Mathematics

mathematicalmathmathematician
The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Convex and discrete geometry were developed to solve problems in number theory and functional analysis but now are pursued with an eye on applications in optimization and computer science. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus.

Three-dimensional space

three-dimensional3Dthree dimensions
In analogy with the conic sections, the set of points whose cartesian coordinates satisfy the general equation of the second degree, namely, :where A, B, C, F, G, H, J, K, L and M are real numbers and not all of A, B, C, F, G and H are zero is called a quadric surface. There are six types of non-degenerate quadric surfaces: The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane and all the lines of ℝ 3 through that conic that are normal to ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.

List of geometry topics

geometriesgeometry
Shear mapping. 2D computer graphics. 2D geometric model. Altitude. Brahmagupta's formula. Bretschneider's formula. Compass and straightedge constructions. Squaring the circle. Complex geometry. Conic section. Focus. Circle. List of circle topics. Thales' theorem. Circumcircle. Concyclic. Incircle and excircles of a triangle. Orthocentric system. Monge's theorem. Power center. Nine-point circle. Circle points segments proof. Mrs. Miniver's problem. Isoperimetric theorem. Annulus. Ptolemaios' theorem. Steiner chain. Eccentricity. Ellipse. Semi-major axis. Hyperbola. Parabola. Matrix representation of conic sections. Dandelin spheres. Curve of constant width. Reuleaux triangle. Frieze group.

Circle

circularcircles360 degrees
The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T. All circles are similar. A circle's circumference and radius are proportional. The area enclosed and the square of its radius are proportional. The constants of proportionality are 2 and, respectively. The circle that is centred at the origin with radius 1 is called the unit circle. Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.

Analytic geometry

analytical geometryCartesian geometrycoordinate geometry
John Casey (1885) Analytic Geometry of the Point, Line, Circle, and Conic Sections, link from Internet Archive. John Casey (1885) Analytic Geometry of the Point, Line, Circle, and Conic Sections, link from Internet Archive. John Casey (1885) Analytic Geometry of the Point, Line, Circle, and Conic Sections, link from Internet Archive.

History of mathematics

historian of mathematicsmathematicshistory
He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere. Apollonius of Perga (c. 262–190 BC) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone. He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").

Archimedes

Archimedes of SyracuseArchimedeanArchimedes Heat Ray
In this treatise Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids. On Floating Bodies (two volumes). In the first part of this treatise, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not self-gravitating, since he assumes the existence of a point towards which all things fall in order to derive the spherical shape. The Quadrature of the Parabola.

Shape

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Other common shapes are points, lines, planes, and conic sections such as ellipses, circles, and parabolas. Among the most common 3-dimensional shapes are polyhedra, which are shapes with flat faces; ellipsoids, which are egg-shaped or sphere-shaped objects; cylinders; and cones. If an object falls into one of these categories exactly or even approximately, we can use it to describe the shape of the object. Thus, we say that the shape of a manhole cover is a disk, because it is approximately the same geometric object as an actual geometric disk.

Cone

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In the Cartesian coordinate system, an elliptic cone is the locus of an equation of the form It is an affine image of the right-circular unit cone with equation From the fact, that the affine image of a conic section is a conic section of the same type (ellipse, parabola,...) one gets: *Any plane section of an elliptic cone is a conic section. Obviously, any right circular cone contains circles. This is also true, but less obvious, in the general case (see circular section). In projective geometry, a cylinder is simply a cone whose apex is at infinity.

Plane (geometry)

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In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have. The plane may be given a spherical geometry by using the stereographic projection. This can be thought of as placing a sphere on the plane (just like a ball on the floor), removing the top point, and projecting the sphere onto the plane from this point). This is one of the projections that may be used in making a flat map of part of the Earth's surface. The resulting geometry has constant positive curvature. Alternatively, the plane can also be given a metric which gives it constant negative curvature giving the hyperbolic plane.

Cylinder

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In the treatise by this name, written c. 225 BCE, Archimedes obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a sphere by exploiting the relationship between a sphere and its circumscribed right circular cylinder of the same height and diameter. The sphere has a volume two-thirds that of the circumscribed cylinder and a surface area two-thirds that of the cylinder (including the bases). Since the values for the cylinder were already known, he obtained, for the first time, the corresponding values for the sphere. The volume of a sphere of radius r is 4⁄3r 3 = 2⁄3 (2r 3 ) . The surface area of this sphere is 4r 2 = 2⁄3 (6r 2 ) .

