Searches , social queries for 3 SPHERE
Search references for 3 SPHERE. Phrases containing 3 SPHERE
See searches and references containing 3 SPHERE!
Search references for 3 SPHERE. Phrases containing 3 SPHERE
See searches and references containing 3 SPHERE!3 SPHERE
Mathematical object
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space
3-sphere
Set of points equidistant from a center
A sphere (from Ancient Greek σφαῖρα (sphaîra) 'ball') is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that
Sphere
Representation of a quantum mechanical system
In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit)
Bloch_sphere
Generalized sphere of dimension n (mathematics)
space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general n {\displaystyle n} -sphere is embedded in an n +
N-sphere
Pathological embedding of the sphere in 3D space
The Alexander horned sphere is a pathological embedding of the 2-sphere into 3-dimensional Euclidean space. The topological object was discovered by J
Alexander_horned_sphere
Topological manifold whose homology coincides with that of a sphere
homology sphere, first constructed by Henri Poincaré. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphere itself) with
Homology_sphere
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in
Hopf_fibration
Group of rotations in 3 dimensions
symmetric. S O ( 3 ) {\displaystyle SO(3)} is doubly covered by the group of unit quaternions, which is isomorphic to the 3-sphere. Since the Haar measure
3D_rotation_group
Group of unitary complex matrices with determinant of 1
norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (uniquely up to sign)
Special_unitary_group
Model of the extended complex plane plus a point at infinity
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the
Riemann_sphere
Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from
Exotic_sphere
Mathematical space
shape of the universe. Just as a sphere looks like a plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe
3-manifold
oriented integral homology 3-spheres, introduced by Andrew Casson. Kevin Walker (1992) found an extension to rational homology 3-spheres, called the Casson–Walker
Casson_invariant
Theorem in geometric topology
/ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere (the hypersphere that bounds the 4-ball in four-dimensional space). Originally
Poincaré_conjecture
represents a reflection hyperplane on 3-dimensional surfaces: the 3-sphere, Euclidean 3-space, and hyperbolic 3-space. Coxeter named them after Édouard
Goursat_tetrahedron
How spheres of various dimensions can wrap around each other
mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples
Homotopy_groups_of_spheres
Four-dimensional analogue of the cube
{4}}=2} edge lengths. An axis-aligned tesseract inscribed in a unit-radius 3-sphere has vertices with coordinates ( ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 ) . {\displaystyle
Tesseract
Geometrical object in four-dimensional space
when a=b it is a square torus. If a2+b2=1, then Ta,b lies in the unit 3-sphere S3 ⊂ R4. The case a=b=1/√2 is a minimal surface in S3 and is often called
Clifford_torus
Entertainment venue in the Las Vegas Valley, United States
Sphere (also known as Sphere at the Venetian Resort or Las Vegas Sphere) is a music and entertainment arena in Paradise, Nevada, United States, east of
Sphere_(venue)
Sculpture in New York City
The Sphere (officially Große Kugelkaryatide N.Y., also known as Sphere at Plaza Fountain, WTC Sphere or Koenig Sphere) is a monumental cast bronze sculpture
The_Sphere
Doughnut-shaped surface of revolution
parameterize the unit 3-sphere as Hopf coordinates. In particular, for certain very specific choices of a square flat torus in the 3-sphere S3, where η = π/4
Torus
Local and global geometry of the universe
locally modeled by a region of a 3-sphere S3. Negative curvature – a drawn triangle's angles add up to less than 180°; such 3-dimensional space is locally
Shape_of_the_universe
Embedding of the circle in three dimensional Euclidean space
(S1) into three-dimensional Euclidean space (R3), or the 3-sphere (S3), since the 3-sphere is compact. Two knots are defined to be equivalent if there
Knot_(mathematics)
Manifold with the same rational homology groups as a sphere
{\displaystyle n} -sphere is an n {\displaystyle n} -dimensional manifold with the same rational homology groups as the n {\displaystyle n} -sphere. These serve
Rational_homology_sphere
Subclass of manifold
the 3-sphere S 3 {\displaystyle S^{3}} . All such manifolds are prime, orientable, and closed. Spherical 3-manifolds are sometimes called elliptic 3-manifolds
Spherical_3-manifold
Function used in computer graphics
three-dimensional rotations, represented as quaternions on an abstract 3-sphere. When the interpolation parameter represents time, spherical linear interpolation
Spherical linear interpolation
Spherical_linear_interpolation
particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1993). It is based on dividing the sphere into two solid tori
Reeb_foliation
Geometric model of the physical space
quadrilateral in R 3 {\displaystyle \mathbb {R} ^{3}} form a parallelogram, and hence are coplanar. A sphere in 3-space (also called a 2-sphere because, like
Three-dimensional_space
Four-dimensional number system
give a group structure on the 3-sphere S3 isomorphic to the groups Spin(3) and SU(2), i.e. the universal cover group of SO(3). The positive and negative
Quaternion
