Constructing an irregular pentagon in this way shows us why not all irregular pentagons can tile the plane: There are certain restrictions on the angles that not all pentagons satisfy.
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The company now occupies roughly as much space worldwide as 38 Pentagons.
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The song was written to teach kids about different shapes like pentagons and octagons.
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Here are the only two possible ways of matching up two such pentagons on that side.
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This just in: Nick Jonas gets all hot and bothered when he sees pentagons and octagons.
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When it comes to tiling the plane, pentagons occupy an area between the inevitable and the impossible.
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Download the "Math Problem With Pentagons" PDF worksheet to practice the concepts and to share with students.
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Dear White People has its fair share of love triangles, rectangles, and maybe even pentagons at this point.
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And it took another mathematician in 2017, Michaël Rao, to computationally verify that no other such pentagons could work.
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Coins are usually circular, but also can be pentagons, spades or even dog tags to be worn around the neck.
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Then one night we came home to find him gluing together a 32-sided sphere of interlocking pentagons and triangles.
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But of all the blocks designed to lie flat on a table or floor, have you ever seen any shaped like pentagons?
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We're starting to see that complicated relationships among the angles and sides make monohedral, edge-to-edge tilings with pentagons particularly complex.
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With pentagons, this adds another dimension of complexity to the already complex problem of finding the right combination of sides and angles.
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First introduced in 1931 at a station in Fukui, the stamps soon proliferated across the country, shaped like circles, squares, pentagons, and hexagons.
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No matter how we arrange them, we'll never get pentagons to snugly match up around a vertex with no gap and no overlap.
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In the 1970s, Marjorie Rice, a homemaker with no mathematical background, ran across a Scientific American column about pentagons that tile the plane.
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To understand the problem with pentagons, let's start with one of the simplest and most elegant of geometric structures: the regular tilings of the plane.
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Three pentagons at a vertex gives us 324 degrees, which leaves a gap of 36 degrees that is too small to fill with another pentagon.
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With triangles and quadrilaterals everything always fits, but when it comes to pentagons, it's a balancing act to get everything to work out just right.
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These presumably would have yielded more complicated configurations of objects in the sky today: triangular arrangements of galaxies, along with quadrilaterals, pentagons, and other shapes.
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That goes to explain why Washington has a large surplus of office space, Whitehead said — roughly equivalent to two Pentagons, or at least 13 million square feet.
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Somehow, lots of individual protein pieces combine to form a capsule, like the hexagons and pentagons of a soccer ball or the triangles on a twenty sided die.
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And it was, in part, the lattice structure of the geodesic dome, a convex polyhedron assembled from hexagons and pentagons, themselves divided into triangles, that would inspire Caspar and Klug's theory.
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Unlike hexagons, however, regular pentagons cannot be built from equilateral triangles, nor can they tessellate a plane: When slid next to each other to tile a surface, gaps and overlaps inevitably arise.
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This, the New Century Global Centre, was to be his crowning accomplishment, the world's largest structure by floor space, the size of 246 football fields, or nearly three Pentagons or eight Louvres.
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The Wyoming School for the Deaf, designed in 1962, featured classrooms shaped like pentagons and octagons instead of rectangles; more sides allowed students to better form a circle around the teacher as he signed lessons to them.
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Spoofing a plan to haul nuclear weapons around the country on railroad cars, he proposed a system of mobile Pentagons, complete with little secretaries of defense and presidents who would crisscross the country to confuse the enemy.
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Everywhere you walk, you will see pentagons throughout the ship," Tadlock said, adding that to her, Memorial Day is about "remembering everyone who came before me and trying to make them proud in everything I do every day.
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His Nobel Prize-winning discovery, which he shared with Richard E. Smalley and Robert F. Curl Jr. of Rice University in Houston, was the Buckminsterfullerene molecule, a cage of 60 carbon atoms made of interlocking pentagons and hexagons.
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Click here to view original GIFYou won't want to actually kick it around without wearing steel-toed shoes, but Russian carpenter Vladimir Zhilenko makes turning a bunch of wooden pentagons into a perfectly-round soccer ball look incredibly easy.
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The two different shades of deep red in "Parting" (2017), along with its glossy surfaces and carefully applied coats of paint, and the tips of the bisected pentagons about to meet, brought such a cinematic scene immediately to mind.
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The 15 types of convex pentagons that admit tilings (not all edge-to-edge) of the plane were discovered by Karl Reinhardt in 1918, Richard Kershner in 1968, Richard James in 1975, Marjorie Rice in 1977, Rolf Stein in 1985, and Casey Mann, Jennifer McLoud-Mann and David Von Derau in 2015.
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The convulsed, predatory "untitled: pinkspree; 2018" (548), an eight-and-a-half-foot-tall unholy marriage of pink-on-black triangles and pentagons, seems to lurch forward, jaws open, from a far corner, while its vulnerable, satchel-sized variation, "untitled: spree (green); 2018" (2018), calls out its aggression from the other side of the room.
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Their calculations indicate that galaxies and other structures are not merely randomly spread out in pairs across the sky; instead, they have a slight tendency to be arranged in more complex configurations: triangles, rectangles, pentagons and all manner of other shapes, which trace back not just to quantum jitter in the Big Bang's clock, but to a much more meaningful turning of the gears.
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The concave pentagons are non-stellated pentagons having at least one angle greater than 180°. The first angle which opens wider than 180° is γ, so γ = 180° (border shown in green at right) is a curve which is the border of the regions of concave pentagons and others, called convex. Pentagons which map exactly to this border have at least two consecutive sides appearing as a double length side, which resembles a pentagon degenerated to a quadrilateral.
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He classified the possible patterns on the surface of an Adidas Telstar soccer ball, i.e. specialThe sides of the pentagons may only encounter hexagons; the hexagons must alternately bifurcate with pentagons and hexagons. tilings with pentagons and hexagons on the sphere.Kolumne Mathematische Unterhaltungen, Spektrum der Wissenschaft, Juli 2006Braungardt, Kotschick Die Klassifikation von Fußballmustern, Math.
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Pentagons that map exactly onto those borders have a line of symmetry. Inside the region of unique mappings there are three types of pentagons: stellated, concave and convex, separated by new borders.
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A similar method can be used to subdivide squares into four congruent non-convex pentagons, or regular hexagons into six congruent non-convex pentagons, and then tile the plane with the resulting unit.
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Pentagons which map exactly to this border have a vertex touching another side.
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Similarly, the company also manufactures drill bits for other angular holes such as pentagons and hexagons.
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In the plane with coordinate axes α and β, α = β is a line dividing the plane in two parts (south border shown in orange in the drawing). δ = β as a curve divides the plane into different sections (north border shown in blue). Both borders enclose a continuous region of the plane whose points map to unique equilateral pentagons. Points outside the region just map to repeated pentagons—that is, pentagons that when rotated or reflected can match others already described.
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This is bounded by twelve irregular pentagons, and is a tetartohedral or quarter-faced form of the hexakis-octahedron.
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The buckyballs are released into a helium or hydrogen gas, which expands at supersonic speeds, carrying the carbon balls with it. The buckyballs achieve energies of around 40 keV without changing their internal dynamics. This material contains hexagons and pentagons that come from the original structures. The pentagons could introduce a band gap.
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This shows that the Penrose tiling has a scaling self- similarity, and so can be thought of as a fractal. Penrose originally discovered the P1 tiling in this way, by decomposing a pentagon into six smaller pentagons (one half of a net of a dodecahedron) and five half- diamonds; he then observed that when he repeated this process the gaps between pentagons could all be filled by stars, diamonds, boats and other pentagons. By iterating this process indefinitely he obtained one of the two P1 tilings with pentagonal symmetry.
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3D model of a small dodecahemidodecahedron In geometry, the small dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U51. It has 18 faces (12 pentagons and 6 decagons), 60 edges, and 30 vertices. Its vertex figure alternates two regular pentagons and decagons as a crossed quadrilateral. It is a hemipolyhedron with six decagonal faces passing through the model center.
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For instance, there are 1812 non-isomorphic fullerenes . Note that only one form of , buckminsterfullerene, has no pair of adjacent pentagons (the smallest such fullerene). To further illustrate the growth, there are 214,127,713 non-isomorphic fullerenes , 15,655,672 of which have no adjacent pentagons. Optimized structures of many fullerene isomers are published and listed on the web.
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Here all the pentagrams have been alternated back into pentagons, and triangles have been inserted to take up the resulting free edges.
