1.
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 higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. 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 set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, 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 often made use of tessellations, both in ordinary Euclidean geometry and in 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 widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions 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

2.
Regular polygon
–
In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is 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. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also 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, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, 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 a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius

3.
Edge (geometry)
–
For edge in graph theory, see Edge In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a segment on the boundary. In a polyhedron or more generally a polytope, an edge is a segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitzs theorem as being exactly the 3-vertex-connected planar graphs. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces, for example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex, more generally, by Balinskis theorem, at least d edges meet at every vertex of a convex polytope. Similarly, in a polyhedron, exactly two faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, archived from the original on 4 February 2007

4.
Vertex (geometry)
–
In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices

5.
Alternation (geometry)
–
In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices. Coxeter labels an alternation by a prefixed by an h, standing for hemi or half, because alternation reduce all polygon faces to half as many sides, it can only be applied for polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a configuration consisting of all even-numbered elements can be alternated. For example, the alternation a vertex figure with 2a. 2b. 2c is a.3. b.3. c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divide in half into degenerate digons, a snub can be seen as an alternation of a truncated regular or truncated quasiregular polyhedron. In general a polyhedron can be snubbed if its truncation has only even-sided faces, all truncated rectified polyhedra can be snubbed, not just from regular polyhedra. The snub square antiprism is an example of a general snub and this alternation operation applies to higher-dimensional polytopes and honeycombs as well, but in general most of the results of this operation will not be uniform. The voids created by the vertices will not in general create uniform facets. Examples, Honeycombs An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb, an alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb. 4-polytope An alternated truncated 24-cell is the snub 24-cell, 4-honeycombs, An alternated truncated 24-cell honeycomb is the snub 24-cell honeycomb. A hypercube can always be alternated into a uniform demihypercube, cube → Tetrahedron → Tesseract → 16-cell → Penteract → demipenteract Hexeract → demihexeract. Coxeter also used the operator a, which contains both halves, so retains the original symmetry, for even-sided regular polyhedra, a represents a compound polyhedron with two opposite copies of h. For odd-sided, greater than 3, regular polyhedra a, becomes a star polyhedron, Norman Johnson extended the use of the altered operator a, b for blended, and c for converted, as, and respectively. The compound polyhedron, stellated octahedron can be represented by a, the star-polyhedron, small ditrigonal icosidodecahedron, can be represented by a, and. Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to take up the free edges. A similar operation can truncate alternate vertices, rather than just removing them, below is a set of polyhedra that can be generated from the Catalan solids. These have two types of vertices which can be alternately truncated, truncating the higher order vertices and both vertex types produce these forms, Conway polyhedral notation Wythoff construction Coxeter, H. S. M

6.
Dual polygon
–
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. The dual of a polygon is an isotoxal polygon. For example, the rectangle and rhombus are duals, in a cyclic polygon, longer sides correspond to larger exterior angles in the dual, and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, for example, the dual of a highly acute isosceles triangle is an obtuse isosceles triangle. In the Dorman Luke construction, each face of a polyhedron is the dual polygon of the corresponding vertex figure. As an example of the duality of polygons we compare properties of the cyclic. This duality is perhaps more clear when comparing an isosceles trapezoid to a kite. The simplest qualitative construction of a polygon is a rectification operation. New edges are formed between these new vertices and that is, the polygon generated by applying it twice is in general not similar to the original polygon. As with dual polyhedra, one can take a circle and perform polar reciprocation in it. Combinatorially, one can define a polygon as a set of vertices, a set of edges, then the dual polygon is obtained by simply switching the vertices and edges. Thus for the triangle with vertices and edges, the triangle has vertices, and edges, where B connects AB & BC. This is not a particularly fruitful avenue, as combinatorially, there is a family of polygons, geometric duality of polygons is more varied. Dual curve Dual polyhedron Self-dual polygon Dual Polygon Applet by Don Hatch

7.
Geometry
–
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space

8.
Polygon
–
In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular

9.
Euclidean geometry
–
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another

10.
Spherical geometry
–
Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry that is not Euclidean, two practical applications of the principles of spherical geometry are navigation and astronomy. In plane geometry, the concepts are points and lines. On a sphere, points are defined in the usual sense, the equivalents of lines are not defined in the usual sense of straight line in Euclidean geometry, but in the sense of the shortest paths between points, which are called geodesics. On a sphere, the geodesics are the circles, other geometric concepts are defined as in plane geometry. Spherical geometry is not elliptic geometry, but is rather a subset of elliptic geometry, for example, it shares with that geometry the property that a line has no parallels through a given point. An important geometry related to that of the sphere is that of the projective plane. Locally, the plane has all the properties of spherical geometry. In particular, it is non-orientable, or one-sided, Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. Higher-dimensional spherical geometries exist, see elliptic geometry, the earliest mathematical work of antiquity to come down to our time is On the rotating sphere by Autolycus of Pitane, who lived at the end of the fourth century BC. The book of unknown arcs of a written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the law of sines. The book On Triangles by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe, however, Gerolamo Cardano noted a century later that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah. L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15,1771, pp. 195–216, Opera Omnia, Series 1, Volume 28, pp. 142–160. L. Euler, De mensura angulorum solidorum, Acta academiae scientarum imperialis Petropolitinae 2,1781, p. 31–54, Opera Omnia, Series 1, vol. L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientarum imperialis Petropolitinae 4,1783, p. 91–96, Opera Omnia, Series 1, vol. L. Euler, Geometrica et sphaerica quaedam, Mémoires de lAcademie des Sciences de Saint-Petersbourg 5,1815, p. 96–114, Opera Omnia, Series 1, vol. L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientarum imperialis Petropolitinae 3,1782, p. 72–86, Opera Omnia, Series 1, vol

11.
Great circle
–
A great circle, also known as an orthodrome or Riemannian circle, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. This partial case of a circle of a sphere is opposed to a circle, the intersection of the sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same circumference as each other, a great circle is the largest circle that can be drawn on any given sphere. Every circle in Euclidean 3-space is a circle of exactly one sphere. For most pairs of points on the surface of a sphere, there is a great circle through the two points. The exception is a pair of points, for which there are infinitely many great circles. The minor arc of a circle between two points is the shortest surface-path between them. In this sense, the arc is analogous to “straight lines” in Euclidean geometry. The length of the arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry. The great circles are the geodesics of the sphere, in higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn+1. To prove that the arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it. Consider the class of all paths from a point p to another point q. Introduce spherical coordinates so that p coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by θ = θ, ϕ = ϕ, a ≤ t ≤ b provided we allow φ to take on arbitrary real values. The infinitesimal arc length in these coordinates is d s = r θ ′2 + ϕ ′2 sin 2 θ d t. So the length of a curve γ from p to q is a functional of the curve given by S = r ∫ a b θ ′2 + ϕ ′2 sin 2 θ d t. Note that S is at least the length of the meridian from p to q, S ≥ r ∫ a b | θ ′ | d t ≥ r | θ − θ |. Since the starting point and ending point are fixed, S is minimized if and only if φ =0, so the curve must lie on a meridian of the sphere φ = φ0 = constant

