Hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of or t. English mathematician John Conway calls it a hextille; the internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane; the other two are the square tiling. The hexagonal tiling is the densest way to arrange circles in two dimensions; the Honeycomb conjecture states that the hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb was investigated by Lord Kelvin, who believed that the Kelvin structure is optimal. However, the less regular Weaire–Phelan structure is better; this structure exists in the form of graphite, where each sheet of graphene resembles chicken wire, with strong covalent carbon bonds. Tubular graphene sheets have been synthesised.
They have many potential applications, due to electrical properties. Silicene is similar. Chicken wire consists of a hexagonal lattice of wires; the hexagonal tiling appears in many crystals. In three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures, they are the densest known sphere packings in three dimensions, are believed to be optimal. Structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite, they differ in the way that the layers are staggered from each other, with the face-centered cubic being the more regular of the two. Pure copper, amongst other materials, forms a face-centered cubic lattice. There are three distinct uniform colorings of a hexagonal tiling, all generated from reflective symmetry of Wythoff constructions; the represent the periodic repeat of one colored tile, counting hexagonal distances as h first, k second. The same counting is used in the Goldberg polyhedra, with a notation h,k, can be applied to hyperbolic tilings for p>6.
The 3-color tiling is a tessellation generated by the order-3 permutohedrons. A chamferred hexagonal tiling replacing edges with new hexagons and transforms into another hexagonal tiling. In the limit, the original faces disappear, the new hexagons degenerate into rhombi, it becomes a rhombic tiling; the hexagons can be dissected into sets of 6 triangles. This process leads to two 2-uniform tilings, the triangular tiling: The hexagonal tiling can be considered an elongated rhombic tiling, where each vertex of the rhombic tiling is stretched into a new edge; this is similar to the relation of the rhombic dodecahedron and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions. It is possible to subdivide the prototiles of certain hexagonal tilings by two, four or nine equal pentagons: This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol, Coxeter diagram, progressing to infinity; this tiling is topologically related to regular polyhedra with vertex figure n3, as a part of sequence that continues into the hyperbolic plane.
It is related to the uniform truncated polyhedra with vertex figure n.6.6. This tiling is a part of a sequence of truncated rhombic polyhedra and tilings with Coxeter group symmetry; the cube can be seen as a rhombic hexahedron. The truncated forms have regular n-gons at the truncated vertices, nonregular hexagonal faces. Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, blue along the original edges, there are 8 forms, 7 which are topologically distinct. There are 3 types of monohedral convex hexagonal tilings, they are all isohedral. Each has parametric variations within a fixed symmetry. Type 2 contains glide reflections, is 2-isohedral keeping chiral pairs distinct. Hexagonal tilings can be made with the identical topology as the regular tiling. With isohedral faces, there are 13 variations. Symmetry given assumes all faces are the same color. Colors here represent the lattice positions.
Single-color lattices are parallelogon hexagons. Other isohedrally-tiled topological hexagonal tilings are seen as quadrilaterals and pentagons that are not edge-to-edge, but interpreted as colinear adjacent edges: The 2-uniform and 3-uniform tessellations have a rotational degree of freedom which distorts 2/3 of the hexagons, including a colinear case that can be seen as a non-edge-to-edge tiling of hexagons and larger triangles, it can be distorted into a chiral 4-colored tri-directional weaved pattern, distorting some hexagons into parallelograms. The weaved pattern with 2 colored faces have rotational 632 symmetry. A chevron pattern has pmg symmetry, lowered to p1 with 3 or 4 colored tiles; the hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing; the gap inside each hexagon allows for one circle, creating the densest packing from the triangular tiling, with each circle contact with the maximum of 6 circles.
There are 2 regular complex apeirogons, sharing the vertices of the
Geometry
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp
Tetrahedron
In geometry, a tetrahedron known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces; the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, may thus be called a 3-simplex. The tetrahedron is one kind of pyramid, a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle, so a tetrahedron is known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper, it has two such nets. For any tetrahedron there exists a sphere on which all four vertices lie, another sphere tangent to the tetrahedron's faces. A regular tetrahedron is one, it is one of the five regular Platonic solids. In a regular tetrahedron, all faces are the same size and shape and all edges are the same length.
Regular tetrahedra alone do not tessellate, but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, a tessellation. The regular tetrahedron is self-dual; the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, two level edges: and Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face level, the vertices are: v1 = v2 = v3 = v4 = with the edge length of sqrt. Still another set of coordinates are based on an alternated cube or demicube with edge length 2; this form has Coxeter diagram and Schläfli symbol h. The tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, the pair together form the stellated octahedron, whose vertices are those of the original cube. Tetrahedron:, Dual tetrahedron:, For a regular tetrahedron of edge length a: With respect to the base plane the slope of a face is twice that of an edge, corresponding to the fact that the horizontal distance covered from the base to the apex along an edge is twice that along the median of a face.
In other words, if C is the centroid of the base, the distance from C to a vertex of the base is twice that from C to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, this point divides each of them in two segments, one of, twice as long as the other. For a regular tetrahedron with side length a, radius R of its circumscribing sphere, distances di from an arbitrary point in 3-space to its four vertices, we have d 1 4 + d 2 4 + d 3 4 + d 4 4 4 + 16 R 4 9 = 2.
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
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
Hosohedron
In geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular n-gonal hosohedron has Schläfli symbol, with each spherical lune having internal angle 2π/n radians. For a regular polyhedron whose Schläfli symbol is, the number of polygonal faces may be found by: N 2 = 4 n 2 m + 2 n − m n The Platonic solids known to antiquity are the only integer solutions for m ≥ 3 and n ≥ 3; the restriction m ≥ 3 enforces. When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons can be represented as spherical lunes, having non-zero area. Allowing m = 2 admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron is represented as n abutting lunes, with interior angles of 2π/n. All these lunes share two common vertices; the digonal faces of a 2n-hosohedron, represents the fundamental domains of dihedral symmetry in three dimensions: Cnv, order 2n.
The reflection domains can be shown as alternately colored lunes as mirror images. Bisecting the lunes into two spherical triangles creates bipyramids and define dihedral symmetry Dnh, order 4n; the tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles. The dual of the n-gonal hosohedron is the n-gonal dihedron; the polyhedron is self-dual, is both a hosohedron and a dihedron. A hosohedron may be modified in the same manner as the other polyhedra to produce a truncated variation; the truncated n-gonal hosohedron is the n-gonal prism. In the limit the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation: Multidimensional analogues in general are called hosotopes. A regular hosotope with Schläfli symbol has each with a vertex figure; the two-dimensional hosotope, is a digon. The term “hosohedron” was coined by H. S. M. Coxeter, derives from the Greek ὅσος “as many”, the idea being that a hosohedron can have “as many faces as desired”.
Polyhedron Polytope McMullen, Peter. S. M. Dover Publications Inc. ISBN 0-486-61480-8 Weisstein, Eric W. "Hosohedron". MathWorld
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