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The Pittco logo has changed many times since the organization's inception. Originally, Pittco used an image of the city of Pittsburgh with "Pittsburgh LAN Coalition" on top of it. The logo eventually changed to the modern one, bearing three pentagons in yellow, blue, and gray to the left of "Pittco". The pentagons reference the "Three Rivers" for which Pittsburgh is known.
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The stellated pentagons have sides intersected by others. A common example of this type of pentagon is the pentagram. A condition for a pentagon to be stellated, or self-intersecting, is to have 2α + β ≤ 180°. So, in the mapping, the line 2α + β = 180° (shown in orange at the north) is the border between the regions of stellated and non- stellated pentagons.
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These are then used to construct the points P1, P2, P3, P4. This detailed procedure involving Carlyle circles for the construction of regular pentagons is given below.
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The Desargues configuration viewed as a pair of mutually inscribed pentagons: each pentagon vertex lies on the line through one of the sides of the other pentagon.
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Buckminsterfullerene is a truncated icosahedron with 60 vertices and 32 faces (20 hexagons and 12 pentagons where no pentagons share a vertex) with a carbon atom at the vertices of each polygon and a bond along each polygon edge. The van der Waals diameter of a molecule is about 1.01 nanometers (nm). The nucleus to nucleus diameter of a molecule is about 0.71 nm. The molecule has two bond lengths.
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The convex pentagons have all of their five angles smaller than 180° and no sides intersecting others. A common example of this type of pentagon is the regular pentagon.
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For all convex pentagons, the sum of the squares of the diagonals is less than 3 times the sum of the squares of the sides.Inequalities proposed in “Crux Mathematicorum”, .
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"However, as will be explained momentarily, differently colored pentagons will be considered to be different types of tiles." ; , shows the edge modifications needed to yield an aperiodic set of prototiles.
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300px The action of the pentagram map on pentagons and hexagons is similar in spirit to classical configuration theorems in projective geometry such as Pascal's theorem, Desargues's theorem and others.
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Many viral capsids are formed by hexameric and pentameric proteins.Virus Capsid Model Such capsids are assigned a triangulation number (T-number) which describe relation between the number of pentagons and hexagons.
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The bilunabirotunda has a weak relationship with the cuboctahedron, as it may be created by replacing four square faces of the cuboctahedron with pentagons. Full rotation of a bilunabirotunda, photo every 15°.
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In geometry, the truncated order-5 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,5}, constructed from one pentagons and two decagons around every vertex.
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In geometry, the snub pentapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{5,5}, constructed from two regular pentagons and three equilateral triangles around every vertex.
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Clathrin cannot bind to membrane or cargo directly and instead uses adaptor proteins to do this. This triskelion will bind to other membrane-attached triskelia to form a rounded lattice of hexagons and pentagons, reminiscent of the panels on a soccer ball, that pulls the membrane into a bud. By constructing different combinations of 5-sided and 6-sided rings, vesicles of different sizes may assemble. The smallest clathrin cage commonly imaged, called a mini-coat, has 12 pentagons and only two hexagons.
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In geometry, the order-5 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,5}, constructed from five pentagons around every vertex. As such, it is self-dual.
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This yields a coarse-resolution equal area grid called the resolution 1 grid. It consists of 20 hexagons on the surface of the sphere and 12 pentagons centered on the 12 vertices of the icosahedron.
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The three- dimensional associahedron K5 is an enneahedron topologically equivalent to the order-4 truncated triangular bipyramid with nine faces (three squares and six pentagons) and fourteen vertices, and its dual is the triaugmented triangular prism.
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The 1978 Tango Durlast consisted of twenty identical hexagonal panels with 'triads' creating the impression of 12 circles around the pentagons. The Adidas Tango Durlast ball was made of genuine leather with a shiny waterproof plastic coating.
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All the equilateral pentagons plotted within the area delimited by the condition α ≥ β ≥ δ. Three regions for each of three types of pentagons are shown: stellated, concave and convex The equilateral pentagon as a function of two variables can be plotted in the two-dimensional plane. Each pair of values (α, β) maps to a single point of the plane and also maps to a single pentagon. The periodicity of the values of α and β and the condition α ≥ β ≥ δ permit the size of the mapping to be limited.
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A well known solution is provided by the triangular tiling with a total compatibility between the local and global rules: the system is said to be "unfrustrated". But now, the interaction energy is supposed to be at a minimum when atoms sit on the vertices of a regular pentagon. Trying to propagate in the long range a packing of these pentagons sharing edges (atomic bonds) and vertices (atoms) is impossible. This is due to the impossibility of tiling a plane with regular pentagons, simply because the pentagon vertex angle does not divide 2.
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Friedrich Haag (20 August 1856 – 8 December 1941) was a pioneering German crystallographer.H. Wondratschek Topics, The Rigaku Journal, Volume 4, Issue 1/2, 1987, p. 33Doris Schattschneider Tiling the plane with congruent pentagons, Math. Mag., 1978, Volume 51, pp.
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3D model of a great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5,5/2} and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path. The discovery of the great dodecahedron is sometimes credited to Louis Poinsot in 1810, though there is a drawing of something very similar to a great dodecahedron in the 1568 book Perspectiva Corporum Regularium by Wenzel Jamnitzer.
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Using these theorems he derived explicit formulas for triangles and tetragons and also gave formulas for pentagons, hexagons, and heptagons. He also presented a classification of problems for tetragons, pentagons, and hexagons. For the second group of problems, Lexell showed that their solutions can be reduced to a few general rules and presented a classification of these problems, solving the corresponding combinatorial problems. In the second article he applied his general method for specific tetragons and showed how to apply his method to a polygon with any number of sides, taking a pentagon as an example.
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3D model of an icosidodecadodecahedron In geometry, the icosidodecadodecahedron (or icosified dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U44. It has 44 faces (12 pentagons, 12 pentagrams and 20 hexagons), 120 edges and 60 vertices. Its vertex figure is a crossed quadrilateral.
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3D model of a medial inverted pentagonal hexecontahedron The medial inverted pentagonal hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron. Its faces are irregular nonconvex pentagons, with one very acute angle.
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Spherical pentagonal hexecontahedron This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations (V3.3.3.3.n). (The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry.
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A Mathematical Solution Book Containing Systematic Solutions to Many of the Most Difficult Problems by Benjamin Franklin Finkel Another name is icosidodecagon, suggesting a (20 and 12)-gon, in parallel to the 32-faced icosidodecahedron, which has 20 triangles and 12 pentagons.
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3D model of a small dodecicosidodecahedron In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (12 triangles, 20 pentagons, and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral.
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Its edges would be of approximately equal length, but the vertices of each face would not necessarily be coplanar. Indeed, K5 is a near-miss Johnson solid: it looks like it might be possible to make from squares and regular pentagons, but it is not.
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The suffix (triangular) -rotunda refers to the complex of three equilateral triangles and three regular pentagons surrounding another equilateral triangle, which bears structural resemblance to the pentagonal rotunda. The triangular hebesphenorotunda is the only Johnson solid with faces of 3, 4, 5 and 6 sides.
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Buckminsterfullerene is the smallest fullerene molecule containing pentagonal and hexagonal rings in which no two pentagons share an edge (which can be destabilizing, as in pentalene). It is also most common in terms of natural occurrence, as it can often be found in soot. The empirical formula of buckminsterfullerene is and its structure is a truncated icosahedron, which resembles an association football ball of the type made of twenty hexagons and twelve pentagons, with a carbon atom at the vertices of each polygon and a bond along each polygon edge. The van der Waals diameter of a buckminsterfullerene molecule is about 1.1 nanometers (nm).
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Its positioning is also a vague reference to the logo of the Pittsburgh Steelers professional American football team, which features three stars to the right of "Steelers". alt=Pittco.org with several atomic molecule signs around it alt=Three pentagons in yellow, blue, and gray with a gray Pittco text next to it, plastic looking with waves alt=Three pentagons in yellow, blue, and gray with a gray Pittco text next to it, flattened alt=A more modern approach to the traditional pentagon logo, with black borders More recent use of the logo shows a drop shadow behind it, but the official logo remains without it.
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3D model of a snub icosidodecadodecahedron In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices. As the name indicates, it belongs to the family of snub polyhedra.
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The great dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the (n-1)-D pentagonal polytope faces of the core nD polytope (pentagons for the great dodecahedron, and line segments for the pentagram) until the figure again closes.
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A variety of polygonal shapes. Some simple shapes can be put into broad categories . For instance, polygons are classified according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into smaller categories; triangles can be equilateral, isosceles, obtuse, acute, scalene, etc.
