Regular polygon
In Euclidean geometry, a regular polygon is a polygon, equiangular and equilateral. Regular polygons may be either star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed; these properties apply to all regular polygons, whether star. A regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle; that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon has an inscribed circle or incircle, tangent to every side at the midpoint, thus a regular polygon is a tangential polygon. A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon; the symmetry group of an n-sided regular polygon is dihedral group Dn: D2, D3, D4...
It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center. If n is then half of these axes pass through two opposite vertices, the other half through the midpoint of opposite sides. If n is odd all axes pass through a vertex and the midpoint of the opposite side. All regular simple polygons are convex; those having the same number of sides are similar. An n-sided convex regular polygon is denoted by its Schläfli symbol. For n < 3, we have two degenerate cases: Monogon Degenerate in ordinary space. Digon. In certain contexts all the polygons considered. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular and the faces will be described as triangle, pentagon, etc. For a regular convex n-gon, each interior angle has a measure of: × 180 degrees, or equivalently 180 n degrees; as the number of sides, n approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides the internal angle is 179.964°.
As the number of sides increase, the internal angle can come close to 180°, the shape of the polygon approaches that of a circle. However the polygon can never become a circle; the value of the internal angle can never become equal to 180°, as the circumference would become a straight line. For this reason, a circle is not a polygon with an infinite number of sides. For n > 2, the number of diagonals is 1 2 n. The diagonals divide the polygon into 1, 4, 11, 24, … pieces OEIS: A007678. For a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from an arbitrary point in the plane to the vertices, we have ∑ i = 1 n d i 4 n + 3 R 4 = 2. For a regular n-gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem; this is a generalization of Viviani's theorem for the n. The circumradius R from the center of a regular polygon to one of the vertices is related to the side length s or to the apothem a by R = s 2 sin = a cos
Rhombille tiling
In geometry, the rhombille tiling known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 120 ° angles. Sets of three rhombi meet at their 120° angles and sets of six rhombi meet at their 60° angles; the rhombille tiling can be seen as a subdivision of a hexagonal tiling with each hexagon divided into three rhombi meeting at the center point of the hexagon. This subdivision represents a regular compound tiling, it can be seen as a subdivision of four hexagonal tilings with each hexagon divided into 12 rhombi. The diagonals of each rhomb are in the ratio 1:√3; this is the dual tiling of the trihexagonal kagome lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, in the face configuration for monohedral tilings it is denoted, it is one of 56 possible isohedral tilings by quadrilaterals, one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.
It is possible to embed the rhombille tiling into a subset of a three-dimensional integer lattice, consisting of the points with |x + y + z| ≤ 1, in such a way that two vertices are adjacent if and only if the corresponding lattice points are at unit distance from each other, more such that the number of edges in the shortest path between any two vertices of the tiling is the same as the Manhattan distance between the corresponding lattice points. Thus, the rhombille tiling can be viewed as an example of an infinite unit distance graph and partial cube; the rhombille tiling can be interpreted as an isometric projection view of a set of cubes in two different ways, forming a reversible figure related to the Necker Cube. In this context it is known as the "reversible cubes" illusion. In the M. C. Escher artworks Metamorphosis I, Metamorphosis II, Metamorphosis III Escher uses this interpretation of the tiling as a way of morphing between two- and three-dimensional forms. In another of his works, Escher played with the tension between the two-dimensionality and three-dimensionality of this tiling: in it he draws a building that has both large cubical blocks as architectural elements and an upstairs patio tiled with the rhombille tiling.