12.
Equator
–
The Equator usually refers to an imaginary line on the Earths surface equidistant from the North Pole and South Pole, dividing the Earth into the Northern Hemisphere and Southern Hemisphere. The Equator is about 40,075 kilometres long, some 78. 7% lies across water and 21. 3% over land, other planets and astronomical bodies have equators similarly defined. Generally, an equator is the intersection of the surface of a sphere with the plane that is perpendicular to the spheres axis of rotation. The latitude of the Earths equator is by definition 0° of arc, the equator is the only line of latitude which is also a great circle — that is, one whose plane passes through the center of the globe. The plane of Earths equator when projected outwards to the celestial sphere defines the celestial equator, in the cycle of Earths seasons, the plane of the equator passes through the Sun twice per year, at the March and September equinoxes. To an observer on the Earth, the Sun appears to travel North or South over the equator at these times, light rays from the center of the Sun are perpendicular to the surface of the Earth at the point of solar noon on the Equator. Locations on the Equator experience the quickest sunrises and sunsets because the sun moves nearly perpendicular to the horizon for most of the year. The Earth bulges slightly at the Equator, the diameter of the Earth is 12,750 kilometres. Because the Earth spins to the east, spacecraft must also launch to the east to take advantage of this Earth-boost of speed, seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earths axis relative to the plane of revolution. During the year the northern and southern hemispheres are inclined toward or away from the sun according to Earths position in its orbit, the hemisphere inclined toward the sun receives more sunlight and is in summer, while the other hemisphere receives less sun and is in winter. At the equinoxes, the Earths axis is not tilted toward the sun, instead it is perpendicular to the sun meaning that the day is about 12 hours long, as is the night, across the whole of the Earth. Near the Equator there is distinction between summer, winter, autumn, or spring. The temperatures are usually high year-round—with the exception of high mountains in South America, the temperature at the Equator can plummet during rainstorms. In many tropical regions people identify two seasons, the wet season and the dry season, but many places close to the Equator are on the oceans or rainy throughout the year, the seasons can vary depending on elevation and proximity to an ocean. The Equator lies mostly on the three largest oceans, the Pacific Ocean, the Atlantic Ocean, and the Indian Ocean. The highest point on the Equator is at the elevation of 4,690 metres, at 0°0′0″N 77°59′31″W and this is slightly above the snow line, and is the only place on the Equator where snow lies on the ground. At the Equator the snow line is around 1,000 metres lower than on Mount Everest, the Equator traverses the land of 11 countries, it also passes through two island nations, though without making a landfall in either. Starting at the Prime Meridian and heading eastwards, the Equator passes through, Despite its name, however, its island of Annobón is 155 km south of the Equator, and the rest of the country lies to the north

13.
Dihedron
–
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. Dihedra have also been called bihedra, flat polyhedra, or doubly covered polygons, a regular dihedron is the dihedron formed by two regular polygons, which may be described by the Schläfli symbol. As a spherical polyhedron, each polygon of such a dihedron fills a hemisphere, the dual of a n-gonal dihedron is the n-gonal hosohedron, where n digon faces share two vertices. A dihedron can be considered a degenerate prism consisting of two n-sided polygons connected back-to-back, so that the object has no depth. The polygons must be congruent, but glued in such a way one is the mirror image of the other. This characterization holds also for the distances on the surface of a dihedron, as a spherical tiling, a dihedron can exist as nondegenerate form, with two n-sided faces covering the sphere, each face being a hemisphere, and vertices around a great circle. The regular polyhedron is self-dual, and is both a hosohedron and a dihedron, in the limit the dihedron becomes an apeirogonal dihedron as a 2-dimensional tessellation, A regular ditope is an n-dimensional analogue of a dihedron, with Schläfli symbol. It has two facets, which share all ridges, in common, polyhedron Polytope Weisstein, Eric W. Dihedron

14.
Hosohedron
–
In geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular n-gonal hosohedron has Schläfli symbol, with each spherical lune having internal angle 2π/n radians, the restriction m ≥3 enforces that the polygonal faces must have at least three sides. When considering polyhedra as a tiling, this restriction may be relaxed, since digons can be represented as spherical lunes. Allowing m =2 admits a new class of regular polyhedra. On a spherical surface, the polyhedron is represented as n abutting lunes, all these lunes share two common vertices. The digonal faces of a 2n-hosohedron, represents the fundamental domains of symmetry in three dimensions, Cnv, order 2n. The reflection domains can be shown as alternately colored lunes as mirror images, bisecting the lunes into two spherical triangles creates bipyramids and define dihedral symmetry Dnh, order 4n. The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the dual of the n-gonal hosohedron is the n-gonal dihedron. The polyhedron is self-dual, and is both a hosohedron and a dihedron, a hosohedron may be modified in the same manner as the other polyhedra to produce a truncated variation. The truncated n-gonal hosohedron is the n-gonal prism, in the limit the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation, Multidimensional analogues in general are called hosotopes. A regular hosotope with Schläfli symbol has two vertices, each with a vertex figure, the two-dimensional hosotope, is a digon. The term “hosohedron” was coined by H. S. M, Coxeter, and possibly derives from the Greek ὅσος “as many”, the idea being that a hosohedron can have “as many faces as desired”. Polyhedron Polytope McMullen, Peter, Schulte, Egon, Abstract Regular Polytopes, Cambridge University Press, ISBN 0-521-81496-0 Coxeter, H. S. M, ISBN 0-486-61480-8 Weisstein, Eric W. Hosohedron

15.
Antipodal point
–
This term applies to opposite points on a circle or any n-sphere. An antipodal point is called an antipode, a back-formation from the Greek loan word antipodes. On a circle, such points are also called diametrically opposite, in other words, each line through the centre intersects the sphere in two points, one for each ray out from the centre, and these two points are antipodal. The Borsuk–Ulam theorem is a result from algebraic topology dealing with pairs of points. It says that any function from Sn to Rn maps some pair of antipodal points in Sn to the same point in Rn. Here, Sn denotes the n-dimensional sphere in -dimensional space, the antipodal map A, Sn → Sn, defined by A = −x, sends every point on the sphere to its antipodal point. It is homotopic to the identity map if n is odd, if one wants to consider antipodal points as identified, one passes to projective space. This article incorporates text from a now in the public domain, Chisholm, Hugh. Hazewinkel, Michiel, ed. Antipodes, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 antipodal