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3D model of a great ditrigonal dodecicosidodecahedron In geometry, the great ditrigonal dodecicosidodecahedron (or great dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U42. It has 44 faces (20 triangles, 12 pentagons, and 12 decagrams), 120 edges, and 60 vertices. Its vertex figure is an isosceles trapezoid.
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3D model of a snub dodecadodecahedron In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U40. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol sr{,5}, as a snub great dodecahedron.
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The metal fasteners, joints, and internal steel frames remain dry, preventing frost and corrosion damage. The concrete resists sun and weathering. Some form of internal flashing or caulking must be placed over the joints to prevent drafts. The 1963 Cinerama Dome was built from precast concrete hexagons and pentagons.
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3D model of a trigyrate rhombicosidodecahedron In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids (J75). It contains 20 triangles, 30 squares and 12 pentagons. It is also a canonical polyhedron. It can be constructed as a rhombicosidodecahedron with three pentagonal cupolae rotated through 36 degrees.
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If two bilunabirotundae are aligned this way on opposite sides of the icosidodecahedron, then two vertices of the bilunabirotundae meet in the very center of the icosidodecahedron. The other two clusters of faces of the bilunabirotunda, the lunes (each lune featuring two triangles adjacent to opposite sides of one square), can be aligned with a congruent patch of faces on the rhombicosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the rhombicosidodecahedron, then a cube can be put between the bilunabirotundae at the very center of the rhombicosidodecahedron. Each of the two pairs of adjacent pentagons (each pair of pentagons sharing an edge) can be aligned with the pentagonal faces of a metabidiminished icosahedron as well.
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One way of constructing the Perles configuration is to start with a regular pentagon and its five diagonals, which form the sides of a smaller regular pentagon within the initial one. The nine points of the configuration consist of four out of the five vertices of each pentagon and the shared center of the two pentagons; the two missing pentagon vertices are chosen to be collinear with the center. The nine lines of the configuration consist of the five lines that are diagonals of the outer pentagon and sides of the inner pentagon, and the four lines that pass through the center and through corresponding pairs of vertices from the two pentagons.
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Defining an aperiodic tiling (the pinwheel tiling) by repeatedly dissecting and inflating a rep-tile. Every square, rectangle, parallelogram, rhombus, or triangle is rep-4. The sphinx hexiamond (illustrated above) is rep-4 and rep-9, and is one of few known self-replicating pentagons. The Gosper island is rep-7.
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A clathrin cage with a single triskelion highlighted in blue. CryoEM map EMD_5119 was rendered in UCSF Chimera and one clathrin triskelion was highlighted. Each cage has 12 pentagons. Mini-coat (left) has 4 hexagons and tetrahedral symmetry as in a truncated triakis tetrahedron. Hexagonal barrel (middle) has 8 hexagons and D6 symmetry.
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In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. The faces in it are considered as two overlapping edges as topological polyhedron.
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Adding a segment between the endpoints of these two new edges cuts off a smaller golden triangle, within which the construction can be repeated.. Some sources add another pentagram, inscribed within the inner pentagon of the largest pentagram of the figure. The other pentagons of the figure do not have inscribed pentagrams..
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Buckminsterfullerene is a type of fullerene with the formula C60. It has a cage-like fused-ring structure (truncated icosahedron) that resembles a soccer ball, made of twenty hexagons and twelve pentagons. Each carbon atom has three bonds. It is a black solid that dissolves in hydrocarbon solvents to produce a violet solution.
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Geometrical frustration occurs when a set of degrees of freedom is incompatible with the space it occupies. A purely geometric example of this is the impossibility of close-packing pentagons in two dimensions. Another is atomic magnetic moments with antiferromagnetic interactions. These moments lower their interaction energy by pointing antiparallel to their neighbors.
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These 12 order-5 vertices can be truncated such that all edges are equal length. The original 30 rhombic faces become non-regular hexagons, and the truncated vertices become regular pentagons. The hexagon faces can be equilateral but not regular with D2 symmetry. The angles at the two vertices with vertex configuration 6.6.
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Buchholz and MacDougall also showed that, in every Robbins pentagon, either all five of the internal diagonals are rational numbers or none of them are. If the five diagonals are rational (the case called a Brahmagupta pentagon by ), then the radius of its circumscribed circle must also be rational, and the pentagon may be partitioned into three Heron triangles by cutting it along any two non-crossing diagonals, or into five Heron triangles by cutting it along the five radii from the circle center to its vertices. Buchholz and MacDougall performed computational searches for Robbins pentagons with irrational diagonals but were unable to find any. On the basis of this negative result they suggested that Robbins pentagons with irrational diagonals may not exist.
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Periodic tiling by the sphinx With pentagons that are not required to be convex, additional types of tiling are possible. An example is the sphinx tiling, an aperiodic tiling formed by a pentagonal rep- tile. The sphinx may also tile the plane periodically, by fitting two sphinx tiles together to form a parallelogram and then tiling the plane by translates of this parallelogram, a pattern that can be extended to any non-convex pentagon that has two consecutive angles adding to 2, thus satisfying the condition(s) of convex Type 1 above. It is possible to divide an equilateral triangle into three congruent non-convex pentagons, meeting at the center of the triangle, and to tile the plane with the resulting three-pentagon unit.
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PROSEA – Gramedia. Jakarta. . The thin and leathery skin is greenish, yellowish to brownish in color, and patterned with pentagons that are either raised protuberances or flat eye facets. The fleshy, edible arils surround the large seeds in a thick layer. These arils are edible raw, or they can be prepared in a number of ways.
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Harold V. McIntosh also describes "nonplanar" flexagons (i.e., ones which cannot be flexed so they lie flat); ones folded from pentagons called pentaflexagons, and from heptagons called heptaflexagons. These should be distinguished from the "ordinary" pentaflexagons and heptaflexagons described above, which are made out of isosceles triangles, and they can be made to lie flat.
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3D model of a small stellated truncated dodecahedron In geometry, the small stellated truncated dodecahedron (or quasitruncated small stellated dodecahedron or small stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U58. It has 24 faces (12 pentagons and 12 decagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol t{,5}, and Coxeter diagram .
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3D model of a rhombidodecadodecahedron In geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U38. It has 54 faces (30 squares, 12 pentagons and 12 pentagrams), 120 edges and 60 vertices. It is given a Schläfli symbol t0,2{,5}, and by the Wythoff construction this polyhedron can also be named a cantellated great dodecahedron.
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3D model of a great dodecahemicosahedron In geometry, the great dodecahemicosahedron (or small dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65. It has 22 faces (12 pentagons and 10 hexagons), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilateral. It is a hemipolyhedron with ten hexagonal faces passing through the model center.
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Wheelchair users cannot visit the tower due to fire regulations. To mark the 2006 FIFA World Cup in Germany, for which the final match was played in the Berlin Olympic Stadium, the sphere was decorated as a football with magenta-coloured pentagons, reflecting the corporate colour of World Cup sponsor and owner of the Fernsehturm, Deutsche Telekom.
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The dragon itself represented the Emperor, Puyi. The medal of the order and the associated order star match in their appearance. The gem has an openwork, green enamel hanger that shows a stylized cloud group consisting of a central cloud vortex and two concentric pentagons. The corresponding rosette is white and shows a blue ring in the middle.
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Collar of the Order. The design was chosen to be an orchid because it was reportedly Puyi's favorite flower. The collar consists of one central large link and 20 small links, interconnected by figured intermediate links in the form of a Buddhist "endless knot". Small chain links are openwork slotted pentagons with rounded corners, symbolizing clouds.
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Geodesic grids may use the dual polyhedron of the geodesic polyhedron, which is the Goldberg polyhedron. Goldberg polyhedra are made up of hexagons and (if based on the icosahedron) 12 pentagons. One implementation that uses an icosahedron as the base polyhedron, hexagonal cells, and the Snyder equal-area projection is known as the Icosahedron Snyder Equal Area (ISEA) grid.
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The Schläfli symbol of a regular polyhedron is {p,q} if its faces are p-gons, and each vertex is surrounded by q faces (the vertex figure is a q-gon). For example, {5,3} is the regular dodecahedron. It has pentagonal (5 edges) faces, and 3 pentagons around each vertex. See the 5 convex Platonic solids, the 4 nonconvex Kepler-Poinsot polyhedra.
