1.
Geometry
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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

2.
Polyhedron
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In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra

3.
Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, 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 F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces

4.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski

5.
Octagonal prism
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In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps. If faces are all regular, it is a semiregular polyhedron, the octagonal prism can also be seen as a tiling on a sphere, In optics, octagonal prisms are used to generate flicker-free images in movie projectors. It is an element of three uniform honeycombs, It is also an element of two four-dimensional uniform 4-polytopes, Weisstein, Eric W. Octagonal prism, interactive model of an Octagonal Prism

6.
Prism (geometry)
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In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases, prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids, a right prism is a prism in which the joining edges and faces are perpendicular to the base faces. This applies if the faces are rectangular. If the joining edges and faces are not perpendicular to the base faces, for example a parallelepiped is an oblique prism of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms. A truncated prism is a prism with nonparallel top and bottom faces, some texts may apply the term rectangular prism or square prism to both a right rectangular-sided prism and a right square-sided prism. A right p-gonal prism with rectangular sides has a Schläfli symbol ×, a right rectangular prism is also called a cuboid, or informally a rectangular box. A right square prism is simply a box, and may also be called a square cuboid. A right rectangular prism has Schläfli symbol ××, an n-prism, having regular polygon ends and rectangular sides, approaches a cylindrical solid as n approaches infinity. The term uniform prism or semiregular prism can be used for a prism with square sides. A uniform p-gonal prism has a Schläfli symbol t, right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being the antiprisms. The dual of a prism is a bipyramid. The volume of a prism is the product of the area of the base, the volume is therefore, V = B ⋅ h where B is the base area and h is the height. The volume of a prism whose base is a regular n-sided polygon with side s is therefore. The surface area of a prism is 2 · B + P · h, where B is the area of the base, h the height. The surface area of a prism whose base is a regular n-sided polygon with side length s and height h is therefore. The rotation group is Dn of order 2n, except in the case of a cube, which has the symmetry group O of order 24. The symmetry group Dnh contains inversion iff n is even, a prismatic polytope is a higher-dimensional generalization of a prism

7.
Square antiprism
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In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an anticube, if all its faces are regular, it is a semiregular polyhedron or uniform polyhedron. One molecule with this geometry is the ion in the salt nitrosonium octafluoroxenate, however. Very few ions are cubical because such a shape would cause large repulsion between ligands, PaF3−8 is one of the few examples, in addition, the element sulfur forms octatomic S8 molecules as its most stable allotrope. The main building block of the One World Trade Center has the shape of a tall tapering square antiprism. It is not a true antiprism because of its taper, the top square has half the area of the bottom one, a twisted prism can be made with the same vertex arrangement. It can be seen as the form with 4 tetrahedrons excavated around the sides. However, after this it can no longer be triangulated into tetrahedra without adding new vertices and it has half of the symmetry of the uniform solution, Dn, +, order 8. The gyroelongated square pyramid is a Johnson solid constructed by augmenting one a square pyramid, similarly, the gyroelongated square bipyramid is a deltahedron constructed by replacing both squares of a square antiprism with a square pyramid. The snub disphenoid is another deltahedron, constructed by replacing the two squares of a square antiprism by pairs of equilateral triangles, the snub square antiprism can be seen as a square antiprism with a chain of equilateral triangles inserted around the middle. The sphenocorona and the sphenomegacorona are other Johnson solids that, like the square antiprism, the square antiprism is first in a series of snub polyhedra and tilings with vertex figure 3.3.4.3. n. Compound of three square antiprisms Weisstein, Eric W. Antiprism, square Antiprism interactive model Virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, A4