M. C. Escher

M.C. EscherEscherMaurits Cornelis Escher
These are the classification of regular tilings using the edge relationships of tiles: two-color and two-motif tilings (counterchange symmetry or antisymmetry); color symmetry (in crystallography); metamorphosis or topological change; covering surfaces with symmetric patterns; Escher's algorithm (for generating patterns using decorated squares); creating tile shapes; local versus global definitions of regularity; symmetry of a tiling induced by the symmetry of a tile; orderliness not induced by symmetry groups; the filling of the central void in Escher's lithograph Print Gallery by H. Lenstra and B. de Smit.

Duality (projective geometry)

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In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways.

Sphere packing

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In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the density of the arrangement.

Hyperbolic geometry

hyperbolic planehyperbolichyperbolic surface
The idea used a conic section or quadric to define a region, and used cross ratio to define a metric. The projective transformations that leave the conic section or quadric stable are the isometries. "Klein showed that if the Cayley absolute is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..." For more history, see article on non-Euclidean geometry, and the references Coxeter and Milnor. The discovery of hyperbolic geometry had important philosophical consequences.

Tessellation

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A Schwarz triangle is a spherical triangle that can be used to tile a sphere. It is possible to tessellate in non-Euclidean geometries such as hyperbolic geometry. A uniform tiling in the hyperbolic plane (which may be regular, quasiregular or semiregular) is an edge-to-edge filling of the hyperbolic plane, with regular polygons as faces; these are vertex-transitive (transitive on its vertices), and isogonal (there is an isometry mapping any vertex onto any other). A uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells.

Euclidean geometry

plane geometryEuclideanEuclidean plane geometry
Apollonius of Perga (c. 262 BCE – c. 190 BCE) is mainly known for his investigation of conic sections. René Descartes (1596–1650) developed analytic geometry, an alternative method for formalizing geometry which focused on turning geometry into algebra. In this approach, a point on a plane is represented by its Cartesian (x, y) coordinates, a line is represented by its equation, and so on. In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.

Metric space

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Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities. In 1906 Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel. However the name is due to Felix Hausdorff. A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a function such that for any, the following holds: |} Given the above three axioms, we also have that for any x, y \in M.

Line (geometry)

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In many models of projective geometry, the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry. In elliptic geometry we see a typical example of this. In the spherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean planes passing through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two points determining a unique line) that make them suitable representations for lines in this geometry.

Gaussian curvature

Gauss curvaturecurvatureLiebmann's theorem
When a surface has a constant zero Gaussian curvature, then it is a developable surface and the geometry of the surface is Euclidean geometry. When a surface has a constant positive Gaussian curvature, then it is a sphere and the geometry of the surface is spherical geometry. When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface and the geometry of the surface is hyperbolic geometry. The two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point.

Synthetic geometry

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With this plethora of possibilities, it is no longer appropriate to speak of "geometry" in the singular. Historically, Euclid's parallel postulate has turned out to be independent of the other axioms. Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry. Other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical or affine geometry. Axioms of continuity and "betweeness" are also optional, for example, discrete geometries may be created by discarding or modifying them.

Two-dimensional space

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There are an infinitude of other curved shapes in two dimensions, notably including the conic sections: the ellipse, the parabola, and the hyperbola. Another mathematical way of viewing two-dimensional space is found in linear algebra, where the idea of independence is crucial. The plane has two dimensions because the length of a rectangle is independent of its width. In the technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described by a linear combination of two independent vectors. The dot product of two vectors A = [A 1, A 2 ] and B = [B 1, B 2 ] is defined as: A vector can be pictured as an arrow.

Real projective plane

projective planeFlat projective planeprojective manifolds
This quotient space of the sphere is homeomorphic with the collection of all lines passing through the origin in R 3. The quotient map from the sphere onto the real projective plane is in fact a two sheeted (i.e. two-to-one) covering map. It follows that the fundamental group of the real projective plane is the cyclic group of order 2; i.e., integers modulo 2. One can take the loop AB from the figure above to be the generator. Because the sphere covers the real projective plane twice, the plane may be represented as a closed hemisphere around whose rim opposite points are similarly identified.

Semi-major and semi-minor axes

semi-major axissemimajor axissemi-major axes
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis (more properly, major semi-axis) is one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (more properly, minor semi-axis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.