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Continuous surjection satisfying a local triviality condition
definition of a fiber bundle from his study of a more particular notion of a sphere bundle, that is a fiber bundle whose fiber is a sphere of arbitrary dimension
Fiber_bundle
topology, a sphere bundle is a fiber bundle in which the fibers are spheres S n {\displaystyle S^{n}} of some dimension n. Similarly, in a disk bundle, the fibers
Sphere_bundle
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered
Hopf_fibration
bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M), UTM, or SM is the unit sphere bundle for the tangent bundle T(M). It is a fiber bundle
Unit_tangent_bundle
Mathematical parametrization of vector spaces by another space
manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball
Vector_bundle
Long exact sequence
space of a sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings given the Euler class of the sphere bundle and vice versa
Gysin_homomorphism
Mathematical object
and mathematics. 1-sphere, 2-sphere, n-sphere tesseract, polychoron, simplex Pauli matrices Hopf bundle, Riemann sphere Poincaré sphere Reeb foliation Clifford
3-sphere
Vector bundle existing over a Grassmannian
bundle over a complex projective space is called the standard line bundle. The sphere bundle of the standard bundle is usually called the Hopf bundle
Tautological_bundle
Fiber bundle whose fibers are group torsors
In the mathematical area of topology, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product
Principal_bundle
Spectral sequence in algebraic topology
rank r vector bundle E {\displaystyle {\mathcal {E}}} which is a finite whitney sum of vector bundles we can construct a sphere bundle S → X {\displaystyle
Serre_spectral_sequence
Algebraic construct classifying topological spaces
S^{7},} not diffeomorphic. Note that any sphere bundle can be constructed from a 4 {\displaystyle 4} -vector bundle, which have structure group S O ( 4 )
Homotopy_group
Special type of principal bundle
three-dimensional sphere, hence principal SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles without their group action are in particular sphere bundles. These
Principal_SU(2)-bundle
Principal fiber bundle
SO(2)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is a special case of a sphere bundle. Circle
Circle_bundle
Tangent spaces of a manifold
some trivial bundle E {\displaystyle E} the Whitney sum T M ⊕ E {\displaystyle TM\oplus E} is trivial. For example, the n-dimensional sphere Sn is framed
Tangent_bundle
{\displaystyle g_{0}} is the standard metric on the n {\displaystyle n} -sphere S n {\displaystyle S^{n}} . It follows that if we define σ ( M ) = sup g
Yamabe_invariant
Topological space associated to a vector bundle
{\displaystyle E_{b}} is an n-dimensional real vector space. We can form an n-sphere bundle Sph ( E ) → B {\displaystyle \operatorname {Sph} (E)\to B} by taking
Thom_space
Generalized sphere of dimension n (mathematics)
{\displaystyle \operatorname {U} (1)} -bundle over the 2 {\displaystyle 2} -sphere, Lie group structure Sp(1) = SU(2). 4-sphere Homeomorphic to the quaternionic
N-sphere
Index of articles associated with the same name
are prime but not irreducible are the trivial 2-sphere bundle over S1 and the twisted 2-sphere bundle over S1. See, for example, Prime decomposition (3-manifold)
Irreducibility_(mathematics)
Model of the extended complex plane plus a point at infinity
drawing used to study Riemann surfaces Hopf bundle – Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibersPages displaying short descriptions
Riemann_sphere
Characterizes closed, orientable, connected 3-manifolds via Dehn surgery on a 3-sphere
3-manifold is obtained by Dehn surgery on a link in the non-orientable 2-sphere bundle over the circle. Similar to the orientable case, the surgery can be
Lickorish–Wallace_theorem
Intrinsic geometric structures in mathematics
unit sphere in E3 S 2 = { a ∈ E 3 : ‖ a ‖ = 1 } . {\displaystyle S^{2}=\{a\in E^{3}\colon \|a\|=1\}.} Its tangent bundle T, unit tangent bundle U and
Riemannian connection on a surface
Riemannian_connection_on_a_surface
Generalization of an orientation of a vector space