A human figure descends from the patio past the cubes, becoming more stylized and two-dimensional as he does so. These works involve only a single three-dimensional interpretation of the tiling, but in Convex and Concave Escher experiments with reversible figures more and includes a depiction of the reversible cubes illusion on a flag within the scene; the rhombille tiling is used as a design for parquetry and for floor or wall tiling, sometimes with variations in the shapes of its rhombi. It appears in ancient Greek floor mosaics from Delos and from Italian floor tilings from the 11th century, although the tiles with this pattern in Siena Cathedral are of a more recent vintage. In quilting, it has been known since the 1850s as the "tumbling blocks" pattern, referring to the visual dissonance caused by its doubled three-dimensional interpretation; as a quilting pattern it has many other names including cubework, heavenly stairs, Pandora's box. It has been suggested that the tumbling blocks quilt pattern was used as a signal in the Underground Railroad: when slaves saw it hung on a fence, they were to box up their belongings and escape.
See Quilts of the Underground Railroad. In these decorative applications, the rhombi may appear in multiple colors, but are given three levels of shading, brightest for the rhombs with horizontal long diagonals and darker for the rhombs with the other two orientations, to enhance their appearance of three-dimensionality. There is a single known instance of implicit rhombille and trihexagonal tiling in English heraldry – in the Geal/e arms; the rhombille tiling may be viewed as the result of overlaying two different hexagonal tilings, translated so that some of the vertices of one tiling land at the centers of the hexagons of the other tiling. Thus, it can be used to define block cellular automata in which the cells of the automaton are the rhombi of a rhombille tiling and the blocks in alternating steps of the automaton are the hexagons of the two overlaid hexagonal tilings. In this context, it is called the "Q*bert neighborhood", after the video game Q*bert which featured an isometric view of a pyramid of cubes as its playing field.
The Q*bert neighborhood may be used to support universal computation via a simulation of billiard ball computers. In condensed matter physics, the rhombille tiling is known as the dice lattice, diced lattice, or dual kagome lattice, it is one of several repeating structures used to investigate Ising models and related systems of spin interactions in diatomic crystals, it has been studied in percolation theory. The rhombille tiling has *632 symmetry, but vertices can be colored with alternating colors on the inner points leading to a *333 symmetry; the rhombille tiling is the dual of the trihexagonal tiling, as such is part of a set of uniform dual tilings. It is a part of a sequence of rhombic polyhedra and tilings with Coxeter group symmetry, starting from the cube, which can be seen as a rhombic hexahedron where the rhombi are squares; the nth element in this sequence has a face configuration of V3.n.3.n. The rhombille tiling is one of many different ways of tiling the plane by congruent rhombi.
Others include a diagonally flattened variation of the square tiling, the tiling used by the Miura-ori folding pattern, the Penrose tiling which
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century, he was born in London, received his BA and PhD from Cambridge, but lived in Canada from age 29. He was always called Donald, from his third name MacDonald, he was most noted for his work on higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10, he felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the Canadian Music Journal. Coxeter went up to Cambridge in 1926 to read mathematics. There he earned his BA in 1928, his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, Solomon Lefschetz.
Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics. In 1934 he spent a further year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto. In 1938 he and P. Du Val, H. T. Flather, John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays published by W. W. Rouse Ball in 1892, he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and a Fellow of the Royal Society in 1950, he met M. C. Escher in 1954 and the two became lifelong friends, he inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, he published twelve books. Since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor, he was made a Fellow of the Royal Society in 1950 and in 1997 he was awarded their Sylvester Medal.
In 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he received the Jeffery–Williams Prize. 1940: Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46: 380-407, MR 2,10 doi:10.1007/BF01181449 1942: Non-Euclidean Geometry, University of Toronto Press, MAA. 1954: "Uniform Polyhedra", Philosophical Transactions of the Royal Society A 246: 401–50 doi:10.1098/rsta.1954.0003 1949: The Real Projective Plane 1957: Generators and Relations for Discrete Groups 1980: Second edition, Springer-Verlag ISBN 0-387-09212-9 1961: Introduction to Geometry 1963: Regular Polytopes, Macmillan Company 1967: Geometry Revisited 1970: Twisted honeycombs 1973: Regular Polytopes, Dover edition, ISBN 0-486-61480-8 1974: Projective Geometry 1974: Regular Complex Polytopes, Cambridge University Press 1981:, Zero-Symmetric Graphs, Academic Press. 1985: Regular and Semi-Regular Polytopes II, Mathematische Zeitschrift 188: 559–591 1987 Projective Geometry ISBN 978-0-387-40623-7 1988: Regular and Semi-Regular Polytopes III, Mathematische Zeitschrift 200: 3–45 1995: F. Arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors: Kaleidoscopes — Selected Writings of H.