16.
Lune (geometry)
–
In plane geometry, a lune is the concave-convex area bounded by two circular arcs, while a convex-convex area is termed a lens. The word lune derives from luna, the Latin word for Moon, formally, a lune is the relative complement of one disk in another. Alternatively, if A and B are disks, then L = A − A ∩ B is a lune, in the 5th century BC, Hippocrates of Chios showed that certain lunes could be exactly squared by straightedge and compass. Arbelos Crescent Gauss–Bonnet theorem Weisstein, Eric W. Lune, the Five Squarable Lunes at MathPages

17.
Digon
–
In geometry, a digon is a polygon with two sides and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved, a regular digon has both angles equal and both sides equal and is represented by Schläfli symbol. It may be constructed on a sphere as a pair of 180 degree arcs connecting antipodal points, the digon is the simplest abstract polytope of rank 2. A truncated digon, t is a square, an alternated digon, h is a monogon. A straight-sided digon is regular even though it is degenerate, because its two edges are the length and its two angles are equal. As such, the regular digon is a constructible polygon, some definitions of a polygon do not consider the digon to be a proper polygon because of its degeneracy in the Euclidean case. A digon as a face of a polyhedron is degenerate because it is a degenerate polygon, but sometimes it can have a useful topological existence in transforming polyhedra. A spherical lune is a digon whose two vertices are antipodal points on the sphere, a spherical polyhedron constructed from such digons is called a hosohedron. The digon is an important construct in the theory of networks such as graphs. Topological equivalences may be established using a process of reduction to a set of polygons. The digon represents a stage in the simplification where it can be removed and substituted by a line segment. The cyclic groups may be obtained as rotation symmetries of polygons, monogon Demihypercube Herbert Busemann, The geometry of geodesics. New York, Academic Press,1955 Coxeter, Regular Polytopes, Dover Publications Inc,1973 ISBN 0-486-61480-8 Weisstein, a. B. Ivanov, Digon, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Media related to Digons at Wikimedia Commons

18.
Herbert Busemann
–
Herbert Busemann was a German-American mathematician specializing in convex and differential geometry. He is the author of Busemanns theorem in Euclidean geometry and geometric tomography and he was a member of the Royal Danish Academy and a winner of the Lobachevsky Medal, the first American mathematician to receive it. He was also a Fulbright scholar in New Zealand in 1952, Herbert Busemann was born in Berlin to a well-to-do family. His father, Alfred Busemann, was a director of Krupp and he studied at University of Munich, Paris, and Rome. He defended his dissertation in University of Göttingen in 1931, where his advisor was Richard Courant and he remained in Göttingen as an assistant until 1933, when he escaped Nazi Germany to Copenhagen. He worked at the University of Copenhagen until 1936, when he left to the United States, there, he got married in 1939 and naturalized in 1943. He advanced to a professor in 1964, and continued working at USC until his retirement in 1970. Over the course of his work at USC, he supervised over 10 Ph. D. students and he is the author of six monographs, two of which were translated into Russian. He received Lobachevsky Medal in 1985 for his book The geometry of geodesics, Busemann was also an active mathematical citizen. At different times, he was the president of the California chapter of Mathematical Association of America, Busemann was also an accomplished linguist, he was able to read and speak in French, German, Spanish, Italian, Russian, and Danish. He could also read Arabic, Latin, Greek and Swedish and he translated a number of papers and monograph, most notably from Russian, a rare language at the time. He was also an accomplished artist and had public exhibitions of his Hard-edge paintings]. He died in Santa Ynez, California on February 3,1994, introduction to algebraic manifolds, Princeton University Press,1939. With Paul J. Kelly, Projective geometry and projective metrics, Academic Press,1953, convex Surfaces, Interscience 1958, Dover,2008. Geometry of Geodesics, Academic Press 1955, Dover,2005, metric methods in Finsler spaces and in the foundations of geometry, Princeton University Press, Oxford University Press,1942. With Bhalchandra Phadke, Spaces with distinguished geodesics, Dekker,1987, recent synthetic differential geometry, Springer 1970. Busemann function Busemann–Petty problem Reinhard Siegmund-Schultze, Mathematicians Fleeing from Nazi Germany, Individual Fates and Global Impact, p.107, Richard J. Gardner, Geometric tomography, p.309, Cambridge University Press,1994. Professor as SC plans study in New Zealand, Los Angeles Times,6 January 1952, professors write Geometry Textbook, Los Angeles Times 10 August 1953, p.1

19.
International Standard Book Number
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

20.
Triangle
–
A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate

21.
Equilateral triangle
–
In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular and they are regular polygons, and can therefore also be referred to as regular triangles. Thus these are properties that are unique to equilateral triangles, the three medians have equal lengths. The three angle bisectors have equal lengths, every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral, in particular, A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide. It is also equilateral if its circumcenter coincides with the Nagel point, for any triangle, the three medians partition the triangle into six smaller triangles. A triangle is equilateral if and only if any three of the triangles have either the same perimeter or the same inradius. A triangle is equilateral if and only if the circumcenters of any three of the triangles have the same distance from the centroid. Morleys trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, a version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, PA, PB, and PC satisfy the inequality that any two of them sum to at least as great as the third. By Eulers inequality, the triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle, specifically. The triangle of largest area of all those inscribed in a circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, the ratio of the area to the square of the perimeter of an equilateral triangle,1123, is larger than that for any other triangle. If a segment splits an equilateral triangle into two regions with equal perimeters and with areas A1 and A2, then 79 ≤ A1 A2 ≤97, in no other triangle is there a point for which this ratio is as small as 2. For any point P in the plane, with p, q, and t from the vertices A, B. For any point P on the circle of an equilateral triangle, with distances p, q. There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral, an equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the group of order 6 D3

22.
Isosceles triangle
–
In geometry, an isosceles triangle is a triangle that has two sides of equal length. By the isosceles triangle theorem, the two angles opposite the sides are themselves equal, while if the third side is different then the third angle is different. By the Steiner–Lehmus theorem, every triangle with two angle bisectors of equal length is isosceles, in an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle, the vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base, if needed, and the term leg is not generally used. A triangle with two equal sides has exactly one axis of symmetry, which goes through the vertex angle. Thus the axis of symmetry coincides with the bisector of the vertex angle, the median drawn to the base, the altitude drawn from the vertex angle. Whether the isosceles triangle is acute, right or obtuse depends on the vertex angle, in Euclidean geometry, the base angles cannot be obtuse or right because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. The Euler line of any triangle goes through the orthocenter, its centroid. In an isosceles triangle with two equal sides, the Euler line coincides with the axis of symmetry. This can be seen as follows, if the vertex angle is acute, then the orthocenter, the centroid, and the circumcenter all fall inside the triangle. In an isosceles triangle the incenter lies on the Euler line, the Steiner inellipse of any triangle is the unique ellipse that is internally tangent to the triangles three sides at their midpoints. For any isosceles triangle with area T and perimeter p, we have 2 p b 3 − p 2 b 2 +16 T2 =0. By substituting the height, the formula for the area of a triangle can be derived from the general formula one-half the base times the height. This is what Herons formula reduces to in the isosceles case, if the apex angle and leg lengths of an isosceles triangle are known, then the area of that triangle is, T =2 = a 2 sin cos . This is derived by drawing a line from the base of the triangle. The bases of two right triangles are both equal to the hypotenuse times the sine of the bisected angle by definition of the term sine. For the same reason, the heights of these triangles are equal to the times the cosine of the bisected angle