Riemannian geometry, the Berger spheres form a special class of examples of Riemannian manifolds diffeomorphic to the 3-sphere. They are named for Marcel Berger
Berger's_sphere
Describes how distinct surgery presentations of a given 3-manifold are related
links in the 3-sphere using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory, he proved that if M and N are 3-manifolds, resulting
Kirby_calculus
Sculpture by Arnaldo Pomodoro, of which several versions exist
sculptor Arnaldo Pomodoro. In 1966, Pomodoro was commissioned to create a 3.5-meter sphere for Expo 67 in Montreal. The success of this sculpture propelled Pomodoro's
Sphere_Within_Sphere
Class of topological space
solid tori together by a homeomorphism of their boundaries. Often the 3-sphere and S 2 × S 1 {\displaystyle S^{2}\times S^{1}} , both of which can be
Lens_space
Complement of a knot in three-sphere
where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise
Knot_complement
Projection of a sphere through its center onto a plane
rectilinear projection, is a perspective projection of a sphere, with center of projection at the sphere's center, onto any plane not passing through the center
Gnomonic_projection
Regular tessellation in 4D Euclidean space
diameter of the spheres (the distance between the centers of kissing spheres) is √2. Just outside this surrounding shell of 24 kissing 3-spheres is another
24-cell_honeycomb
Arrangement of spheres within a space
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical
Sphere_packing
Existence of geodesic circles on surfaces
theorem, states that every Riemannian manifold with the topology of a sphere has at least three simple closed geodesics (i.e. three embedded geodesic
Theorem of the three geodesics
Theorem_of_the_three_geodesics
Linear stacking of regular tetrahedra that form helices
helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3-sphere surface of the 600-cell, one of the six regular convex polychora. Buckminster
Boerdijk–Coxeter_helix
Proposition in mathematics that is unproven
3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states that: Every simply connected, closed 3-manifold
Conjecture
theorem) states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere. The conjecture was posed by Heinz Hopf in 1926
Spherical space form conjecture
Spherical_space_form_conjecture
Three-dimensional packing problem
Sphere packing in a sphere is a three-dimensional packing problem with the objective of packing a given number of equal spheres inside a unit sphere. It
Sphere_packing_in_a_sphere
Model of objects in the sky consisting of a framework of rings
armillary sphere (variations are known as spherical astrolabe, armilla, or armil) is a model of objects in the sky (on the celestial sphere), consisting
Armillary_sphere
American holding company in New York City
Sphere Entertainment Co. is an American entertainment holding company based in New York City, and controlled by the family of Charles Dolan. It owns the
Sphere_Entertainment
Quaternion of norm 1 (unit quaternion)
versors, with quaternion multiplication, forms a group, and appears as a 3-sphere in the 4-dimensional quaternion algebra. Sir William Rowan Hamilton wrote
Versor
Hypothetical megastructure around a star
A Dyson sphere is a hypothetical megastructure that encompasses a star and captures a large percentage of its power output. The concept is a thought experiment
Dyson_sphere
Theorem that the diffeomorphism group of the 3-sphere has the homotopy-type of O(4)
after Stephen Smale, is the statement that the diffeomorphism group of the 3-sphere has the homotopy-type of its isometry group, the orthogonal group O(4)
Smale_conjecture
Special orthogonal group
3D-space. While they are still 2D surfaces, they are embedded in the 3-sphere. The 3-sphere can be stereographically projected onto the whole Euclidean 3D-space
Rotations in 4-dimensional Euclidean space
Rotations_in_4-dimensional_Euclidean_space
Area on a sphere bounded by two semicircles joined at antipodal points
In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points. It is an
Spherical_lune
2022 video game
20 April 2023. "New Victoria 3 DLC is coming, and some players get it free". PCGamesN. 25 October 2023. "Victoria 3: Sphere of Influence Review - A Diplomatic
Victoria_3
Operation used to modify three-dimensional topological spaces
{\displaystyle \pm 1/r} , then the surgered 3-manifold is still the 3-sphere. If M {\displaystyle M} is the 3-sphere, L {\displaystyle L} is the right-handed
Dehn_surgery
Mathematical descriptions of a rotation group
thus cannot be represented as a point on the sphere. This will not be the case with a general rotation in 3-space, which do form a closed set under composition
Charts_on_SO(3)
Concept in euclidean geometry
vertex-centered 3-spheres and cell-inscribed 3-spheres will both fit at once, forming the unique regular body-centered cubic lattice of equal-sized spheres (in any
Tesseractic_honeycomb
2007 video game
(Australia and Europe) in 2008. A remake, titled Odin Sphere Leifthrasir, was released on PlayStation 3, PlayStation 4, and PlayStation Vita in 2016: Atlus
Odin_Sphere
Volume space bounded by a sphere
bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball
Ball_(mathematics)
Geometric space with four dimensions
by a sphere in three dimensions ( V = 4 3 π r 3 {\textstyle V={\frac {4}{3}}\pi r^{3}} ). One might guess that the volume enclosed by the sphere in four-dimensional
Four-dimensional_space
"spherium" model consists of two electrons trapped on the surface of a sphere of radius R {\displaystyle R} . It has been used by Berry and collaborators