8.
Antiprism
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In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles. Antiprisms are a subclass of the prismatoids and are a type of snub polyhedra, Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals. In the case of a regular n-sided base, one considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained when the line connecting the centers is perpendicular to the base planes. As faces, it has the two bases and, connecting those bases, 2n isosceles triangles. A uniform antiprism has, apart from the faces, 2n equilateral triangles as faces. As a class, the uniform antiprisms form a series of vertex-uniform polyhedra. For n =2 we have as degenerate case the regular tetrahedron as a digonal antiprism, the dual polyhedra of the antiprisms are the trapezohedra. Let a be the edge-length of a uniform antiprism, then the volume is V = n 4 cos 2 π2 n −1 sin 3 π2 n 12 sin 2 π n a 3 and the surface area is A = n 2 a 2. There are a set of truncated antiprisms, including a lower-symmetry form of the truncated octahedron. These can be alternated to create snub antiprisms, two of which are Johnson solids, and the snub triangular antiprism is a lower form of the icosahedron. The symmetry group contains inversion if and only if n is odd, uniform star antiprisms are named by their star polygon bases, and exist in prograde and retrograde solutions. Crossed forms have intersecting vertex figures, and are denoted by inverted fractions, p/ instead of p/q, in the retrograde forms but not in the prograde forms, the triangles joining the star bases intersect the axis of rotational symmetry. Some retrograde star antiprisms with regular star polygon bases cannot be constructed with equal edge lengths, star antiprism compounds also can be constructed where p and q have common factors, thus a 10/4 antiprism is the compound of two 5/2 star antiprisms. Prism Apeirogonal antiprism Grand antiprism – a four-dimensional polytope One World Trade Center, California, University of California Press Berkeley. Chapter 2, Archimedean polyhedra, prisma and antiprisms Weisstein, Eric W. Antiprism, archived from the original on 4 February 2007. Archived from the original on 4 February 2007, nonconvex Prisms and Antiprisms Paper models of prisms and antiprisms

9.
Square cupola
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In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids. It can be obtained as a slice of the rhombicuboctahedron, as in all cupolae, the base polygon has twice as many edges and vertices as the top, in this case the base polygon is an octagon. A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform and they were named by Norman Johnson, who first listed these polyhedra in 1966. The following formulae for volume, surface area, and circumradius can be used if all faces are regular and it can be obtained as a slice of the nonconvex great rhombicuboctahedron or quasirhombicuboctahedron, analogously to how the square cupola may be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the polygon has twice as many edges and vertices as the top. It may be seen as a cupola with a square base, so that the squares and triangles connect across the bases in the opposite way to the square cupola. Eric W. Weisstein, Square cupola at MathWorld

10.
Johnson solid
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In geometry, a Johnson solid is a strictly convex polyhedron, which is not uniform, and each face of which is a regular polygon. There is no requirement that each face must be the same polygon, an example of a Johnson solid is the square-based pyramid with equilateral sides, it has 1 square face and 4 triangular faces. As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees, since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid is an example that actually has a degree-5 vertex. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3,4,5,6,8. In 1966, Norman Johnson published a list which included all 92 solids and he did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnsons list was complete, however, it is not vertex-transitive, as it has different isometry at different vertices, making it a Johnson solid rather than an Archimedean solid. The naming of Johnson Solids follows a flexible & precise descriptive formula, from there, a series of prefixes are attached to the word to indicate additions, rotations and transformations, Bi- indicates that two copies of the solid in question are joined base-to-base. For cupolae and rotundae, the solids can be joined so that like either faces or unlike faces meet, using this nomenclature, an octahedron can be described as a square bipyramid, a cuboctahedron as a triangular gyrobicupola, and an icosidodecahedron as a pentagonal gyrobirotunda. Elongated indicates a prism is joined to the base of the solid in question, a rhombicuboctahedron can thus be described as an elongated square orthobicupola. Gyroelongated indicates an antiprism is joined to the base of the solid in question or between the bases in the case of Bi- solids, an icosahedron can thus be described as a gyroelongated pentagonal bipyramid. Augmented indicates a pyramid or cupola is joined to one or more faces of the solid in question, diminished indicates a pyramid or cupola is removed from one or more faces of the solid in question. Gyrate indicates a cupola mounted on or featured in the solid in question is rotated such that different edges match up, the last three operations — augmentation, diminution, and gyration — can be performed multiple times certain large solids. Bi- & Tri- indicate a double and treble operation respectively, for example, a bigyrate solid has two rotated cupolae, and a tridiminished solid has three removed pyramids or cupolae. In in certain solids, a distinction is made between solids where altered faces are parallel and solids where altered faces are oblique. Para- indicates the former, that the solid in question has altered parallel faces, for example, a parabiaugmented solid has had two parallel faces augmented, and a metabigyrate solid has had 2 oblique faces gyrated. The last few Johnson solids have names based on certain polygon complexes from which they are assembled and these names are defined by Johnson with the following nomenclature, A lune is a complex of two triangles attached to opposite sides of a square. Spheno- indicates a complex formed by two adjacent lunes