Edge Tessellation
In geometry, an edge tessellation is a partition of the plane into non-overlapping polygons (a tessellation) with the property that the reflection of any of these polygons across any of its edges is another polygon in the tessellation. All of the resulting polygons must be convex, and congruent to each other. There are eight possible edge tessellations in Euclidean geometry, but others exist in non-Euclidean geometry. The eight Euclidean edge tessellations are: In the first four of these, the tiles have no obtuse angles, and the degrees of the vertices are all even. Because the degrees are even, the sides of the tiles form lines through the tiling, so each of these four tessellations can alternatively be viewed as an arrangement of lines. In the second four, each tile has at least one obtuse angle at which the degree is three, and the sides of tiles that meet at that angle do not extend to lines in the same way. These tessellations were considered by 19th-century inventor Dav ...
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Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ...
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Tiling Dual Semiregular V3-6-3-6 Quasiregular Rhombic
Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester Theodor Tiling (1818–1871), physician and botanist *Reinhold Tiling (1893–1933), German rocket pioneer Other uses *Neuronal tiling *Tile drainage, an agriculture practice that removes excess water from soil *Tiling (crater), a small, undistinguished crater on the far side of the Moon See also *Brickwork *Packing (other) *Tiling puzzle Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given ...

Kaleidoscope
A kaleidoscope () is an optical instrument with two or more reflecting surfaces (or mirrors) tilted to each other at an angle, so that one or more (parts of) objects on one end of these mirrors are shown as a regular symmetrical pattern when viewed from the other end, due to repeated reflection. These reflectors are usually enclosed in a tube, often containing on one end a cell with loose, colored pieces of glass or other transparent (and/or opaque) materials to be reflected into the viewed pattern. Rotation of the cell causes motion of the materials, resulting in an ever-changing view being presented. Etymology Coined by its Scottish inventor David Brewster, "kaleidoscope" is derived from the Ancient Greek word καλός (''kalos''), "beautiful, beauty", εἶδος (''eidos''), "that which is seen: form, shape" and σκοπέω (''skopeō''), "to look to, to examine", hence "observation of beautiful forms." It was first published in the patent that was granted on July 10 ...
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David Brewster
Sir David Brewster KH PRSE FRS FSA Scot FSSA MICE (11 December 178110 February 1868) was a British scientist, inventor, author, and academic administrator. In science he is principally remembered for his experimental work in physical optics, mostly concerned with the study of the polarization of light and including the discovery of Brewster's angle. He studied the birefringence of crystals under compression and discovered photoelasticity, thereby creating the field of optical mineralogy.A. D. Morrison-Low (2004) "Brewster, Sir David (1781–1868)" in ''Oxford Dictionary of National Biography'' For this work, William Whewell dubbed him the "father of modern experimental optics" and "the Johannes Kepler of optics." A pioneer in photography, Brewster invented an improved stereoscope, which he called "lenticular stereoscope" and which became the first portable 3D-viewing device. He also invented the stereoscopic camera, two types of polarimeters, the polyzonal lens, the li ...
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Arrangement Of Lines
In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orchestration in that the latter process is limited to the assignment of notes to instruments for performance by an orchestra, concert band, or other musical ensemble. Arranging "involves adding compositional techniques, such as new thematic material for introductions, transitions, or modulations, and endings. Arranging is the art of giving an existing melody musical variety".(Corozine 2002, p. 3) In jazz, a memorized (unwritten) arrangement of a new or pre-existing composition is known as a ''head arrangement''. Classical music Arrangement and transcriptions of classical and serious music go back to the early history of this genre. Eighteenth century J.S. Bach frequently made arrangements of his own and other composers' pieces. ...
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Vertex (geometry)
In geometry, a vertex (in plural form: vertices or vertexes) is a point (geometry), point where two or more curves, line (geometry), lines, or edge (geometry), edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedron, polyhedra are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two Line (mathematics)#Ray, rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection (Euclidean geometry), intersection of Edge (geometry), edges, face (geometry), faces or facets of the object. In a polygon, a vertex is called "convex set, convex" if the internal an ...
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Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G, denoted by \Delta(G), and the minimum degree of a graph, denoted by \delta(G), are the maximum and minimum of its vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is entitled negative deg(v). Handshaking lemma ...
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Triakis Triangular Tiling
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex. As the name implies this tiling is constructed by a truncation operation applies to a hexagonal tiling, leaving dodecagons in place of the original hexagons, and new triangles at the original vertex locations. It is given an extended Schläfli symbol of ''t''. Conway calls it a truncated hextille, constructed as a truncation operation applied to a hexagonal tiling (hextille). There are 3 regular and 8 semiregular tilings in the plane. Uniform colorings There is only one uniform coloring of a truncated hexagonal tiling. (Naming the colors by indices around a vertex: 122.) Topologically identical tilings The dodecagonal faces can be distorted into different geometries, such as: Related polyhedra and tilings Wythoff constructions from hexagonal and triangular tilings Like the uniform polyhedra there are eight ...
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Deltoidal Trihexagonal Tiling
In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr. John Conway calls it a rhombihexadeltille.Conway, 2008, p288 table It can be considered a cantellated by Norman Johnson's terminology or an expanded hexagonal tiling by Alicia Boole Stott's operational language. There are 3 regular and 8 semiregular tilings in the plane. Uniform colorings There is only one uniform coloring in a rhombitrihexagonal tiling. (Naming the colors by indices around a vertex (3.4.6.4): 1232.) With edge-colorings there is a half symmetry form (3*3) orbifold notation. The hexagons can be considered as truncated triangles, t with two types of edges. It has Coxeter diagram , Schläfli symbol s2. The bicolored square can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, a triangular tiling results, constructed as ...
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Rhombille Tiling
In geometry, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles, and sets of six rhombi meet at their 60° angles. Properties The rhombille tiling can be seen as a subdivision of a hexagonal tiling with each hexagon divided into three rhombi meeting at the center point of the hexagon. This subdivision represents a regular compound tiling. It can also be seen as a subdivision of four hexagonal tilings with each hexagon divided into 12 rhombi. The diagonals of each rhomb are in the ratio 1:. This is the dual tiling of the trihexagonal tiling or kagome lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, and in the face configuration for monohedral tilings it is denoted .6.3.6 It is al ...
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Hexagonal Tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling). English mathematician John Conway called 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 triangular tiling and the square tiling. Applications 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 (or rather, soap bubbles) was investigated by Lord Kelvin, who believed that the Kelvin structure (or body-centered cubic lattice) is optimal. However, the less regular Weaire–Phel ...
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Tiling Dual Semiregular V3-12-12 Triakis Triangular
Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester Theodor Tiling (1818–1871), physician and botanist *Reinhold Tiling (1893–1933), German rocket pioneer Other uses *Neuronal tiling *Tile drainage, an agriculture practice that removes excess water from soil *Tiling (crater), a small, undistinguished crater on the far side of the Moon See also *Brickwork *Packing (other) *Tiling puzzle Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given ...