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Plan of Millbank Prison: six pentagons with a tower at the centre are arranged around a chapel. Annotated floor plan of Eastern State Penitentiary in 1836 An 1880s architectural drawing by John Frederick Adolphus McNair depicting a proposed prison at Outram, Singapore that was never built. Presidio Modelo in Cuba, photo taken 2005. Inside one of the buildings of the Presidio Modelo.
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Its name comes from a topological construction from the snub dodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are computed to be the same distance from the center. The 80 of the triangles are equilateral, and 60 triangles from the pentagons are isosceles. It is a (2,1) geodesic polyhedron, made of all triangles.
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Cairo pentagonal tiling Top view (top) and side view (bottom) of penta- graphane. Yellow and blue spheres show two types of carbon atoms, while red balls correspond to hydrogens. Penta-graphene is a hypothetical carbon allotrope composed entirely of carbon pentagons and resembling the Cairo pentagonal tiling. Penta-graphene was proposed in 2014 on the basis of analyses and simulations.
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Figurate numbers representing pentagons (including five) are called pentagonal numbers. Five is also a square pyramidal number. Five is the only prime number to end in the digit 5 because all other numbers written with a 5 in the ones place under the decimal system are multiples of five. As a consequence of this, 5 is in base 10 a 1-automorphic number.
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3D model of a (uniform) pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even- numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for a total of 12 faces. Hence, it is a non-regular dodecahedron.
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There, within one polymeric chain, all the halide atoms lie in one graphite-like plane and form planar pentagons around the protactinium ions. The coordination 7 of protactinium originates from the 5 halide atoms and two bonds to protactinium atoms belonging to the nearby chains. These compounds easily hydrolyze in water. The pentachloride melts at 300 °C and sublimates at even lower temperatures.
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Some metallocenes consist of metal plus two cyclooctatetraenide anions (, abbreviated cot2−), namely the lanthanocenes and the actinocenes (uranocene and others). Metallocenes are a subset of a broader class of compounds called sandwich compounds . In the structure shown at right, the two pentagons are the cyclopentadienyl anions with circles inside them indicating they are aromatically stabilized. Here they are shown in a staggered conformation.
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Three such pentagons can easily fit at a common vertex, but a gap remains between two edges. It is this kind of discrepancy which is called "geometric frustration". There is one way to overcome this difficulty. Let the surface to be tiled be free of any presupposed topology, and let us build the tiling with a strict application of the local interaction rule.
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3D model of an icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.
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Jennifer McLoud-Mann is an American mathematician known for her 2015 discovery, with Casey Mann and undergraduate student David Von Derau, of the 15th and last class of convex pentagons to tile the plane. She is a professor of mathematics at the University of Washington Bothell, where she chairs the division of engineering and mathematics. Beyond tiling, her research interests include knot theory and combinatorics.
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Retrieved: 2013-10-14. Most modern Association footballs are stitched from 32 panels of waterproofed leather or plastic: 12 regular pentagons and 20 regular hexagons. The 32-panel configuration is the spherical polyhedron corresponding to the truncated icosahedron; it is spherical because the faces bulge from the pressure of the air inside. The first 32-panel ball was marketed by Select in the 1950s in Denmark.
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These figures have pentagrams (star pentagons) as faces or vertex figures. The small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, but pentagrammic vertex figures. In all cases, two faces can intersect along a line that is not an edge of either face, so that part of each face passes through the interior of the figure.
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Altamaha Tech's Dodecahedron The Dodecahedron is the official symbol of Altamaha Technical College. The twelve faceted sides are pentagons that represent the multiple facets of technical education, and is a reminder to stay focused on the future and to be multifaceted in the school's programs and services. The inspiration for this choice was the design for the deep space probes in the novel Contact by Carl Sagan.
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The pentagram map is the identity on the moduli space of pentagons. This is to say that there is always a projective transformation carrying a pentagon to its image under the pentagram map. The map T^2 is the identity on the space of labeled hexagons. Here T is the second iterate of the pentagram map, which acts naturally on labeled hexagons, as described above.
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The structure shown, is built of 36 triskelia, one of which is shown in blue. Another common assembly is a truncated icosahedron. To enclose a vesicle, exactly 12 pentagons must be present in the lattice. In a cell, clathrin triskelion in the cytoplasm binds to an adaptor protein that has bound membrane, linking one of its three feet to the membrane at a time.
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Robbins pentagons were named by after David P. Robbins, who had previously given a formula for the area of a cyclic pentagon as a function of its edge lengths. Buchholz and MacDougall chose this name by analogy with the naming of Heron triangles after Hero of Alexandria, the discoverer of Heron's formula for the area of a triangle as a function of its edge lengths.
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A logo for the new union was launched on 30 June 2017. The logo features a starburst of pentagons with the name of the union written at the centre. Despite being a proper noun, the name is stylised in the logo in all lower case letters as "national education union" rather than "National Education Union". The union uses the strapline "together we'll shape the future of education".
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Schattschneider won the Mathematical Association of America's Carl B. Allendoerfer Award for excellence in expository writing in Mathematics Magazine in 1979, for her article "Tiling the plane with congruent pentagons".. In 1993, she won the MAA's Award for Distinguished Teaching of College or University Mathematics.. In 2012 she became a fellow of the American Mathematical Society.List of Fellows of the American Mathematical Society, retrieved 2013-07-13.
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At the top of the cup, there is a silver ball with light-silver stars and pentagons. In the middle, the CONMEBOL logo is held together by two hoists, while the phrase "COPA SUDAMERICANA" can be read from the top. The trophy carries room for a 24 badges. The badges would be placed at the top base of the pedestal one underneath another and span 8 columns.
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Renga is a "theatrical co-op" game. It is designed to be played in a movie theater with up to one hundred players, with each player holding a laser pointer. Players must pilot a space ship by hitting one of four arrows, which causes the ship to move. At the same time, players must also defend the ship from objects such as pentagons and hexagons.
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In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball,HOW TO BUILD YOUR OWN HYPERBOLIC SOCCER BALL MODEL is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional soccer ball (truncated icosahedron) with heptagons in place of pentagons. It has Schläfli symbol of t{3,7}.
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Therefore, the resulting cone angle should have only certain, discrete values α = 2 arcsin(1 − n/6) = 112.9°, 83.6°, 60.0°, 38.9°, and 19.2° for n = 1, ..., 5, respectively. The graphene sheet is composed solely of carbon hexagons which can not form a continuous cone cap. As in the fullerenes, pentagons have to be added to form a curved cone tip, and their number is correspondingly n = 1, ..., 5.
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3D model of a triangular hebesphenorotunda In geometry, the triangular hebesphenorotunda is one of the Johnson solids (J92). . It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Most evident is the cluster of three pentagons and four triangles on one side of the solid.
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Furthermore, the angle of an Isosceles Triangle or the number of sides of a (regular) Polygon may be altered during life by deeds or surgical adjustments. An equilateral triangle is a member of the craftsman class. Squares and Pentagons are the "gentlemen" class, as doctors, lawyers, and other professions. Hexagons are the lowest rank of nobility, all the way up to (near) Circles, who make up the priest class.
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Reinhardt had previously considered the question of anisohedral convex polygons, showing that there were no anisohedral convex hexagons but being unable to show there were no such convex pentagons, while finding the five types of convex pentagon tiling the plane isohedrally. Kershner gave three types of anisohedral convex pentagon in 1968; one of these tiles using only direct isometries without reflections or glide reflections, so answering a question of Heesch.
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Rhombic dodecahedron The rhombic dodecahedron is a zonohedron with twelve rhombic faces and octahedral symmetry. It is dual to the quasiregular cuboctahedron (an Archimedean solid) and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space. The rhombic dodecahedron can be seen as a degenerate pyritohedron where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces.
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Equilateral pentagon built with four equal circles disposed in a chain. An equilateral pentagon is a polygon with five sides of equal length. However, its five internal angles can take a range of sets of values, thus permitting it to form a family of pentagons. In contrast, the regular pentagon is unique up to similarity, because it is equilateral and it is equiangular (its five angles are equal).
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The tiles are shapes bounded by three horocyclic segments (two of which are part of the same horocycle), and two line segments. All tiles are congruent. Although they are modeled by squares or rectangles of the Poincaré model, the tiles have five sides rather than four, and are not hyperbolic polygons, because their horocyclic edges are not straight. Alternatively, a combinatorially equivalent tiling uses hyperbolic pentagons that connect the same vertices in the same pattern.
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The dodecahedron is a regular polyhedron with Schläfli symbol {5,3}, having 3 pentagons around each vertex. In geometry, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions.