23.
Quadrilateral
–
In Euclidean plane geometry, a quadrilateral is a polygon with four edges and four vertices or corners. Sometimes, the quadrangle is used, by analogy with triangle. The origin of the quadrilateral is the two Latin words quadri, a variant of four, and latus, meaning side. Quadrilaterals are simple or complex, also called crossed, simple quadrilaterals are either convex or concave. The interior angles of a simple quadrilateral ABCD add up to 360 degrees of arc and this is a special case of the n-gon interior angle sum formula × 180°. All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges, any quadrilateral that is not self-intersecting is a simple quadrilateral. In a convex quadrilateral, all angles are less than 180°. Irregular quadrilateral or trapezium, no sides are parallel, trapezium or trapezoid, at least one pair of opposite sides are parallel. Isosceles trapezium or isosceles trapezoid, one pair of sides are parallel. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, parallelogram, a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of length, that opposite angles are equal. In other words, parallelograms include all rhombi and all rhomboids, rhombus or rhomb, all four sides are of equal length. An equivalent condition is that the diagonals bisect each other. Rhomboid, a parallelogram in which adjacent sides are of unequal lengths, not all references agree, some define a rhomboid as a parallelogram which is not a rhombus. Rectangle, all four angles are right angles, an equivalent condition is that the diagonals bisect each other and are equal in length. Square, all four sides are of length, and all four angles are right angles. An equivalent condition is that opposite sides are parallel, that the diagonals bisect each other. A quadrilateral is a if and only if it is both a rhombus and a rectangle

24.
Square
–
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter

25.
Rectangle
–
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as a quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle, a rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle, a rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin rectangulus, which is a combination of rectus and angulus, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with sides equal in length. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons, a convex quadrilateral with successive sides a, b, c, d whose area is 12. A rectangle is a case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is Simple, The boundary does not cross itself, star-shaped, The whole interior is visible from a single point, without crossing any edge. De Villiers defines a more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles, quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia, a rectangle is cyclic, all corners lie on a single circle. It is equiangular, all its corner angles are equal and it is isogonal or vertex-transitive, all corners lie within the same symmetry orbit. It has two lines of symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus, as shown in the table below, the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa

26.
Rhombus
–
In Euclidean geometry, a rhombus is a simple quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length, every rhombus is a parallelogram and a kite. A rhombus with right angles is a square, the word rhombus comes from Greek ῥόμβος, meaning something that spins, which derives from the verb ῥέμβω, meaning to turn round and round. The word was used both by Euclid and Archimedes, who used the term solid rhombus for two right circular cones sharing a common base, the surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones. This is a case of the superellipse, with exponent 1. Every rhombus has two diagonals connecting pairs of vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals and it follows that any rhombus has the following properties, Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular, that is, a rhombus is an orthodiagonal quadrilateral, the first property implies that every rhombus is a parallelogram. Thus denoting the common side as a and the diagonals as p and q, not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite, every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus. A rhombus is a tangential quadrilateral and that is, it has an inscribed circle that is tangent to all four sides. As for all parallelograms, the area K of a rhombus is the product of its base, the base is simply any side length a, K = a ⋅ h. The inradius, denoted by r, can be expressed in terms of the p and q as. The dual polygon of a rhombus is a rectangle, A rhombus has all sides equal, a rhombus has opposite angles equal, while a rectangle has opposite sides equal. A rhombus has a circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex angles, the diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle, a rhombohedron is a three-dimensional figure like a cube, except that its six faces are rhombi instead of squares. The rhombic dodecahedron is a polyhedron with 12 congruent rhombi as its faces

27.
Parallelogram
–
In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length, by comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped, rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles Rectangle – A parallelogram with four angles of equal size. Rhombus – A parallelogram with four sides of equal length, square – A parallelogram with four sides of equal length and angles of equal size. A simple quadrilateral is a if and only if any one of the following statements is true. Two pairs of opposite angles are equal in measure, one pair of opposite sides are parallel and equal in length. Each diagonal divides the quadrilateral into two congruent triangles, the sum of the squares of the sides equals the sum of the squares of the diagonals. It has rotational symmetry of order 2, the sum of the distances from any interior point to the sides is independent of the location of the point. Thus all parallelograms have all the properties listed above, and conversely, if just one of statements is true in a simple quadrilateral. Opposite sides of a parallelogram are parallel and so will never intersect, the area of a parallelogram is twice the area of a triangle created by one of its diagonals. The area of a parallelogram is also equal to the magnitude of the cross product of two adjacent sides. Any line through the midpoint of a parallelogram bisects the area, any non-degenerate affine transformation takes a parallelogram to another parallelogram. A parallelogram has rotational symmetry of order 2, if it also has exactly two lines of reflectional symmetry then it must be a rhombus or an oblong. If it has four lines of symmetry, it is a square. The perimeter of a parallelogram is 2 where a and b are the lengths of adjacent sides, unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area. The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square. If two lines parallel to sides of a parallelogram are constructed concurrent to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area, the diagonals of a parallelogram divide it into four triangles of equal area. All of the formulas for general convex quadrilaterals apply to parallelograms

28.
Trapezoid
–
The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, the first recorded use of the Greek word translated trapezoid was by Marinus Proclus in his Commentary on the first book of Euclids Elements. This article uses the term trapezoid in the sense that is current in the United States, in many other languages using a word derived from the Greek for this figure, the form closest to trapezium is used. A right trapezoid has two adjacent right angles, right trapezoids are used in the trapezoidal rule for estimating areas under a curve. An acute trapezoid has two adjacent acute angles on its longer base edge, while an obtuse trapezoid has one acute, an acute trapezoid is also an isosceles trapezoid, if its sides have the same length, and the base angles have the same measure. An obtuse trapezoid with two pairs of sides is a parallelogram. A parallelogram has central 2-fold rotational symmetry, a Saccheri quadrilateral is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean plane. A Lambert quadrilateral in the plane has 3 right angles. A tangential trapezoid is a trapezoid that has an incircle, there is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Some define a trapezoid as a quadrilateral having one pair of parallel sides. Others define a trapezoid as a quadrilateral with at least one pair of parallel sides, the latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral ill-defined and this article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals, under the inclusive definition, all parallelograms are trapezoids. Rectangles have mirror symmetry on mid-edges, rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices. Four lengths a, c, b, d can constitute the sides of a non-parallelogram trapezoid with a and b parallel only when | d − c | < | b − a | < d + c. The quadrilateral is a parallelogram when d − c = b − a =0, the angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal. The diagonals cut each other in mutually the same ratio, the diagonals cut the quadrilateral into four triangles of which one opposite pair are similar. The diagonals cut the quadrilateral into four triangles of which one pair have equal areas

