a row of spheres. The centers will all lie on a straight line, their x-coordinate will vary by 2r since the distance between each center of the spheres are touching is 2r. The y-coordinate and z-coordinate wil
Loewner's torus inequality
In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the area of an arbitrary Riemannian metric on the 2-torus. In 1949 Charles Loewner proved that every metric on the 2-torus T 2 satisfies the optimal inequality sys 2 ≤ 2 3 area , where "sys" is its systole, i.e. least length of a noncontractible loop. The constant appearing on the right hand side is the Hermite constant γ 2 in dimension 2, so that Loewner's torus inequality can be rewritten as sys 2 ≤ γ 2 area ; the inequality was first mentioned in the literature in Pu. The boundary case of equality is attained if and only if the metric is flat and homothetic to the so-called equilateral torus, i.e. torus whose group of deck transformations is the hexagonal lattice spanned by the cube roots of unity in C. Given a doubly periodic metric on R 2, there is a nonzero element g ∈ Z 2 and a point p ∈ R 2 such that dist 2 ≤ 2 3 area , where F is a fundamental domain for the action, while dist is the Riemannian distance, namely least length of a path joining p and g. p. Loewner's torus inequality can be proved most by using the computational formula for the variance, E − 2 = v a r.
Namely, the formula is applied to the probability measure defined by the measure of the unit area flat torus in the conformal class of the given torus. For the random variable X, one takes the conformal factor of the given metric with respect to the flat one; the expected value E of X 2 expresses the total area of the given metric. Meanwhile, the expected value E of X can be related to the systole by using Fubini's theorem; the variance of X can be thought of as the isosystolic defect, analogous to the isoperimetric defect of Bonnesen's inequality. This approach therefore produces the following version of Loewner's torus inequality with isosystolic defect: a r e a − 3 2 2 ≥ v a r, where ƒ is the conformal factor of the metric with respect to a unit area flat metric in its conformal class. Whether or not the inequality 2 ≤ γ 2 a r e a is satisfied by all surfaces of nonpositive Euler characteristic is unknown. For orientable surfaces of genus 2 and genus 20 and above, the answer is affirmative, see work by Katz and Sabourau below.
Pu's inequality for the real projective plane Gromov's systolic inequality for essential manifolds Gromov's inequality for complex projective space Eisenstein integer Systoles of surfaces Horowitz, Charles. "Loewner's torus inequality with isosystolic defect". Journal of Geometric Analysis. 19: 796–808. ArXiv:0803.0690. Doi:10.1007/s12220-009-9090-y. MR 2538936. Katz, Mikhail G.. Systolic geometry and topology. Mathematical Surveys and Monographs. 137. With an appendix by J. Solomon. Providence, RI: American Mathematical Society. Doi:10.1090/surv/137. ISBN 978-0-8218-4177-8. MR 2292367. Katz, Mikhail G.. "Entropy of systolically extremal surfaces and asymptotic bounds". Ergodic Theory Dynam. Systems. 25: 1209–1220. ArXiv:math. DG/0410312. Doi:10.1017/S0143385704001014. MR 2158402. Katz, Mikhail G.. "Hyperelliptic surfaces are Loewner". Proc. Amer. Math. Soc. 134: 1189–1195. ArXiv:math. DG/0407009. Doi:10.1090/S0002-9939-05-08057-3. MR 2196056. Pu, Pao Ming. "Some inequalities in certain nonorientable Riemannian manifolds".
Pacific J. Math. 2: 55–71. MR 0048886