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Matila Ghyka, The Geometry of Art and Life (1977), p.68 The boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. It can be thought of as the 4-dimensional analog of the regular dodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedra, with 3 around each edge.
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The regular dodecahedron is topologically related to a series of tilings by vertex figure n3. The regular dodecahedron can be transformed by a truncation sequence into its dual, the icosahedron: The regular dodecahedron is a member of a sequence of otherwise non-uniform polyhedra and tilings, composed of pentagons with face configurations (V3.3.3.3.n). (For n > 6, the sequence consists of tilings of the hyperbolic plane.) These face-transitive figures have (n32) rotational symmetry.
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A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 6 pentagons), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts. It has 6 pentagonal faces, 15 edges, and 10 vertices.
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Equilateral pentagon built with four equal circles disposed in a chain. In geometry an equilateral pentagon is a polygon with five sides of equal length. Its five internal angles, in turn, can take a range of sets of values, thus permitting it to form a family of pentagons. The requirement is that all angles must add up to 540 degrees and must be between 0 and 360 degrees but not equal to 180 degrees.
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3D model of a gyroelongated square bipyramid In geometry, the gyroelongated square bipyramid, heccaidecadeltahedron, or tetrakis square antiprism is one of the Johnson solids (J17). As the name suggests, it can be constructed by gyroelongating an octahedron (square bipyramid) by inserting a square antiprism between its congruent halves. It is one of the eight strictly-convex deltahedra. The dual of the gyroelongated square bipyramid is a square truncated trapezohedron with 10 faces: 8 pentagons and 2 square.
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3D model of a bilunabirotunda In geometry, the bilunabirotunda is one of the Johnson solids (J91). It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Either one of the two clusters of two pentagons and two triangles can be aligned with a congruent patch of faces on the icosidodecahedron.
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The highest symmetry forms are D2d, order 8, while if the underlying rectangular cuboid is distorted into a rhombohedron, the symmetry is reduced to 2-fold rotational symmetry, C2, order 2. It has all 3-valence vertices and its dual has all triangular faces, including the snub disphenoid as a deltahedron with all equilateral triangles.Dual of Snub Disphenoid (J84) However the dual of the snub disphenoid is not space-filling because the pentagons are not right-angled.
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Some considered that setting out also involved the > use of equilateral or Pythagorean triangles, pentagons, and octagons. Two > authors believe the Golden Section (or at least its approximation) was used, > but its use in medieval times is not supported by most architectural > historians. The Australian architectural historian John James made a detailed study of the Cathedral of Chartres. In his work The Master Masons of Chartres he says that Bronze, one of the master masons, used the golden ratio.
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The most common variety of n-flake is two-dimensional (in terms of its topological dimension) and is formed of polygons. The four most common special cases are formed with triangles, squares, pentagons, and hexagons, but it can be extended to any polygon. Its boundary is the von Koch curve of varying types - depending on the n-gon - and infinitely many Koch curves are contained within. The fractals occupy zero area yet have an infinite perimeter.
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Spherical pentagonal icositetrahedron This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations (V3.3.3.3.n). (The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry. The pentagonal icositetrahedron is second in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n. The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.
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Snyder equal-area projection is used in the ISEA (Icosahedral Snyder Equal Area) discrete global grids. The first projection studies was conducted by John P. Snyder in the 1990s. Snyder, J. P. (1992), “An Equal-Area Map Projection for Polyhedral Globes”, Cartographica, 29(1), 10-21. urn:doi:10.3138/27H7-8K88-4882-1752. It is a modified Lambert azimuthal equal-area projection, most adequate to the polyhedral globe, a truncated icosahedron with 32 same-area faces (20 hexagons and 12 pentagons).
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This diagram is only an approximation of the ionic contributions to the membrane potential. Other ions including sodium, chloride, calcium, and others play a more minor role, even though they have strong concentration gradients, because they have more limited permeability than potassium. Key: pentagons – sodium ions; squares – potassium ions; circles – chloride ions; rectangles – membrane- impermeable anions (these arise from a variety of sources including proteins). The large structure with an arrow represents a transmembrane potassium channel and the direction of net potassium movement.
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In 1855, the U.S. inventor Charles Goodyear – who had patented vulcanised rubber – exhibited a spherical football, with an exterior of vulcanised rubber panels, at the Paris Exhibition Universelle. The ball was to prove popular in early forms of football in the U.S.soccerballworld.com, (no date) "Charles Goodyear's Soccer Ball" Downloaded 30/11/06. The iconic ball with a regular pattern of hexagons and pentagons (see truncated icosahedron) did not become popular until the 1960s, and was first used in the World Cup in 1970.
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15th monohedral convex pentagonal type, discovered in 2015 In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane and the sphere; the latter produces a tiling topologically equivalent to the dodecahedron.
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A full snub dodecahedral antiprism or omnisnub dodecahedral antiprism can be defined as an alternation of an truncated icosidodecahedral prism, represented by ht0,1,2,3{5,3,2}, or , although it cannot be constructed as a uniform 4-polytope. It has 184 cells: 2 snub dodecahedrons connected by 30 tetrahedrons, 12 pentagonal antiprisms, and 20 octahedrons, with 120 tetrahedrons in the alternated gaps. It has 120 vertices, 480 edges, and 544 faces (24 pentagons and 40+480 triangles). It has [5,3,2]+ symmetry, order 120.
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Mosque of Ibn Tulun: window with girih- style 10-point stars (at rear), with floral roundels in hexagons forming a frieze at front Jali are pierced stone screens with regularly repeating patterns. They are characteristic of Indo-Islamic architecture, for example in the Mughal dynasty buildings at Fatehpur Sikri and the Taj Mahal. The geometric designs combine polygons such as octagons and pentagons with other shapes such as 5- and 8-pointed stars. The patterns emphasized symmetries and suggested infinity by repetition.
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3D model of a truncated icosahedron In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges. It is the Goldberg polyhedron GPV(1,1) or {5+,3}1,1, containing pentagonal and hexagonal faces. This geometry is associated with footballs (soccer balls) typically patterned with white hexagons and black pentagons.
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The compound polyhedron known as the stellated octahedron can be represented by a{4,3} (an altered cube), and , 40px. The star polyhedron known as the small ditrigonal icosidodecahedron can be represented by a{5,3} (an altered dodecahedron), and , 40px. Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to take up the resulting free edges. The star polyhedron known as the great ditrigonal icosidodecahedron can be represented by a{5/2,3} (an altered great stellated dodecahedron), and , 40px.
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The is team No. 5 in the A-Rank ranking. They are known as "the face of Border", due to their frequent appearances in Border propaganda and promotional material. Because they spend so much time doing publicity for Border, they are believed by many to be inferior to the other A-rank teams, though this has been shown to be untrue. Their emblem has a design composed of 5 small black pentagons with a white star symbol inside each, arranged so as to form another star symbol.
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850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems, though never connected it to the series of numbers named after him. Luca Pacioli named his book Divina proportione (1509) after the ratio, and explored its properties including its appearance in some of the Platonic solids. Leonardo da Vinci, who illustrated the aforementioned book, called the ratio the sectio aurea ('golden section').
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Roman dodecahedron Omnidirectional sound source Regular dodecahedral objects have found some practical applications, and have also played a role in the visual arts and in philosophy. Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons."Florian Cajori, A History of Mathematics (1893) In Theaetetus, a dialogue of Plato, Plato was able to prove that there are just five uniform regular solids; these later became known as the platonic solids. Timaeus (c.
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The light chains bind primarily to the proximal leg portion of the heavy chain with some interaction near the trimerization domain. The β-propeller at the 'foot' of clathrin contains multiple binding sites for interaction with other proteins. When triskelia assemble together in solution, they can interact with enough flexibility to form 6-sided rings (hexagons) that yield a flat lattice, or 5-sided rings (pentagons) that are necessary for curved lattice formation. When many triskelions connect, they can form a basket-like structure.
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These matching rules can be imposed by decorations of the edges, as with the Wang tiles. Penrose's tiling can be viewed as a completion of Kepler's finite Aa pattern. A non- Penrose tiling by pentagons and thin rhombs in the early 18th-century Pilgrimage Church of Saint John of Nepomuk at Zelená hora, Czech Republic Penrose subsequently reduced the number of prototiles to two, discovering the kite and dart tiling (tiling P2 below) and the rhombus tiling (tiling P3 below). The rhombus tiling was independently discovered by Robert Ammann in 1976.