29.
Kite (geometry)
–
In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of sides, but they are opposite to each other rather than adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape, kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object. A kite, as defined above, may be convex or concave. A concave kite is called a dart or arrowhead, and is a type of pseudotriangle. If all four sides of a kite have the same length, if a kite is equiangular, meaning that all four of its angles are equal, then it must also be equilateral and thus a square. A kite with three equal 108° angles and one 36° angle forms the hull of the lute of Pythagoras. The kites that are cyclic quadrilaterals are exactly the ones formed from two congruent right triangles. That is, for these kites the two angles on opposite sides of the symmetry axis are each 90 degrees. These shapes are called right kites and they are in fact bicentric quadrilaterals, among all the bicentric quadrilaterals with a given two circle radii, the one with maximum area is a right kite. The tiling that it produces by its reflections is the deltoidal trihexagonal tiling, among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, 5π/12. Its four vertices lie at the three corners and one of the midpoints of the Reuleaux triangle. In non-Euclidean geometry, a Lambert quadrilateral is a kite with three right angles. A quadrilateral is a if and only if any one of the following conditions is true. One diagonal is the bisector of the other diagonal. One diagonal is a line of symmetry, one diagonal bisects a pair of opposite angles. The kites are the quadrilaterals that have an axis of symmetry along one of their diagonals, if crossings are allowed, the list of quadrilaterals with axes of symmetry must be expanded to also include the antiparallelograms. Every kite is orthodiagonal, meaning that its two diagonals are at angles to each other

30.
Pentagon
–
In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the angles in a simple pentagon is 540°. A pentagon may be simple or 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, and rotational symmetry of order 5. The diagonals of a regular pentagon are in the golden ratio to its sides. The area of a regular convex pentagon with side length t is given by A = t 225 +1054 =5 t 2 tan 4 ≈1.720 t 2. A pentagram or pentangle is a regular star pentagon and its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio. The area of any polygon is, A =12 P r where P is the perimeter of the polygon. Substituting the regular pentagons values for P and r gives the formula A =12 ×5 t × t tan 2 =5 t 2 tan 4 with side length t, like every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the r of the inscribed circle. Like every regular polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, the regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon, one method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwells Polyhedra. The top panel shows the construction used in Richmonds method to create the side of the inscribed pentagon, the circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius and this point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the axis at point Q. A horizontal line through Q intersects the circle at point P, to determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras theorem and two sides, the hypotenuse of the triangle is found as 5 /2

31.
Hexagon
–
In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations

32.
Heptagon
–
In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is also referred to as the septagon, using sept- together with the Greek suffix -agon meaning angle. A regular heptagon, in all sides and all angles are equal, has internal angles of 5π/7 radians. The area of a regular heptagon of side length a is given by, the apothem is half the cotangent of π /7, and the area of each of the 14 small triangles is one-fourth of the apothem. This expression cannot be rewritten without complex components, since the indicated cubic function is casus irreducibilis. As 7 is a Pierpont prime but not a Fermat prime and this type of construction is called a neusis construction. It is also constructible with compass, straightedge and angle trisector, the impossibility of straightedge and compass construction follows from the observation that 2 cos 2 π7 ≈1.247 is a zero of the irreducible cubic x3 + x2 − 2x −1. Consequently, this polynomial is the polynomial of 2cos, whereas the degree of the minimal polynomial for a constructible number must be a power of 2. An approximation for practical use with an error of about 0. 2% is shown in the drawing and it is attributed to Albrecht Dürer. Let A lie on the circumference of the circumcircle, then B D =12 B C gives an approximation for the edge of the heptagon. Example to illustrate the error, At a circumscribed circle radius r =1 m, since 7 is a prime number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z7, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the heptagon, john Conway labels these by a letter and group order. Full symmetry of the form is r14 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g7 subgroup has no degrees of freedom but can seen as directed edges. However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis. A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon and angles π /7,2 π /7, thus its sides coincide with one side and two particular diagonals of the regular heptagon. Two kinds of star heptagons can be constructed from regular heptagons, labeled by Schläfli symbols, blue, and green star heptagons inside a red heptagon

33.
Octagon
–
In geometry, an octagon is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol and can also be constructed as a truncated square, t. A truncated octagon, t is a hexadecagon, t, the sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°, the midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. A regular octagon is a figure with sides of the same length. It has eight lines of symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol, the internal angle at each vertex of a regular octagon is 135°. The area of an octagon of side length a is given by A =2 cot π8 a 2 =2 a 2 ≃4.828 a 2. In terms of the circumradius R, the area is A =4 sin π4 R2 =22 R2 ≃2.828 R2. In terms of the r, the area is A =8 tan π8 r 2 =8 r 2 ≃3.314 r 2. These last two coefficients bracket the value of pi, the area of the unit circle. The area can also be expressed as A = S2 − a 2, where S is the span of the octagon, or the second-shortest diagonal, and a is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside and then takes the corner triangles and places them with right angles pointed inward, the edges of this square are each the length of the base. Given the length of a side a, the span S is S = a 2 + a + a 2 = a ≈2.414 a. The area is then as above, A =2 − a 2 =2 a 2 ≈4.828 a 2, expressed in terms of the span, the area is A =2 S2 ≈0.828 S2. Another simple formula for the area is A =2 a S, more often the span S is known, and the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above, a ≈ S /2.414, the two end lengths e on each side, as well as being e = a /2, may be calculated as e = /2. The circumradius of the octagon in terms of the side length a is R = a

34.
Nonagon
–
In geometry, a nonagon /ˈnɒnəɡɒn/ is a nine-sided polygon or 9-gon. The name nonagon is a hybrid formation, from Latin, used equivalently, attested already in the 16th century in French nonogone. The name enneagon comes from Greek enneagonon, and is more correct. A regular nonagon is represented by Schläfli symbol and has angles of 140°. Although a regular nonagon is not constructible with compass and straightedge and it can be also constructed using neusis, or by allowing the use of an angle trisector. The following is a construction of a nonagon using a straightedge. The regular enneagon has Dih9 symmetry, order 18, there are 2 subgroup dihedral symmetries, Dih3 and Dih1, and 3 cyclic group symmetries, Z9, Z3, and Z1. These 6 symmetries can be seen in 6 distinct symmetries on the enneagon, john Conway labels these by a letter and group order. Full symmetry of the form is r18 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g9 subgroup has no degrees of freedom but can seen as directed edges. The regular enneagon can tessellate the euclidean tiling with gaps and these gaps can be filled with regular hexagons and isosceles triangles. In the notation of symmetrohedron this tiling is called H with H representing *632 hexagonal symmetry in the plane, the K9 complete graph is often drawn as a regular enneagon with all 36 edges connected. This graph also represents an orthographic projection of the 9 vertices and 36 edges of the 8-simplex and they Might Be Giants have a song entitled Nonagon on their childrens album Here Come the 123s. It refers to both an attendee at a party at which everybody in the party is a many-sided polygon, slipknots logo is also a version of a nonagon, being a nine-pointed star made of three triangles. King Gizzard & the Lizard Wizard have an album titled Nonagon Infinity, temples of the Bahai Faith are required to be nonagonal. The U. S. Steel Tower is an irregular nonagon, enneagram Trisection of the angle 60°, Proximity construction Weisstein, Eric W. Nonagon

