153 relations: Affine sphere, Albert Einstein, Alexander horned sphere, Analytic geometry, Angle, Antipodal point, Apollonius of Perga, Arc length, Archimedes, Astronomical object, Australia, Axiom, Ball (mathematics), Boundary (topology), Cartesian coordinate system, Cavalieri's principle, Celestial spheres, Centre (geometry), Channel surface, Circle, Circle of a sphere, Circumference, Circumscribed circle, Collinearity, Compact space, Cone, Coplanarity, Cross section (geometry), Cube, Curvature, Cylinder, David Hilbert, Density, Derivative, Diameter, Diffeomorphism, Differential form, Dihedral angle, Dimension, Directional statistics, Disc integration, Discrete space, Disk (mathematics), Dupin cyclide, Dyson sphere, Earth, Ellipse, Embedding, Equator, Euclidean distance, ..., Euclidean geometry, Euclidean space, Euler angles, Filling area conjecture, Focal surface, Fused quartz, Gamma function, Gaussian curvature, Geodesic, Geoid, Geometry, Gravity Probe B, Great circle, Great-circle distance, Greek language, Gyroscope, Hand with Reflecting Sphere, Heine–Borel theorem, Helicoid, Hoberman sphere, Homeomorphism, Homology sphere, Homotopy groups of spheres, Homotopy sphere, Hypersphere, Inscribed figure, Integer, Integral, Isoperimetric inequality, John Wiley & Sons, Kilogram, Knot (mathematics), Latitude, Lénárt sphere, Line (geometry), Locus (mathematics), Longitude, M. C. Escher, Mathematics, Mean curvature, Meridian (geography), Metric space, Minimal surface, Nanometre, Napkin ring problem, Natural number, Norm (mathematics), Normal (geometry), Orb (optics), Orthogonality, Parallel postulate, Plane (geometry), Point (geometry), Point at infinity, Principal curvature, Pseudosphere, Pythagorean theorem, Radius, Real projective plane, Reuleaux tetrahedron, Riemann sphere, Riemannian circle, Rotation around a fixed axis, Rotation group SO(3), Semi-major and semi-minor axes, Similarity (geometry), Soap bubble, Solid angle, Solid geometry, Specific surface area, Sphere, Sphere eversion, Sphere packing, Spherical cap, Spherical coordinate system, Spherical Earth, Spherical lune, Spherical sector, Spherical segment, Spherical shell, Spherical trigonometry, Spherical wedge, Spheroid, Stephan Cohn-Vossen, Surface (topology), Surface area, Surface of revolution, Surface tension, Tangent indicatrix, Three-dimensional space, Topological manifold, Topology, Torus, Transfinite number, Trigonometry, Umbilical point, Unit sphere, Up to, Volume, Volume element, Volume of an n-ball, Zoll surface, 3-sphere. Expand index (103 more) » « Shrink index
In mathematics, and especially differential geometry, an affine sphere is a hypersurface for which the affine normals all intersect in a single point.
Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).
The Alexander horned sphere is a pathological object in topology discovered by.
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it — so situated that a line drawn from the one to the other passes through the center of the sphere and forms a true diameter.
Apollonius of Perga (Ἀπολλώνιος ὁ Περγαῖος; Apollonius Pergaeus; late 3rdearly 2nd centuries BC) was a Greek geometer and astronomer known for his theories on the topic of conic sections.
Determining the length of an irregular arc segment is also called rectification of a curve.
Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.
An astronomical object or celestial object is a naturally occurring physical entity, association, or structure that exists in the observable universe.
Australia, officially the Commonwealth of Australia, is a sovereign country comprising the mainland of the Australian continent, the island of Tasmania and numerous smaller islands.
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
In mathematics, a ball is the space bounded by a sphere.
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows.
The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus, and others.
In geometry, a centre (or center) (from Greek κέντρον) of an object is a point in some sense in the middle of the object.
A channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its directrix.
A circle is a simple closed shape.
A circle of a sphere is a circle that lies on a sphere.
In geometry, the circumference (from Latin circumferentia, meaning "carrying around") of a circle is the (linear) distance around it.
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon.
In geometry, collinearity of a set of points is the property of their lying on a single line.
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all.
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces.
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.
A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.
David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.
The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume.
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle.
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.
A dihedral angle is the angle between two intersecting planes.
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Rn), axes (lines through the origin in Rn) or rotations in Rn.
Disc integration, also known in integral calculus as the disc method, is a means of calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution.
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense.
In geometry, a disk (also spelled disc).
In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone.
A Dyson sphere is a hypothetical megastructure that completely encompasses a star and captures a large percentage of its power output.
Earth is the third planet from the Sun and the only astronomical object known to harbor life.
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
An equator of a rotating spheroid (such as a planet) is its zeroth circle of latitude (parallel).
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.
In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the surfaces that fill a closed curve of given length without introducing shortcuts between its points.
For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at the point of tangency.
Fused quartz or fused silica is glass consisting of silica in amorphous (non-crystalline) form.
In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point: For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature 0 everywhere.
In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".
The geoid is the shape that the surface of the oceans would take under the influence of Earth's gravity and rotation alone, in the absence of other influences such as winds and tides.
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
Gravity Probe B (GP-B) was a satellite-based mission which launched on 20 April 2004 on a Delta II rocket.
A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere.
The great-circle distance or orthodromic distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior).
Greek (Modern Greek: ελληνικά, elliniká, "Greek", ελληνική γλώσσα, ellinikí glóssa, "Greek language") is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean and the Black Sea.
A gyroscope (from Ancient Greek γῦρος gûros, "circle" and σκοπέω skopéō, "to look") is a device used for measuring or maintaining orientation and angular velocity.