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The most common way presented to construct a regular polyhedron is via a fold-out net. To obtain a fold-out net of a polyhedron, one takes the surface of the polyhedron and cuts it along just enough edges so that the surface may be laid out flat. This gives a plan for the net of the unfolded polyhedron. Since the Platonic solids have only triangles, squares and pentagons for faces, and these are all constructible with a ruler and compass, there exist ruler-and-compass methods for drawing these fold-out nets.
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Some interesting fold-out nets of the cube, octahedron, dodecahedron and icosahedron are available here. Numerous children's toys, generally aimed at the teen or pre-teen age bracket, allow experimentation with regular polygons and polyhedra. For example, klikko provides sets of plastic triangles, squares, pentagons and hexagons that can be joined edge-to-edge in a large number of different ways. A child playing with such a toy could re-discover the Platonic solids (or the Archimedean solids), especially if given a little guidance from a knowledgeable adult.
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Nonuniform variants with [4,3,4] symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra to produce a nonuniform honeycomb with truncated octahedra and hexagonal prisms (as ditrigonal trapezoprisms). Its vertex figure is a C2v-symmetric triangular bipyramid. This honeycomb can then be alternated to produce another nonuniform honeycomb with pyritohedral icosahedra, octahedra (as triangular antiprisms), and tetrahedra (as sphenoids). Its vertex figure has C2v symmetry and consists of 2 pentagons, 4 rectangles, 4 isosceles triangles (divided into two sets of 2), and 4 scalene triangles.
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The visible parts of each face comprise five isosceles triangles which touch at five points around the pentagon. We could treat these triangles as 60 separate faces to obtain a new, irregular polyhedron which looks outwardly identical. Each edge would now be divided into three shorter edges (of two different kinds), and the 20 false vertices would become true ones, so that we have a total of 32 vertices (again of two kinds). The hidden inner pentagons are no longer part of the polyhedral surface, and can disappear.
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Shortly afterwards he travelled to Zurich to meet Calatrava, and ask him to design a residential building based on the idea of a structure of twisting cubes. Illustration of the general structure of the Turning Torso. This is a solid, immobile building constructed in nine segments of five-story pentagons that twist relative to each other as it rises; the topmost segment is twisted 90 degrees clockwise with respect to the ground floor. Each floor consists of an irregular pentagonal shape rotating around the vertical core, which is supported by an exterior steel framework.
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Irregular dodecahedron The Weaire–Phelan structure differs from Kelvin's in that it uses two kinds of cells, although they have equal volume. From a topological and symmetrical point of view, one is a pyritohedron, an irregular dodecahedron with pentagonal faces, possessing tetrahedral symmetry (Th). Tetrakaidecahedron The second is a form of truncated hexagonal trapezohedron, a species of tetrakaidecahedron with two hexagonal and twelve pentagonal faces, in this case only possessing two mirror planes and a rotoreflection symmetry. Like the hexagons in the Kelvin structure, the pentagons in both types of cells are slightly curved.
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The first name is from the regular-faced gyrobifastigium but elongated with 4 triangles expanded into pentagons. The name of the gyrobifastigium comes from the Latin fastigium, meaning a sloping roof.. In the standard naming convention of the Johnson solids, bi- means two solids connected at their bases, and gyro- means the two halves are twisted with respect to each other. The gyrobifastigium is first in a series of gyrobicupola, so this solid can also be called an elongated digonal gyrobicupola. Geometrically it can also be constructed as the dual of a digonal gyrobianticupola.
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In a sense one can understand them as affirming the existence of general bridges from thoughts to things. Both however can, like the postulates concerning specific constructions, be understood as "finiteness principles" affirming the existence of new arithmoi. Mayberry's “corrected” Euclid would thus underpin the sister disciplines of Geometry and Arithmetic with Common Notions, applicable to both, supplemented by two sets of Postulates, one for each discipline. Indeed, in so far as Geometry does rely on the notion of arithmos – it does so even in defining triangles, quadrilaterals, pentagons etc.
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The discoverers of the allotrope named the newfound molecule after Buckminster Fuller, who designed many geodesic dome structures that look similar to C60 and who had died the previous year in 1983 before discovery in 1984. This is slightly misleading, however, as Fuller's geodesic domes are constructed only by further dividing hexagons or pentagons into triangles, which are then deformed by moving vertices radially outward to fit the surface of a sphere. Geometrically speaking, buckminsterfullerene is a naturally-occurring example of a Goldberg polyhedron. A common, shortened name for buckminsterfullerene is "buckyballs".
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In La@C72, the lanthanum appears to stabilize the C72 carbon cage. A 1998 study by Stevenson et al. verified the presence of La@C72 as well as La2@C72, but empty-cage C72 was absent, based on laser desorption mass spectrometry and UV−vis spectroscopy. A 2008 study by Lu et al. showed that La2C72 do not adhere to the isolated pentagon rule (IPR), but has two pairs of fused pentagons at each pole of the cage and that the two La atoms reside close to the two fused-pentagon pairs.
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This is the opposite of what happens in the case of buckminsterfullerenes in which carbon sheets are given positive curvature by the inclusion of pentagons. The large-scale structure of carbon nanofoam is similar to that of an aerogel, but with 1% of the density of previously produced carbon aerogels—or only a few times the density of air at sea level. Unlike carbon aerogels, carbon nanofoam is a poor electrical conductor. The nanofoam contains numerous unpaired electrons, which Rode and colleagues propose is due to carbon atoms with only three bonds that are found at topological and bonding defects.
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The prison's burial ground, with the Houses of Parliament in the background. The image was published in 1862. Map of 1867, showing the prison and its surroundings The plan of the prison comprised a circular chapel at the centre of the site, surrounded by a three-storey hexagon made up of the governor's quarters, administrative offices and laundries, surrounded in turn by six pentagons of cell blocks. The buildings of each pentagon were set around a cluster of five small courtyards (with a watchtower at the centre) used as airing-yards, and in which prisoners undertook labour.
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Penrose's first tiling uses pentagons and three other shapes: a five- pointed "star" (a pentagram), a "boat" (roughly 3/5 of a star) and a "diamond" (a thin rhombus). To ensure that all tilings are non-periodic, there are matching rules that specify how tiles may meet each other, and there are three different types of matching rule for the pentagonal tiles. Treating these three types as different prototiles gives a set of six prototiles overall. It is common to indicate the three different types of pentagonal tiles using three different colors, as in the figure above right.
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Tie and Navette tiling (in red on a Penrose background) The three variants of the Penrose tiling are mutually locally derivable. Selecting some subsets from the vertices of a P1 tiling allows to produce other non-periodic tilings. If the corners of one pentagon in P1 are labeled in succession by 1,3,5,2,4 an unambiguous tagging in all the pentagons is established, the order being either clockwise or counterclockwise. Points with the same label define a tiling by Robinson triangles while points with the numbers 3 and 4 on them define the vertices of a Tie-and-Navette tiling.
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As well as building design, Gwynne also advised on furnishings and landscaping to create a complete ensemble. His use of plastic finishes included a special grass paper which was also his own product. "People seem to recognise my work as being from my hand in spite of the strong influence of client and site", he wrote in 1984. His later houses became more curvilinear, with rounded corners; one in Blackheath (22 Park Gate, designed for Leslie Bilsby) is designed as a series of linked pentagons, a space-age capsule that references the proportions of neighbouring Regency buildings.
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Under some conditions, carbon crystallizes as lonsdaleite, a hexagonal crystal lattice with all atoms covalently bonded and properties similar to those of diamond. Fullerenes are a synthetic crystalline formation with a graphite-like structure, but in place of flat hexagonal cells only, some of the cells of which fullerenes are formed may be pentagons, nonplanar hexagons, or even heptagons of carbon atoms. The sheets are thus warped into spheres, ellipses, or cylinders. The properties of fullerenes (split into buckyballs, buckytubes, and nanobuds) have not yet been fully analyzed and represent an intense area of research in nanomaterials.
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A Ho–Mg–Zn quasicrystal in the shape of a dodecahedron. A holmium–magnesium–zinc (Ho–Mg–Zn) quasicrystal is a quasicrystal made of an alloy of the three metals holmium, magnesium and zinc that has the shape of a regular dodecahedron, a Platonic solid with 12 five-sided faces. Unlike the similar pyritohedron shape of some cubic-system crystals such as pyrite, this quasicrystal has faces that are true regular pentagons. The crystal is part of the R-Mg-Zn family of crystals, where R=Y, Gd, Tb, Dy, Ho or Er. They were first discovered in 1994.