35.
Decagon
–
In geometry, a decagon is a ten-sided polygon or 10-gon. A regular decagon has all sides of length and each internal angle will always be equal to 144°. Its Schläfli symbol is and can also be constructed as a pentagon, t. By simple trigonometry, d =2 a, and it can be written algebraically as d = a 5 +25. The side of a regular decagon inscribed in a circle is −1 +52 =1 ϕ. As 10 =2 ×5, a power of two times a Fermat prime, it follows that a regular decagon is constructible using compass and straightedge, or by an edge-bisection of a regular pentagon. An alternative method is as follows, Construct a pentagon in a circle by one of the shown in constructing a pentagon. Extend a line from each vertex of the pentagon through the center of the circle to the side of that same circle. Where each line cuts the circle is a vertex of the decagon, the five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon, both in the construction with given circumcircle as well as with given side length is the golden ratio dividing a line segment by exterior division the determining construction element. In the construction with given circumcircle the circular arc around G with radius GE3 produces the segment AH, a M ¯ M H ¯ = A H ¯ A M ¯ =1 +52 = Φ ≈1.618. In the construction with side length the circular arc around D with radius DA produces the segment E10F. E1 E10 ¯ E1 F ¯ = E10 F ¯ E1 E10 ¯ = R a =1 +52 = Φ ≈1.618, the regular decagon has Dih10 symmetry, order 20. There are 3 subgroup dihedral symmetries, Dih5, Dih2, and Dih1, and 4 cyclic group symmetries, Z10, Z5, Z2, and Z1. These 8 symmetries can be seen in 10 distinct symmetries on the decagon, john Conway labels these by a letter and group order. Full symmetry of the form is r20 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g10 subgroup has no degrees of freedom but can seen as directed edges

36.
Hendecagon
–
In geometry, a hendecagon or 11-gon is an eleven-sided polygon. A regular hendecagon is represented by Schläfli symbol, a regular hendecagon has internal angles of 147.27 degrees. The area of a regular hendecagon with side length a is given by A =114 a 2 cot π11 ≃9.36564 a 2, as 11 is not a Fermat prime, the regular hendecagon is not constructible with compass and straightedge. Because 11 is not a Pierpont prime, construction of a regular hendecagon is still impossible even with the usage of an angle trisector and it can, however, be constructed via neusis construction. Close approximations to the regular hendecagon can be constructed, however, for instance, the ancient Greek mathematicians approximated the side length of a hendecagon inscribed in a unit circle as being 14/25 units long. The following construction description is given by T, the regular hendecagon has Dih11 symmetry, order 22. Since 11 is a number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z11. These 4 symmetries can be seen in 4 distinct symmetries on the hendecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r22 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g11 subgroup has no degrees of freedom but can seen as directed edges. The Canadian dollar coin, the loonie, is similar to, but not exactly, the cross-section of a loonie is actually a Reuleaux hendecagon. Anthony dollar has a hendecagonal outline along the inside of its edges

37.
Dodecagon
–
In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a figure with sides of the same length. It has twelve lines of symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a hexagon, t, or a twice-truncated triangle. The internal angle at each vertex of a regular dodecagon is 150°, as 12 =22 ×3, regular dodecagon is constructible using compass and straightedge, Coxeter states that every parallel-sided 2m-gon can be divided into m/2 rhombs. For the dodecagon, m=6, and it can be divided into 15 rhombs and this decomposition is based on a Petrie polygon projection of a 6-cube, with 15 of 240 faces. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons, the regular dodecagon has Dih12 symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries, each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges, the interior of such an dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes, a regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a antiprism with the same D5d, symmetry. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons, the regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensionare the 24-cell, snub 24-cell, 6-6 duoprism, in 6 dimensions 6-cube, 6-orthoplex,221,122. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell, a dodecagram is a 12-sided star polygon, represented by symbol. There is one regular star polygon, using the same vertices, but connecting every fifth point. There are also three compounds, is reduced to 2 as two hexagons, and is reduced to 3 as three squares, is reduced to 4 as four triangles, and is reduced to 6 as six degenerate digons. Deeper truncations of the regular dodecagon and dodecagrams can produce intermediate star polygon forms with equal spaced vertices. A truncated hexagon is a dodecagon, t=, a quasitruncated hexagon, inverted as, is a dodecagram, t=

38.
Tridecagon
–
In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon. As 13 is a Pierpont prime but not a Fermat prime, however, it is constructible using neusis, or an angle trisector. An approximate construction of a regular tridecagon using straightedge and compass is shown here, another possible animation of an approximate construction, also possible with using straightedge and compass. For details, see, Wikibooks, Tridecagon, construction description The regular tridecagon has Dih13 symmetry, since 13 is a prime number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z13, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r26 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g13 subgroup has no degrees of freedom but can seen as directed edges. The regular tridecagon is used as the shape of the Czech 20 korun coin, a tridecagram is a 13-sided star polygon. There are 5 regular forms given by Schläfli symbols, and, the regular tridecagon is the Petrie polygon 12-simplex, Weisstein, Eric W. Tridecagon

39.
Tetradecagon
–
In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon. A regular tetradecagon has Schläfli symbol and can be constructed as a quasiregular truncated heptagon, t, however, it is constructible using neusis, as for example in the following illustration with use of the angle trisector. Based on the circle r =1 Constructed side length of the tetradecagon in GeoGebra a =0.445041867912629. Side length of the tetradecagon a s h o u l d =2 ⋅ sin =0.4450418679126288089, for details, see, Wikibooks, Tetradecagon, construction description The regular tetradecagon has Dih14 symmetry, order 28. There are 3 subgroup dihedral symmetries, Dih7, Dih2, and Dih1, and 4 cyclic group symmetries, Z14, Z7, Z2, and Z1. These 8 symmetries can be seen in 10 distinct symmetries on the tetradecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r28 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g14 subgroup has no degrees of freedom but can seen as directed edges. These two forms are duals of each other and have half the order of the regular tetradecagon. The regular tetradecagon is used as the shape of some commemorative gold and silver Malaysian coins, a tetradecagram is a 14-sided star polygon, represented by symbol. There are two regular polygons, and, using the same vertices, but connecting every third or fifth points. There are also three compounds, is reduced to 2 as two heptagons, while and are reduced to 2 and 2 as two different heptagrams, and finally is reduced to seven digons. Deeper truncations of the regular heptagon and heptagrams can produce isogonal intermediate tetradecagram forms with equally spaced vertices, other truncations can form double covering polyons 2, namely, t==2, t==2, and t==2. Regular skew tetradecagons exist as Petrie polygon for many higher-dimensional polytopes, shown in these orthogonal projections, including, Weisstein