Hand with Reflecting Sphere also known as Self-Portrait in Spherical Mirror is a lithograph print by Dutch artist M. C. Escher, first printed in January 1935.
In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space Rn, the following two statements are equivalent.
The helicoid, after the plane and the catenoid, is the third minimal surface to be known.
A Hoberman sphere is an isokinetic structure patented by Chuck Hoberman that resembles a geodesic dome, but is capable of folding down to a fraction of its normal size by the scissor-like action of its joints.
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1.
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other.
In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere.
In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its center.
An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid.
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.
In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume.
John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.
The kilogram or kilogramme (symbol: kg) is the base unit of mass in the International System of Units (SI), and is defined as being equal to the mass of the International Prototype of the Kilogram (IPK, also known as "Le Grand K" or "Big K"), a cylinder of platinum-iridium alloy stored by the International Bureau of Weights and Measures at Saint-Cloud, France.
In mathematics, a knot is an embedding of a circle S^1 in 3-dimensional Euclidean space, R3 (also known as E3), considered up to continuous deformations (isotopies).
In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface.
A Lénárt sphere is a teaching and educational research model for spherical geometry.
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.
In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.
Longitude, is a geographic coordinate that specifies the east-west position of a point on the Earth's surface.
Maurits Cornelis Escher (17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically-inspired woodcuts, lithographs, and mezzotints.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
A (geographical) meridian (or line of longitude) is the half of an imaginary great circle on the Earth's surface, terminated by the North Pole and the South Pole, connecting points of equal longitude.
In mathematics, a metric space is a set for which distances between all members of the set are defined.
In mathematics, a minimal surface is a surface that locally minimizes its area.
The nanometre (International spelling as used by the International Bureau of Weights and Measures; SI symbol: nm) or nanometer (American spelling) is a unit of length in the metric system, equal to one billionth (short scale) of a metre (m).
In geometry, the napkin-ring problem involves finding the volume of a "band" of specified height around a sphere, i.e. the part that remains after a hole in the shape of a circular cylinder is drilled through the center of the sphere.
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.
In photography, an orb is a typically circular artifact on an image, created as a result of flash photography illuminating a mote of dust or other particle.
In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry.
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.
In modern mathematics, a point refers usually to an element of some set called a space.
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point.
In geometry, the term pseudosphere is used to describe various surfaces with constant negative Gaussian curvature.
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.
In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length.
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface.
The Reuleaux tetrahedron is the intersection of four balls of radius s centered at the vertices of a regular tetrahedron with side length s. The spherical surface of the ball centered on each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron.
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.
In metric space theory and Riemannian geometry, the Riemannian circle is a great circle equipped with its great-circle distance.
Rotation around a fixed axis or about a fixed axis of revolution or motion with respect to a fixed axis of rotation is a special case of rotational motion.
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.
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.
Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other.
A soap bubble is an extremely thin film of soapy water enclosing air that forms a hollow sphere with an iridescent surface.
In geometry, a solid angle (symbol) is a measure of the amount of the field of view from some particular point that a given object covers.
In mathematics, solid geometry is the traditional name for the geometry of three-dimensional Euclidean space.
Specific surface area (SSA) is a property of solids defined as the total surface area of a material per unit of mass, (with units of m2/kg or m2/g) or solid or bulk volume (units of m2/m3 or m−1).
A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space.
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space.
In geometry, a spherical cap, spherical dome, or spherical segment of one base is a portion of a sphere cut off by a plane.
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
The earliest reliably documented mention of the spherical Earth concept dates from around the 6th century BC when it appeared in ancient Greek philosophy but remained a matter of speculation until the 3rd century BC, when Hellenistic astronomy established the spherical shape of the Earth as a physical given.
In spherical geometry, a spherical lune is an area on a sphere bounded by two half great circles which meet at antipodal points, and is an example of a digon, θ, with dihedral angle θ. The word "lune" derives from luna, the Latin word for Moon.
In geometry, a spherical sector is a portion of a sphere defined by a conical boundary with apex at the center of the sphere.
In geometry, a spherical segment is the solid defined by cutting a sphere with a pair of parallel planes.
In geometry, a spherical shell is a generalization of an annulus to three dimensions.
Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by a number of intersecting great circles on the sphere.
In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune (termed the wedge's base).
A spheroid, or ellipsoid of revolution, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters.
Stefan or Stephan Cohn-Vossen (28 May 1902 – 25 June 1936) was a mathematician, who was responsible for Cohn-Vossen's inequality and the Cohn-Vossen transformation is also named for him.
In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.
The surface area of a solid object is a measure of the total area that the surface of the object occupies.
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Surface tension is the elastic tendency of a fluid surface which makes it acquire the least surface area possible.
In differential geometry, the tangent indicatrix of a closed space curve is a curve on the unit sphere intimately related to the curvature of the original curve.
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).
In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below.
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.
Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles.
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical.
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point.
In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates.
In geometry, a ball is a region in space comprising all points within a fixed distance from a given point; that is, it is the region enclosed by a sphere or hypersphere.
In mathematics, a Zoll surface, named after Otto Zoll, is a surface homeomorphic to the 2-sphere, equipped with a Riemannian metric all of whose geodesics are closed and of equal length.
In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere.
2-sphere, Globose, Géosphère, Hemispherical, Maschler space, Orb (shape), Sphere (geometry), Spheres, Spheric section, Spherical, Spherical product, Spherical surface, Spherical volume, Surface area of a sphere, Surface area of the sphere, S², Topological sphere, Two-dimensional sphere, Volume Of A Sphere, Volume of a sphere, Volume of sphere, X^2+y^2+z^2=r^2, ⁴⁄₃πr³.