orientation to the unit sphere bundle of E. Just as a real vector bundle is classified by the real infinite Grassmannian, oriented bundles are classified by
Orientation of a vector bundle
Orientation_of_a_vector_bundle
Concept in differential geometry
name sphere space, but in 1940 Whitney changed the name to sphere bundle. The theory of fibered spaces, of which vector bundles, principal bundles, topological
Fibered_manifold
Vector bundle of rank 1
Complex line bundles are closely related to circle bundles. There are some celebrated ones, for example the Hopf fibrations of spheres to spheres. In algebraic
Line_bundle
Characteristic class of oriented, real vector bundles
^{n+1}} has trivial normal bundle, thus the tangent bundle of the sphere plus a trivial line bundle is the tangent bundle of Euclidean space, restricted
Euler_class
Mathematical result in differential geometry
example, complex vector spaces with the trivial bundle together with certain bundles over even dimensional spheres. So the index theorem can be proved by checking
Atiyah–Singer_index_theorem
Theorem that the diffeomorphism group of the 3-sphere has the homotopy-type of O(4)
As of 2021, the proof remains unpublished in a mathematical journal. Sphere bundle Hatcher, Allen E. (May 1983). "A Proof of the Smale Conjecture, Diff(S3)
Smale_conjecture
Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
first exotic spheres were constructed by John Milnor (1956) in dimension n = 7 {\displaystyle n=7} as S 3 {\displaystyle S^{3}} -bundles over S 4 {\displaystyle
Exotic_sphere
Straight path on a curved surface or a Riemannian manifold
double tangent bundle TTM into horizontal and vertical bundles: T T M = H ⊕ V . {\displaystyle TTM=H\oplus V.} The double tangent bundle can be visualized
Geodesic
Function in mathematics
manifold) Connection (principal bundle) Connection (vector bundle) Connection (affine bundle) Connection (composite bundle) Connection (algebraic framework)
Connection_(mathematics)
Concept in algebraic geometry
canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle
Canonical_bundle
Limit of spheres in algebraic topology
principal bundles. With the usual definition S n = { x ∈ R n + 1 | ‖ x ‖ 2 = 1 } {\displaystyle S^{n}=\{x\in \mathbb {R} ^{n+1}|\|x\|_{2}=1\}} of the sphere with
Infinite-dimensional_sphere
Mathematics glossary
see spectrum (topology). sphere bundle A sphere bundle is a fiber bundle whose fibers are spheres. sphere spectrum The sphere spectrum is a spectrum consisting
Glossary of algebraic topology
Glossary_of_algebraic_topology
Concept in geometry
{\displaystyle H} or E {\displaystyle E} are necessarily trivial. The unit sphere bundle Z = S ( E ) {\displaystyle Z=S(E)} inside E {\displaystyle E} corresponds
Quaternionic_manifold
How spheres of various dimensions can wrap around each other
specific bundle, each group homomorphism πi(S1) → πi(S3), induced by the inclusion S1 → S3, maps all of πi(S1) to zero, since the lower-dimensional sphere S1
Homotopy_groups_of_spheres
In Euclidean geometry, the bundle theorem is a statement about six circles and eight points in the Euclidean plane. In general incidence geometry, it is
Bundle_theorem
Special type of principal bundle
one-dimensional sphere, hence principal U ( 1 ) {\displaystyle \operatorname {U} (1)} -bundles without their group action are in particular circle bundles. These
Principal_U(1)-bundle
British-Lebanese mathematician (1929–2019)
defined as the group of stable fiber homotopy equivalence classes of sphere bundles; this was later studied in detail by J. F. Adams in a series of papers
Michael_Atiyah
Concept in differential geometry
holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles. In each of these cases, the holonomy
Holonomy
Way to create new manifolds out of disk bundles
{\displaystyle D(\tau _{S^{2k}})} denote the disk bundle associated to the tangent bundle of the 2k-sphere. If we plumb eight copies of D ( τ S 2 k ) {\displaystyle
Plumbing_(mathematics)
Sphere with radius one, usually centered on the origin of the space
the unit sphere in the dual number plane. Ball n {\displaystyle n} -sphere Sphere Superellipse Unit circle Unit disk Unit tangent bundle Unit square
Unit_sphere
Mathematical operation on vector bundles
the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces. The dual bundle of a vector bundle π : E → X