40.
Pentadecagon
–
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. A regular pentadecagon is represented by Schläfli symbol, a regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by A =154 a 2 cot π15 =15 a 28 ≃17.6424 a 2. A regular triangle, decagon, and pentadecagon can completely fill a plane vertex. The point H divides the radius A M ¯ in golden ratio, compared with the first animation are in the following two images the two circular arcs rotated 90° counterclockwise shown. They do not use the segment C G ¯, but rather they use segment M G ¯ as radius A H ¯ for the circular arc. A compass and straightedge construction for a side length.618. Dih15 has 3 dihedral subgroups, Dih5, Dih3, and Dih1, and four more cyclic symmetries, Z15, Z5, Z3, and Z1, with Zn representing π/n radian rotational symmetry. On the pentadecagon, there are 8 distinct symmetries, john Conway labels these symmetries with a letter and order of the symmetry follows the letter. He gives r30 for the full symmetry, Dih15. These lower symmetries allows degrees of freedoms in defining irregular pentadecagons, only the g15 subgroup has no degrees of freedom but can seen as directed edges. There are three regular polygons, constructed from the same 15 vertices of a regular pentadecagon, but connected by skipping every second, fourth. There are also three regular star figures, the first being a compound of three pentagons, the second a compound of five triangles, and the third a compound of three pentagrams. Calculation of the circumradius Weisstein, Eric W. Pentadecagon

41.
Hexadecagon
–
In mathematics, a hexadecagon or 16-gon is a sixteen-sided polygon. A regular hexadecagon is a hexadecagon in which all angles are equal and its Schläfli symbol is and can be constructed as a truncated octagon, t, and a twice-truncated square tt. A truncated hexadecagon, t, is a triacontadigon, as 16 =24, a regular hexadecagon is constructible using compass and straightedge, this was already known to ancient Greek mathematicians. Each angle of a regular hexadecagon is 157.5 degrees, the area of a regular hexadecagon with edge length t is A =4 t 2 cot π16 =4 t 2. Since the area of the circumcircle is π R2, the regular hexadecagon fills approximately 97. 45% of its circumcircle, the regular hexadecagon has Dih16 symmetry, order 32. There are 4 dihedral subgroups, Dih8, Dih4, Dih2, and Dih1, and 5 cyclic subgroups, Z16, Z8, Z4, Z2, and Z1, on the regular hexadecagon, there are 14 distinct symmetries. John Conway labels full symmetry as r32 and no symmetry is labeled a1, the dihedral symmetries are divided depending on whether they pass through vertices or edges Cyclic symmetries in the middle column are labeled as g for their central gyration orders. These two forms are duals of each other and have half the order of the regular hexadecagon. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g16 subgroup has no degrees of freedom but can seen as directed edges. A skew hexadecagon is a polygon with 24 vertices and edges. The interior of such an hexadecagon is not generally defined, a skew zig-zag hexadecagon has vertices alternating between two parallel planes. A regular skew hexadecagon is vertex-transitive with equal edge lengths, in 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and side edges of a octagonal antiprism with the same D8d, symmetry, order 32. The octagrammic antiprism, s and octagrammic crossed-antiprism, s also have regular skew octagons, there are three regular star polygons, using the same vertices, but connecting every third, fifth or seventh points. There are also three compounds, is reduced to 2 as two octagons, is reduced to 4 as four squares and reduces to 2 as two octagrams, and finally is reduced to 8 as eight digons. Deeper truncations of the octagon and octagram can produce isogonal intermediate hexadecagram forms with equally spaced vertices. A truncated octagon is a hexadecagon, t=, a quasitruncated octagon, inverted as, is a hexadecagram, t=. A truncated octagram is a hexadecagram, t= and a quasitruncated octagram, inverted as, is a hexadecagram, hexadecagrams are included in the Girih patterns in the Alhambra. An octagonal star can be seen as a concave hexadecagon, Weisstein, Eric W. Hexadecagon

42.
Heptadecagon
–
In geometry, a heptadecagon or 17-gon is a seventeen-sided polygon. A regular heptadecagon is represented by the Schläfli symbol, as 17 is a Fermat prime, the regular heptadecagon is a constructible polygon, this was shown by Carl Friedrich Gauss in 1796 at the age of 19. This proof represented the first progress in regular polygon construction in over 2000 years, constructing a regular heptadecagon thus involves finding the cosine of 2 π /17 in terms of square roots, which involves an equation of degree 17—a Fermat prime. Gauss book Disquisitiones Arithmeticae gives this as,16 cos 2 π17 = −1 +17 +34 −217 +217 +317 −34 −217 −234 +217. The explicit construction of a heptadecagon was given by Herbert William Richmond in 1893, the following method of construction uses Carlyle circles, as shown below. Make OI one-fourth of OB, and the angle OIE one-fourth of OIA, another more recent construction is given by Callagy. The regular heptadecagon has Dih17 symmetry, order 34, since 17 is a prime number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z17, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r34 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g17 subgroup has no degrees of freedom but can seen as directed edges. A heptadecagram is a 17-sided star polygon, there are seven regular forms given by Schläfli symbols, and. The regular heptadecagon is the Petrie polygon for one higher-dimensional regular convex polytope, projected in an orthogonal projection. – Describes the algebraic aspect, by Gauss, contains a description of the construction. Heptadecagon trigonometric functions heptadecagon building New R&D center for SolarUK BBC video of New R&D center for SolarUK Eisenbud, Heptadecagon Heptadecagon, a construction with only one point N, a variation of the design according to H. W. Richmond

43.
Octadecagon
–
An octadecagon or 18-gon is an eighteen-sided polygon. A regular octadecagon has a Schläfli symbol and can be constructed as a truncated enneagon, t. As 18 =2 ×32, a regular octadecagon cannot be constructed using a compass, however, it is constructible using neusis, or an angle trisection with a tomahawk. The following approximate construction is similar to that of the enneagon. It is also feasible with exclusive use of compass and straightedge, the regular octadecagon has Dih18 symmetry, order 36. There are 5 subgroup dihedral symmetries, Dih9, and, and 6 cyclic group symmetries and these 15 symmetries can be seen in 12 distinct symmetries on the octadecagon. John Conway labels these by a letter and group order, full symmetry of the regular form is r36 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g18 subgroup has no degrees of freedom but can seen as directed edges. A regular triangle, nonagon, and octadecagon can completely surround a point in the plane, one of 17 different combinations of polygons with this property. The regular octadecagon can tessellate the plane with concave hexagonal gaps, and another tiling mixes in nonagons and octagonal gaps. The first tiling is related to a hexagonal tiling. An octadecagram is an 18-sided star polygon, represented by symbol, there are two regular star polygons, and, using the same points, but connecting every fifth or seventh points. Deeper truncations of the regular enneagon and enneagrams can produce isogonal intermediate octadecagram forms with equally spaced vertices, other truncations form double coverings, t==2, t==2, t==2. The regular octadecagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in these orthogonal projections from Coxeter planes, octadecagon Weisstein