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Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold. It is a quotient space of the order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.
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Semesterberichte, Bd. 54, 2007, S. 53–68,Kotschick The topology and combinatorics of soccer balls, American Scientist, July/August 2006 In the case of the sphere, there is only the standard football (12 black pentagons, 20 white hexagons, with a pattern corresponding to an icosahedral root) provided that "precisely three edges meet at every vertex". If more than three faces meet at some vertex, then there is a method to generate infinite sequences of different soccer balls by a topological construction called a branched covering. Kotschick's analysis also applies to fullerenes and polyhedra that Kotschick calls generalized soccer balls.
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Likewise, 20 yards past the goal line at the Liberty Bowl puts one several rows up into the end zone stands. AstroTurf sections were added around the grass field to accommodate the required width, while the end zones became half AstroTurf/half grass pentagons in order to bring the field to the required length. However, the end zones were nowhere near regulation; they averaged around 9 yards in length, and no other American stadium had end zones shorter than 15 yards. The stands jutted into the corners of the end zones, creating a distinct safety hazard.
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It is common for the White (gote) and Black (sente) player to be indicated at the beginning of the notation string with either black and white triangles () or shogi-piece-shaped pentagons (), such as ' or '. However, this is not obligatory: several books notate shogi moves without explicit indication of which player is making the moves. (See the adjacent image for an example.) In such cases, knowing which player the move refers to can be determined by the context in the book. This white/black convention is more common when the moves are not numbered (which is also optional to notate).
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By 1999, chaos theory became a profound influence on Kidner's work and geometric abstraction in the form of Penrose pentagons reprinted on paper became a critical tool as a metaphor for ordering the chaos in the world. This was Kidner's response to the many dystopian world events, such as global warming, wars, ethnic cleansing, terrorism and intense nationalism. Kidner was intrigued by the fact that his pentagon patterns looked chaotic, like whirlpools that appeared and receded as he viewed them. His use of colour in these works was often random and whereas in his former work, colour was used to clarify the grid, now it subverted it.
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190 In 1919 the badges were standardized as four shapes: Circular (Battleships & Battle Cruisers), Pentagonal (Cruisers), Shield (Destroyers) and Diamond (all other types and shore establishments). Testing was carried out to ensure that the badges were designed appropriately to identify ships. Cardboard mockups were created, gilded, and installed on a police launch, which was observed on patrol of the Thames by a captured German submarine moored outside the Palace of Westminster. It was decided to use different shapes to identify different types of vessel: circles for battleships, pentagons for cruisers, 'U'-shaped shields for destroyers, and diamonds for auxiliary units, including depot ships, small war vessels, and aircraft carriers.
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A non-Desargues (103103) configuration. As a projective configuration, the Desargues configuration has the notation (103103), meaning that each of its ten points is incident to three lines and each of its ten lines is incident to three points. Its ten points can be viewed in a unique way as a pair of mutually inscribed pentagons, or as a self-inscribed decagon . The Desargues graph, a 20-vertex bipartite symmetric cubic graph, is so called because it can be interpreted as the Levi graph of the Desargues configuration, with a vertex for each point and line of the configuration and an edge for every incident point-line pair.
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As a result, defending an edge against a center attack is very difficult. Schensted and Titus attacked this problem with successive versions of the game board, culminating in the present "official" board with three pentagons inserted among the hexagons. They noted that were players to play on a hemisphere rather than a plane with hexagons, with the equator divided into three "sides" (each 1/3 the circumference of the hemisphere), the distance from the "north pole" of the hemisphere to the equator was 1/4 the circumference, and thus the distance ratio improved from 1/3 to 3/4. This made defending a side from a center attack much more plausible.
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In mathematics, the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane. The pentagram map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a new polygon from these intersections. Richard Schwartz introduced the pentagram map for a general polygon in a 1992 paper though it seems that the special case, in which the map is defined for pentagons only, goes back to an 1871 paper of Alfred Clebsch and a 1945 paper of Theodore Motzkin. The pentagram map is similar in spirit to the constructions underlying Desargues' theorem and Poncelet's porism.
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Released in 2012, the Nike Zoom KD V brought back the midtop upper with Hyperfuse and the Adaptive Fit lacing system. The midsole featured Nike Zoom forefoot, but is shaped like a bar to increase flexibility, and for the first time in the KD line, a 180 Air Max unit in the heel, placed in a lightweight Phylon midsole which also acts as a heel cup. The outsole featured storytelling pattern in the form of pentagons, which has five sides, and five is the number of the KD V, also is the second number in his jersey. An elite version released on April 20, 2013.
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Carbon nanofoam is the fifth known allotrope of carbon, discovered in 1997 by Andrei V. Rode and co-workers at the Australian National University in Canberra. It consists of a low-density cluster-assembly of carbon atoms strung together in a loose three-dimensional web. Each cluster is about 6 nanometers wide and consists of about 4000 carbon atoms linked in graphite-like sheets that are given negative curvature by the inclusion of heptagons among the regular hexagonal pattern. This is the opposite of what happens in the case of buckminsterfullerenes, in which carbon sheets are given positive curvature by the inclusion of pentagons.
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Assignment of these peaks to certain carbon functionalization types is somewhat uncertain and still under debates. For example, one of interpretations goes as following: non-oxygenated ring contexts (284.8 eV), C-O (286.2 eV), C=O (287.8 eV) and O-C=O (289.0 eV). Another interpretation using density functional theory calculation goes as following: C=C with defects such as functional groups and pentagons (283.6 eV), C=C (non- oxygenated ring contexts) (284.3 eV), sp3C-H in the basal plane and C=C with functional groups (285.0 eV), C=O and C=C with functional groups, C-O (286.5 eV), and O-C=O (288.3 eV).
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As a result, even with the addition of AstroTurf cutouts to widen and lengthen the field, it was still narrower and shorter than all other CFL fields, including other US fields which were not regulation. In order to shoehorn even an approximation of a Canadian field onto the playing surface, the end zones became half-grass/half-Astroturf pentagons that were only nine yards long in the middle and seven yards long at the sidelines. CFL rules call for a 20-yard end zone, and no other stadium had end zones shorter than 15 yards. The stands jutted into the corners of the end zones, creating a clear safety hazard.
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The balls used in association football and team handball are perhaps the best-known example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life. The ball comprises the same pattern of regular pentagons and regular hexagons, but it is more spherical due to the pressure of the air inside and the elasticity of the ball. This ball type was introduced to the World Cup in 1970 (starting in 2006, this iconic design has been superseded by alternative patterns). Geodesic domes are typically based on triangular facetings of this geometry with example structures found across the world, popularized by Buckminster Fuller.
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The two sphere-shaped pennants in the probe had surfaces covered by 72 pentagonal elements in a pattern similar to that later used by association footballs. In the centre was an explosive charge designed to shatter the sphere, sending the pentagonal shields in all directions. Each pentagonal element was made of titanium alloy; the centre regular pentagon had the State Emblem of the Soviet Union with the Cyrillic letters СССР ("USSR") engraved below and was surrounded by five non- regular pentagons which were each engraved with СССР СЕНТЯБРЬ 1959 ("USSR SEPTEMBER 1959"). The third pennant was similar engravings on aluminium strips which were embossed on the last stage of the Luna 2 rocket.
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Although the procedure for the synthesis of the C60 fullerene is well established (generation of a large current between two nearby graphite electrodes in an inert atmosphere) a 2002 study described an organic synthesis of the compound starting from simple organic compounds.The numbers in image correspond to the way the new carbon carbon bonds are formed. :Multistep fullerene synthesis In the final step a large polycyclic aromatic hydrocarbon consisting of 13 hexagons and three pentagons was submitted to flash vacuum pyrolysis at 1100 °C and 0.01 Torr. The three carbon chlorine bonds served as free radical incubators and the ball was stitched up in a no-doubt complex series of radical reactions.
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As a result of origami study through the application of geometric principles, methods such as Haga's theorem have allowed paperfolders to accurately fold the side of a square into thirds, fifths, sevenths, and ninths. Other theorems and methods have allowed paperfolders to get other shapes from a square, such as equilateral triangles, pentagons, hexagons, and special rectangles such as the golden rectangle and the silver rectangle. Methods for folding most regular polygons up to and including the regular 19-gon have been developed. A regular n-gon can be constructed by paper folding if and only if n is a product of distinct Pierpont primes, powers of two, and powers of three.