Dual_bundle
Type of differentiable manifold
{\displaystyle p} . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on
Parallelizable_manifold
Topological space that locally resembles Euclidean space
normal bundle, but instead there is an intrinsic stable normal bundle. The n-sphere Sn is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere)
Manifold
How many linearly independent smooth nowhere-zero vector fields can be on an n-sphere
)-dimensional sphere. In detail, the question applies to the 'round spheres' and to their tangent bundles: in fact since all exotic spheres have isomorphic
Vector_fields_on_spheres
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
consequence, given the general theory of fibre bundles, is that every Hilbert bundle is a trivial bundle. The result on the contractibility of S∞ gives
Kuiper's_theorem
Mathematical concept
unit sphere and then identifying under the natural action of U(1) one obtains CPn. For n = 1 this construction yields the classical Hopf bundle S 3 →
Complex_projective_space
French mathematician (1923–2002)
University of Paris. His thesis, titled Espaces fibrés en sphères et carrés de Steenrod (Sphere bundles and Steenrod squares), was written under the direction
René_Thom
_{+}(T^{*}X)} be the co-sphere bundle of X , {\displaystyle X,} that is, the oriented projectivization of the cotangent bundle. Let P ( X ) {\displaystyle
Valuation_(geometry)
integration. Now, suppose π {\displaystyle \pi } is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence 0 → K → Ω ∗ ( E )
Integration_along_fibers
Gromov–Hausdorff metric to a two-dimensional sphere of constant curvature 4. The Hopf fibration S3 → S2 is a principal U(1)-bundle. Furthermore, relative to the standard
Berger's_sphere
Fiber bundle Principal bundle Frame bundle Hopf bundle Associated bundle Vector bundle Tangent bundle Cotangent bundle Line bundle Jet bundle Sheaf (mathematics)
List of differential geometry topics
List_of_differential_geometry_topics
System of moving vectors in differential geometry
g. on a sphere). Parallel transport of tangent vectors is a special case of a more general construction involving an arbitrary vector bundle E {\displaystyle
Parallel_transport
Study of angle-preserving transformations of a geometric space
dt^{2}+2t\,dt\,d\rho ,} where gij is the metric on the sphere. In these terms, a section of the bundle N+ consists of a specification of the value of the
Conformal_geometry
Collection of maps
typically a bundle of maps of Earth or of a continent or region of Earth. Advances in astronomy have also resulted in atlases of the celestial sphere or of
Atlas
M\times S^{k}} . Even more generally, the double of a disc bundle over a manifold is a sphere bundle over the same manifold. More concretely, the double of
Double_(manifold)
Possibility of a consistent definition of "clockwise" in a mathematical space
also be expressed in terms of the tangent bundle. The tangent bundle is a vector bundle, so it is a fiber bundle with structure group GL ( n , R ) {\displaystyle
Orientability
exotic sphere which can be expressed as a biquotient of a compact Lie group. It can be expressed as a S 3 {\displaystyle S^{3}} -fiber bundle over S 4
Gromoll–Meyer_sphere
Topological construct
construction is a way of constructing fiber bundles, particularly vector bundles on spheres. Consider the sphere S n {\displaystyle S^{n}} as the union of
Clutching_construction
Construct allowing differentiation of tangent vector fields of manifolds
simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry
Affine_connection
Topological space
of circles. In other words, it is a S 1 {\displaystyle S^{1}} -bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber
Seifert_fiber_space
Santaló's formula describes how to integrate a function on the unit sphere bundle of a Riemannian manifold by first integrating along every geodesic separately
Santaló's_formula
Branch of mathematics
tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of the sphere; it applies to any kind of
Topology
Branch of geometry
Mathematicians Redefine the Sphere". Quanta Magazine. Retrieved 2023-11-07. Hoffman, William C. (1989-08-01). "The visual cortex is a contact bundle". Applied Mathematics