44.
Enneadecagon
–
In geometry an enneadecagon or 19-gon is a nineteen-sided polygon. It is also known as an enneakaidecagon or a nonadecagon, a regular enneadecagon is represented by Schläfli symbol. The radius of the circumcircle of the regular enneadecagon with side length t is R = t 2 csc 18019, the area, where t is the edge length, is 194 t 2 cot π19 ≃28.4652 t 2. As 19 is a Pierpont prime but not a Fermat prime, however, it is constructible using neusis, or an angle trisector. Another animation of an approximate construction, based on the unit circle r =1 Constructed side length of the enneadecagon in GeoGebra a =0.329189180561468. Side length of the enneadecagon a s h o u l d =2 ⋅ sin =0.329189180561467788, the regular enneadecagon has Dih19 symmetry, order 38. Since 19 is a number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z19. These 4 symmetries can be seen in 4 distinct symmetries on the enneadecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r38 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g19 subgroup has no degrees of freedom but can seen as directed edges. A enneadecagram is a 19-sided star polygon, there are eight regular forms given by Schläfli symbols, and. The regular enneadecagon is the Petrie polygon for one higher-dimensional polytope, projected in an orthogonal projection, enneadecagon Weisstein

45.
Icosagon
–
In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagons interior angles is 3240 degrees, the regular icosagon has Schläfli symbol, and can also be constructed as a truncated decagon, t, or a twice-truncated pentagon, tt. One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°, the area of a regular icosagon with edge length t is A =5 t 2 ≃31.5687 t 2. In terms of the radius R of its circumcircle, the area is A =5 R22, since the area of the circle is π R2, the Big Wheel on the popular US game show The Price Is Right has an icosagonal cross-section. The Globe, the outdoor theater used by William Shakespeares acting company, was discovered to have built on an icosagonal foundation when a partial excavation was done in 1989. As a golygonal path, the swastika is considered to be an irregular icosagon, a regular square, pentagon, and icosagon can completely fill a plane vertex. E20 E1 ¯ E1 F ¯ = E20 F ¯ E20 E1 ¯ =1 +52 = φ ≈1.618 The regular icosagon has Dih20 symmetry, order 40. There are 5 subgroup dihedral symmetries, and, and 6 cyclic group symmetries and these 10 symmetries can be seen in 16 distinct symmetries on the icosagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order, full symmetry of the regular form is r40 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g20 subgroup has no degrees of freedom but can seen as directed edges. These two forms are duals of each other and have half the order of the regular icosagon. An icosagram is a 20-sided star polygon, represented by symbol, there are three regular forms given by Schläfli symbols, and. There are also five regular star figures using the vertex arrangement,2,4,5,2,4. Deeper truncations of the regular decagon and decagram can produce isogonal intermediate icosagram forms with equally spaced vertices, a regular icosagram, can be seen as a quasitruncated decagon, t=. Similarly a decagram, has a quasitruncation t=, and finally a simple truncation of a decagram gives t=

46.
Icositetragon
–
In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragons interior angles is 3960 degrees, the regular icositetragon is represented by Schläfli symbol and can also be constructed as a truncated dodecagon, t, or a twice-truncated hexagon, tt, or thrice-truncated triangle, ttt. One interior angle in a regular icositetragon is 165°, meaning that one exterior angle would be 15°, the area of a regular icositetragon is, A =6 t 2 cot π24 =6 t 2. The icositetragon appeared in Archimedes polygon approximation of pi, along with the hexagon, dodecagon, tetracontaoctagon, as 24 =23 ×3, a regular icositetragon is constructible using a compass and straightedge. As a truncated dodecagon, it can be constructed by an edge-bisection of a regular dodecagon, the regular icositetragon has Dih24 symmetry, order 48. There are 7 subgroup dihedral symmetries, and, and 8 cyclic group symmetries and these 16 symmetries can be seen in 22 distinct symmetries on the icositetragon. John Conway labels these by a letter and group order, the full symmetry of the regular form is r48 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g24 subgroup has no degrees of freedom but can seen as directed edges. A regular triangle, octagon, and icositetragon can completely fill a plane vertex, an icositetragram is a 24-sided star polygon. There are 3 regular forms given by Schläfli symbols, and, there are also 7 regular star figures using the same vertex arrangement,2,3,4,6,8,3, and 2. There are also isogonal icositetragrams constructed as deeper truncations of the regular dodecagon and these also generate two quasitruncations, t=, and t=. A skew icositetragon is a polygon with 24 vertices and edges. The interior of such an icositetragon is not generally defined, a skew zig-zag icositetragon has vertices alternating between two parallel planes. A regular skew icositetragon is vertex-transitive with equal edge lengths, in 3-dimensions it will be a zig-zag skew icositetragon and can be seen in the vertices and side edges of a dodecagonal antiprism with the same D12d, symmetry, order 48. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons, the regular icositetragon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes, including, Weisstein, Eric W. Icositetragon. Naming Polygons and Polyhedra polygon icosatetragon

47.
Triacontagon
–
In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagons interior angles is 5040 degrees, the regular triacontagon is a constructible polygon, by an edge-bisection of a regular pentadecagon, and can also be constructed as a truncated pentadecagon, t. A truncated triacontagon, t, is a hexacontagon, one interior angle in a regular triacontagon is 168°, meaning that one exterior angle would be 12°. The regular triacontagon has Dih30 dihedral symmetry, order 60, represented by 30 lines of reflection, Dih30 has 7 dihedral subgroups, Dih15, and. It also has eight more cyclic symmetries as subgroups, and, john Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d with mirror lines through vertices, p with mirror lines through edges and these lower symmetries allows degrees of freedoms in defining irregular triacontagons. Only the g30 subgroup has no degrees of freedom but can seen as directed edges, a triacontagram is a 30-sided star polygon. There are 3 regular forms given by Schläfli symbols, and, there are also isogonal triacontagrams constructed as deeper truncations of the regular pentadecagon and pentadecagram, and inverted pentadecagrams, and. Other truncations form double coverings, t==2, t==2, t==2, the regular triacontagon is the Petrie polygon for three 8-dimensional polytopes with E8 symmetry, shown in orthogonal projections in the E8 Coxeter plane. It is also the Petrie polygon for two 4-dimensional polytopes, shown in the H4 Coxeter plane, the regular triacontagram is also the Petrie polygon for the great grand stellated 120-cell and grand 600-cell