Spherium
Loop seen as a trivial knot
To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic (that is, deformable) to a geometrically round
Unknot
Sphere touching all of a polyhedron's vertices
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere
Circumscribed_sphere
Geometry founded on spheres
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by
Lie_sphere_geometry
Subfield of mathematical topology
problems. Rubinstein and Thompson's 3-sphere recognition algorithm. This is an algorithm that takes as input a triangulated 3-manifold and determines whether
Computational_topology
Type of knot in knot theory
Horst Schubert, using the fact that the 2-sheeted branched cover of the 3-sphere over the knot is a lens space. The names rational knot and rational link
2-bridge_knot
Seven mathematical problems with a US$1 million prize for each solution
manifold which is closed and simply-connected must be homeomorphic to the 3-sphere. Although the conjecture is usually stated in this form, it is equivalent
Millennium_Prize_Problems
Topics referred to by the same term
Steiner The three spheres of right, a concept in Hegelian philosophy a 3-sphere, an n-sphere whose surface is three-dimensional three spheres inequality, a
Three_spheres
Asymmetry of classical and quantum action
we may consider the 3-sphere at infinity to be a single point, as the gauge transformations vanish there anyway. If the 3-sphere at infinity is identified
Anomaly_(physics)
Mathematical game
mind that the balloon is S 2 {\displaystyle S^{2}} , the 2-sphere; it's not the 3-sphere of rotations.) To further simplify, one can start with S 1 {\displaystyle
Tangloids
Non-orientable surface with one edge
constructed by sweeping a great circle through a great-circular motion in the 3-sphere, and the Sudanese Möbius strip is obtained by sweeping a semicircle instead
Möbius_strip
Simplest non-trivial closed knot with three crossings
the trefoil can also be obtained as the intersection in C2 of the unit 3-sphere S3 with the complex plane curve of zeroes of the complex polynomial z2 + w3
Trefoil_knot
Sphere with radius one, usually centered on the origin of the space
In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space.
Unit_sphere
Topological operation of turning a sphere inside-out without creasing
In differential topology, sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (the word eversion means
Sphere_eversion
Knot that can't be tied in a string of constant diameter
D^{2}} into the 3-sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain. In knot theory and 3-manifold theory
Wild_knot
Concept in differential topology
Brieskorn homology spheres Σ ( 2 , 5 , 7 ) {\displaystyle \Sigma (2,5,7)} , Σ ( 3 , 4 , 5 ) {\displaystyle \Sigma (3,4,5)} , and Σ ( 2 , 3 , 13 ) {\displaystyle
Mazur_manifold
Theorem in topology
mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot
Smith_conjecture
Type of n-manifold in topology
sphere in R 3 {\displaystyle \mathbb {R} ^{3}} that does not bound a ball. Thus the stipulation that the sphere be smooth is necessary. The 3-sphere S
Prime_manifold
Russian mathematician (born 1966)
throughout the 20th century regarded as a key problem in topology. On the 3-sphere, defined as the set of points at unit length from the origin in four-dimensional
Grigori_Perelman
Chinese camera company
consumer cameras, including X2, X3, X4, ONE X/X2, ONE RS/R, ONE, GO 2/3, Sphere, EVO and Nano S. A new interface was introduced in September 2021, for
Insta360
Orientable surface whose boundary is a knot or link
or link in Euclidean 3-space (or in the 3-sphere). A Seifert surface is a compact, connected, oriented surface S embedded in 3-space whose boundary is
Seifert_surface
On when elements of the 2nd homotopy group of a 3-manifold can be embedded spheres
topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos (1957) gives conditions for elements of the second homotopy group of a 3-manifold
Sphere_theorem_(3-manifolds)
How many times curves wind around each other
removed (this can be seen by interpreting 3-space as the 3-sphere with the point at infinity removed, and the 3-sphere as two solid tori glued along the boundary)
Linking_number
Branch of mathematics
4-sphere, is also diffeomorphic to it. That is, does the 4-sphere admit only one smooth structure? This conjecture is true in dimensions 1, 2, and 3, by
Differential_topology
Four-dimensional analog of the dodecahedron
cell (hull 1). A section is a flat 3-dimensional hyperplane slice through the 3-sphere: a 2-sphere (ordinary sphere). It is dimensionally analogous to
120-cell
On the intersection form of a smooth, closed 4-manifold with a spin structure
the choice of M. Homology 3-spheres have a unique spin structure so we can define the Rokhlin invariant of a homology 3-sphere to be the element sign
Rokhlin's_theorem
4-dimensional figure
of the sphere and the apex). The surface volume of the spherical base is the same as for any sphere, 4 3 π r 3 {\textstyle {\frac {4}{3}}\pi r^{3}} . Therefore
Hypercone
curvature is covered by the 3-sphere, moreover the group of covering transformations are isometries of the 3-sphere. If the original 3-manifold had in fact a
Thurston elliptization conjecture
Thurston_elliptization_conjecture
Two tame knots with homeomorphic complements are the same or mirror images
theorem): no nontrivial Dehn surgery on a nontrivial knot in the 3-sphere can yield the 3-sphere. The theorem was proved by Cameron Gordon and John Luecke.