S. M. Coxeter. John Wiley and Sons ISBN 0-471-01003-0 1999: The Beauty of Geometry: Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler; the Coxeter Legacy: Reflections and Projections. Providence, R. I.: American Mathematical Society. ISBN 978-0-8218-3722-1. OCLC 62282754. Roberts, Siobhan. King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. New York: Walker & Company. ISBN 978-0-8027-1499-2. OCLC 71436884. Archival papers held at University of Toronto Archives and Records Management Services Harold Scott MacDonald Coxeter at the Mathematics Genealogy Project H. S. M. Coxeter, Erich W. Ellers, Branko Grünbaum, Peter McMullen, Asia Ivic Weiss Notices of the AMS: Volume 50, Number 10. Www.donaldcoxeter.com www.math.yorku.ca/dcoxeter webpages dedicated to him Jaron's World: Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C. Escher video of a lecture by H. S. M. Coxeter, April 28, 2000
Tessellation
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to a variety of geometries. A periodic tiling has a repeating pattern; some special kinds include regular tilings with regular polygonal tiles all of the same shape, semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons; such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings.
Tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles. Decorative mosaic tilings made of small squared blocks called tesserae were employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made an early documented study of tessellations, he wrote about semiregular tessellations in his Harmonices Mundi. Some two hundred years in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries.
Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Aleksei Shubnikov and Nikolai Belov, Heinrich Heesch and Otto Kienzle. In Latin, tessella is a small cubical piece of stone or glass used to make mosaics; the word "tessella" means "small square". It corresponds to the everyday term tiling, which refers to applications of tessellations made of glazed clay. Tessellation in two dimensions called planar tiling, is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules; these rules can be varied. Common ones are that there must be no gaps between tiles, that no corner of one tile can lie along the edge of another; the tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile.
There are only three shapes that can form such regular tessellations: the equilateral triangle and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps. Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner. Irregular tessellations can be made from other shapes such as pentagons, polyominoes and in fact any kind of geometric shape; the artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, these can be used to decorate physical surfaces such as church floors. More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries.
These tiles may be any other shapes. Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane; the Conway criterion is a sufficient but not necessary set of rules for deciding if a given shape tiles the plane periodically without reflections: some tiles fail the criterion but still tile the plane. No general rule has been found for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations. Mathematically, tessellations can be extended to spaces other than the Euclidean plane; the Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more dimensions, he further defined the Schläfli symbol notation to make it easy to describe polytopes.
For example, the Schläfli symbol for an equilateral triangle is. The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is. Other methods exist for describing polygonal tilings; when the tessellation
Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, 14 vertices of two types, it is a Catalan solid, the dual polyhedron of the cuboctahedron. The rhombic dodecahedron is a zonohedron, its polyhedral dual is the cuboctahedron. The long diagonal of each face is √2 times the length of the short diagonal, so that the acute angles on each face measure arccos, or 70.53°. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B; the rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron and the rhombic triacontahedron. The rhombic dodecahedron can be used to tessellate three-dimensional space.
It can be stacked to fill a space. This polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice, it is the Brillouin zone of body centered cubic crystals. Some minerals such as garnet form a rhombic dodecahedral crystal habit. Honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of, a hexagonal prism capped with half a rhombic dodecahedron; the rhombic dodecahedron appears in the unit cells of diamond and diamondoids. In these cases, four vertices are absent; the graph of the rhombic dodecahedron is nonhamiltonian. A rhombic dodecahedron can be dissected with its center into 4 trigonal trapezohedra; these rhombohedra are the cells of a trigonal trapezohedral honeycomb. This is analogous to the dissection of a regular hexagon dissected into rhombi, tiled in the plane as a rhombille. If the edge length of a rhombic dodecahedron is a, the radius of an inscribed sphere is r i = 6 3 a ≈ 0.816 496 5809 a, OEIS: A157697the radius of the midsphere is r m = 2 2 3 a ≈ 0.942 809 041 58 a, OEIS: A179587.and the radius of the circumscribed sphere is r o = 2 3 3 a ≈ 1.154 700 538 a, OEIS: A020832.