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The configuration in which these three ordinary lines are replaced by a single line cannot be realized in the Euclidean plane, but forms a finite projective space known as the Fano plane. Because of this connection, the Kelly–Moser example has also been called the non-Fano configuration. The other counterexample, due to McKee,. consists of two regular pentagons joined edge-to-edge together with the midpoint of the shared edge and four points on the line at infinity in the projective plane; these 13 points have among them 6 ordinary lines. Modifications of Böröczky's construction lead to sets of odd numbers of points with 3\lfloor n/4\rfloor ordinary lines.
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A polygon with Heesch number 5, the highest finite such number known, found by Casey Mann Casey Mann is an American mathematician, specialising in discrete and computational geometry, in particular tessellation and knot theory. He is Professor for Mathematics at University of Washington Bothell, and received the PhD at the University of Arkansas in 2001. He is known for his 2015 discovery, with Jennifer McLoud-Mann and undergraduate student David Von Derau, of the 15th and last class of convex pentagons to tile the plane. Mann is also known for his work on the Heesch's problem, to which he contributed the polygon with the largest known Heesch number, which was covered in a numberphile video.
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According to Mario Livio, Ancient Greek mathematicians first studied what we now call the golden ratio, because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (an irrational number), surprising Pythagoreans. Euclid's Elements () provides several propositions and their proofs employing the golden ratio, and contains its first known definition which proceeds as follows: Michael Maestlin, the first to write a decimal approximation of the ratio The golden ratio was studied peripherally over the next millennium. Abu Kamil (c.
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The domed enclosure of the Jameh Mosque of Isfahan, built in 1086-7 by Nizam al-Mulk, was the largest masonry dome in the Islamic world at that time, had eight ribs, and introduced a new form of corner squinch with two quarter domes supporting a short barrel vault. In 1088 Tāj-al-Molk, a rival of Nizam al-Mulk, built another dome at the opposite end of the same mosque with interlacing ribs forming five-pointed stars and pentagons. This is considered the landmark Seljuk dome, and may have inspired subsequent patterning and the domes of the Il-Khanate period. The use of tile and of plain or painted plaster to decorate dome interiors, rather than brick, increased under the Seljuks.
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The domed enclosure of the Jameh Mosque of Isfahan, built in 1086-7 by Nizam al-Mulk, was the largest masonry dome in the Islamic world at that time, had eight ribs, and introduced a new form of corner squinch with two quarter domes supporting a short barrel vault. In 1088 Tāj-al-Molk, a rival of Nizam al-Mulk, built another dome at the opposite end of the same mosque with interlacing ribs forming five-pointed stars and pentagons. This is considered the landmark Seljuk dome, and may have inspired subsequent patterning and the domes of the Il-Khanate period. The use of tile and of plain or painted plaster to decorate dome interiors, rather than brick, increased under the Seljuks.
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There also exist eight other (103103) configurations (that is, sets of points and lines in the Euclidean plane with three lines per point and three points per line) that are not incidence-isomorphic to the Desargues configuration, one of which is shown at right. In all of these configurations, each point has three other points that are not collinear with it. But in the Desargues configuration, these three points are always collinear with each other (if the chosen point is the center of perspectivity, then the three points form the axis of perspectivity) while in the other configuration shown in the illustration these three points form a triangle of three lines. As with the Desargues configuration, the other depicted configuration can be viewed as a pair of mutually inscribed pentagons.
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The book is organized in chapters related to various soccer subjects ("The ball", "The field", "The player", etc.). Inside each chapter we find the answers to questions such as what it would be like to play soccer in each of the planets of the Solar System, why the ball is more than one meter ahead of where we think it really is in a 100 km/h kick (due to delays in the transmission of the nervous impulse), or why soccer balls are made of 12 pentagons and 20 hexagons, etc. This book is also available in e-reader format. Introduction to BioMEMS (CRC Press, 2012) covers the whole breadth of BioMEMS (Biomedical MicroElectroMechanicalSystems), including classical microfabrication, microfluidics, tissue engineering, cell-based and noncell- based devices, and implantable systems.
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Piero della Francesca's image of a truncated icosahedron from his book De quinque corporibus regularibus The truncated icosahedron was known to Archimedes, who classified the 13 Archimedean solids in a lost work. All we know of his work on these shapes comes from Pappus of Alexandria, who merely lists the numbers of faces for each: 12 pentagons and 20 hexagons, in the case of the truncated icosahedron. The first known image and complete description of a truncated icosahedron is from a rediscovery by Piero della Francesca, in his 15th-century book De quinque corporibus regularibus, which included five of the Archimedean solids (the five truncations of the regular polyhedra). The same shape was depicted by Leonardo da Vinci, in his illustrations for Luca Pacioli's plagiarism of della Francesca's book in 1509.
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Buckminsterfullerene, C60 In 1985, on the basis of the Sussex studies and the stellar discoveries, laboratory experiments (with co- workers James R. Heath, Sean C. O'Brien, Yuan Liu, Robert Curl and Richard Smalley at Rice University) which simulated the chemical reactions in the atmospheres of the red giant stars demonstrated that stable C60 molecules could form spontaneously from a condensing carbon vapour. The co-investigators directed lasers at graphite and examined the results. The C60 molecule is a molecule with the same symmetry pattern as a football, consisting of 12 pentagons and 20 hexagons of carbon atoms. Kroto named the molecule buckminsterfullerene, after Buckminster Fuller who had conceived of the geodesic domes, as the dome concept had provided a clue to the likely structure of the new species.
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The Adidas Telstar became the standard design for representing footballs in different media Law 2 of the game specifies that the ball is an air-filled sphere with a circumference of , a weight of , inflated to a pressure of 0.6 to 1.1 atmospheres () "at sea level", and covered in leather or "other suitable material". The weight specified for a ball is the dry weight, as older balls often became significantly heavier in the course of a match played in wet weather. The standard ball is a Size 5, although smaller sizes exist: Size 3 is standard for team handball and Size 4 in futsal and other small-field variants. Other sizes are used in underage games or as novelty items. Most modern footballs are stitched from 32 panels of waterproofed leather or plastic: 12 regular pentagons and 20 regular hexagons.
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Its three-part name derives from its key features: "flow-following" indicates that its vertical coordinates are based on both terrain and potential temperature (isentropic sigma coordinates, previously used in the now-discontinued rapid update cycle model), and "finite-volume" describes the method used for calculating horizontal transport. The "icosahedral" portion describes the model's most uncommon feature: whereas most grid-based forecast models have historically used rectangular grid points (a less than ideal arrangement for a planet that is a slightly oblate spheroid), the FIM instead fits Earth to a Goldberg polyhedron with icosahedral symmetry, with twelve evenly spaced pentagons (including two at the poles) anchoring a grid of hexagons. In November 2016, the ESRL announced it was no longer pursuing the FIM as a replacement for the GFS and would be instead developing the FV3, which uses some of the FIM's principles except on a square grid. The FIM will continue to be run for experimental purposes until FV3 commences.
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Schattschneider was born in Staten Island; her mother, Charlotte Lucile Ingalls Wood, taught Latin and was herself the daughter of a Staten Island school principal, and her father, Robert W. Wood, Jr., worked as a bridge engineer for New York City.. Her family moved to Lake Placid, New York during World War II, while her father served as an engineer for the U. S. Army; she began her schooling in Lake Placid, but returned to Staten Island after the war. She did her undergraduate studies in mathematics at the University of Rochester, and earned a Ph.D. in 1966 from Yale University under the joint supervision of Tsuneo Tamagawa and Ichirô Satake; her thesis, in abstract algebra, concerned semisimple algebraic groups. She taught at Northwestern University and the University of Illinois at Chicago Circle before joining the faculty of Moravian College in 1968, where she remained for 34 years until her retirement.Author biography from "Tiling the Plane with Congruent Pentagons", Mathematics Magazine, 1978.
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The Desargues configuration viewed as a pair of mutually inscribed pentagons: each pentagon vertex lies on the line through one of the sides of the other pentagon. The ten lines involved in Desargues's theorem (six sides of triangles, the three lines and , and the axis of perspectivity) and the ten points involved (the six vertices, the three points of intersection on the axis of perspectivity, and the center of perspectivity) are so arranged that each of the ten lines passes through three of the ten points, and each of the ten points lies on three of the ten lines. Those ten points and ten lines make up the Desargues configuration, an example of a projective configuration. Although Desargues's theorem chooses different roles for these ten lines and points, the Desargues configuration itself is more symmetric: any of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity.
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