Gordon–Luecke_theorem
Dense arrangement of congruent spheres in an infinite, regular arrangement
fraction of space occupied by spheres – that can be achieved by a lattice packing is π 3 2 ≈ 0.74048 {\textstyle {\frac {\pi }{3{\sqrt {2}}}}\approx 0.74048}
Close-packing of equal spheres
Close-packing_of_equal_spheres
Mathematical proofs published by Archimedes
On the Sphere and Cylinder (Greek: Περὶ σφαίρας καὶ κυλίνδρου; Perì sphaíras kaì kulíndrou) is a treatise that was published by Archimedes in two volumes
On_the_Sphere_and_Cylinder
3 surface) Möbius strip Real projective plane, RP2 Sphere, S2 Surface of genus g Torus Double torus 3-sphere, S3 3-torus, T3 Poincaré homology sphere
List_of_manifolds
4-dimensional object
constructed from the 3-sphere by "slicing" off the bulge of the 3-sphere on either side of the ridge. The analog of this on the 2-sphere is to draw minor
Duocylinder
2-sphere together with its interior. Lininger showed in 1965 that the union of a crumpled cube and an open 3-ball glued along their boundaries is a 3-sphere
Crumpled_cube
Region in which an astronomical body dominates the attraction of satellites
In celestial mechanics, the Hill sphere is a common model for the calculation of a gravitational sphere of influence. It is the most commonly used model
Hill_sphere
Fundamental group of a knot complement
{\displaystyle \pi _{1}\left(\mathbb {R} ^{3}\setminus K\right).} Other conventions consider knots to be embedded in the 3-sphere, in which case the knot group is
Knot_group
Special mathematical functions defined on the surface of a sphere
spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many
Spherical_harmonics
vertices of the dual 24-cell with norm squared 2, projected to the unit 3-sphere. These 48 vertices correspond to the binary octahedral group 2O or <2,3
Truncated_24-cells
Simplest nontrivial knot link
continuous function from the 3-sphere (a three-dimensional surface in four-dimensional Euclidean space) into the more familiar 2-sphere, with the property that
Hopf_link
Concept in algebraic topology
groups and the same homology groups as the n-sphere, and so every homotopy sphere is necessarily a homology sphere. The topological generalized Poincaré conjecture
Homotopy_sphere
Type of mathematical link
In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e.
Hyperbolic_link
World War II era Pan-Asian union under the Empire of Japan
The Greater East Asia Co-Prosperity Sphere (Japanese: 大東亜共栄圏, Hepburn: Dai Tōa Kyōeiken; Kyujitai: 大東亞共榮圈), also known as the GEACPS, was a pan-Asian union
Greater East Asia Co-Prosperity Sphere
Greater_East_Asia_Co-Prosperity_Sphere
Theorem in topology
statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained
Property_P_conjecture
Metal sphere found in 1974 in Florida, United States
The Betz mystery sphere is a metal sphere with an approximate diameter of 8 inches (20 cm) weighing nearly 22 pounds (10 kg) uncovered in 1974 by a family
Betz_mystery_sphere
travel, tourism, insurance
3 SPHERE
3 SPHERE
3 SPHERE
3 SPHERE
3 SPHERE
3 SPHERE
3 SPHERE
3 SPHERE
3 SPHERE
travel, tourism, insurance