The area A and the volume V of the rhombic dodecahedron of edge length a are: A = 8 2 a 2 ≈ 11.313 7085 a 2 V = 16 3 9 a 3 ≈ 3.079 201 44 a 3 The rhombic dodecahedron has four special orthogonal projections along its axes of symmetry, centered on a face, an edge, the two types of vertex and fourfold. The last two correspond to the B2 and A2 Coxeter planes; the eight vertices where three faces meet at their obtuse angles have Cartesian coordinates: The coordinates of the six vertices where four faces meet at their acute angles are:, The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron, with permutation of coordinates and with parameter h = 1. The rhombic dodecahedron is a parallelohedron, a space-filling polyhedron, being the dual to the tetroctahedrille or half cubic honeycomb, described by two Coxeter diagrams: and. With D3d symmetry, it can be seen as an elongated trigonal trapezohedron. Other symmetry constructions of the rhombic dodecahedron are space-filling, as parallelotopes they are similar to variations of space-filling truncated octahedra.
For example, with 4 square faces, 60-degree rhombic faces, D4h dihedral symmetry, order 16. It be seen as a cuboctahedron with square pyramids augmented on the bottom. In 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the Bilinski dodecahedron, it has the same different geometry. The rhombic faces in this form have the golden ratio. Another topologically equivalent variation, sometimes called a deltoidal dodecahedron or trapezoidal dodecahedron, is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites, it has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by. Is the rhombic solution; as approaches 1/2, approaches infinity. (
Pentagon
In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be self-intersecting. A self-intersecting regular pentagon is called a pentagram. A regular pentagon has Schläfli symbol and interior angles are 108°. A regular pentagon has five lines of reflectional symmetry, rotational symmetry of order 5; the diagonals of a convex regular pentagon are in the golden ratio to its sides. Its height and width are given by Height = 5 + 2 5 2 ⋅ Side ≈ 1.539 ⋅ Side, Width = Diagonal = 1 + 5 2 ⋅ Side ≈ 1.618 ⋅ Side, Diagonal = R 5 + 5 2 = 2 R cos 18 ∘ = 2 R cos π 10 ≈ 1.902 R, where R is the radius of the circumcircle. The area of a convex regular pentagon with side length t is given by A = t 2 25 + 10 5 4 = 5 t 2 tan 4 ≈ 1.720 t 2. A pentagram or pentangle is a regular star pentagon, its Schläfli symbol is. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.
When a regular pentagon is circumscribed by a circle with radius R, its edge length t is given by the expression t = R 5 − 5 2 = 2 R sin 36 ∘ = 2 R sin π 5 ≈ 1.176 R, its area is A = 5 R 2 4 5 + 5 2. The area of any regular polygon is: A = 1 2 P r where P is the perimeter of the polygon, r is the inradius. Substituting the regular pentagon's values for P and r gives the formula A = 1 2 ⋅ 5 t ⋅ t tan 2 = 5 t 2 tan 4 with side length “f” Like every regular convex polygon, the regular convex pentagon has an inscribed circle; the apothem, the radius r of the inscribed circle, of a regular pentagon is related to the side length t by r = t 2 tan = t 2 5 − 20 ≈ 0.6882 ⋅ t. Like every regular convex polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C PA + PD = PB + PC + PE; the regular pentagon is constructible with compass and straightedge. A variety of methods are known for constructing a regular pentagon.
Some are discussed below. One method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwell's "Polyhedra."The top panel shows the construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius, its center is located at point