Contact_geometry
Differentiable function whose derivative is everywhere injective
normal bundles plus trivial bundles, and thus if the stable normal bundle has cohomological dimension k, it cannot come from an (unstable) normal bundle of
Immersion_(mathematics)
Property of a mathematical space
surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional
Dimension
Tensor in differential geometry
curvature form of the canonical line bundle. The canonical line bundle is the top exterior power of the bundle of holomorphic Kähler differentials: κ
Ricci_curvature
Type of topological space
obtained by identifying antipodal points of the unit n {\displaystyle n} -sphere, S n {\displaystyle S^{n}} , in R n + 1 {\displaystyle \mathbb {R}
Real_projective_space
Assignment of a vector to each point in a subset of Euclidean space
vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field. Given a subset
Vector_field
of Grothendieck (1957), that holomorphic vector bundles on the Riemann sphere are sums of line bundles, is now often called the Birkhoff–Grothendieck theorem
Vector bundles on algebraic curves
Vector_bundles_on_algebraic_curves
Set of topological invariants
vector bundles of different ranks over the same base space X. It can also happen when E and F have the same rank: the tangent bundle of the 2-sphere S 2
Stiefel–Whitney_class
Branch of mathematics
differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle. Loosely speaking, this structure by itself is sufficient only
Differential_geometry
Affine connection on the tangent bundle of a manifold
the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric and is torsion-free
Levi-Civita_connection
Method for constructing vector bundles
to construct instantons over the 4-sphere. Barth, Wolf; Hulek, Klaus (1978), "Monads and moduli of vector bundles", Manuscripta Mathematica, 25 (4): 323–347
Horrocks_construction
2 {\displaystyle S^{2}} bundle over S 1 , {\displaystyle S^{1},} or it is irreducible, which means that any embedded 2-sphere bounds a ball. So the theorem
Prime decomposition of 3-manifolds
Prime_decomposition_of_3-manifolds
powers of the line bundles whose product is used in the iterated sphere-bundle construction of the Bott tower. Canonical complex line bundles ρ i {\displaystyle
Quasitoric_manifold
Characteristic classes of vector bundles
Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics
Chern_class
Group of unitary complex matrices with determinant of 1
arbitrary point in the sphere is isomorphic to SU(2), which topologically is a 3-sphere. It then follows that SU(3) is a fiber bundle over the base S5 with
Special_unitary_group
Standard or referential form
partition Canonical one-form, a special 1-form defined on the cotangent bundle T*M of a manifold M Canonical symplectic form, the exterior derivative of
Canonical
Manifold upon which it is possible to perform calculus
bundle. One can also define the tangent bundle as the bundle of 1-jets from R (the real line) to M. One may construct an atlas for the tangent bundle
Differentiable_manifold
Geometric surface
curvature −1/R2 at each point. Its name comes from the analogy with the sphere of radius R, which is a surface of curvature 1/R2. Examples include the
Pseudosphere
Manifold with the same rational homotopy groups as a sphere
{\displaystyle \mathbb {R} P^{n}} is a rational homotopy sphere for all n > 0 {\displaystyle n>0} . The fiber bundle S 0 → S n → R P n {\displaystyle S^{0}\rightarrow
Rational_homotopy_sphere
Manifold
that is, the tangent bundle is equipped with a linear complex structure. Concretely, this is an endomorphism of the tangent bundle whose square is −I;
Complex_manifold
{\displaystyle S^{2}\times S^{1}} and the non-orientable fiber bundle of the 2-sphere over the circle S 1 {\displaystyle S^{1}} are both prime but not
Prime_manifold
JavaScript software stack
which relies on that concept. LAMP (software bundle) List of all Apache/MySQL/PHP stacks LYME (software bundle) – a stack based on Erlang
MEAN_(solution_stack)
Element of the representation ring
1016/0040-9383(65)90040-6. Bott, Raoul (1962), "A note on the KO-theory of sphere-bundles", Bulletin of the American Mathematical Society, 68 (4): 395–400, doi:10
Bott_cannibalistic_class
Operator generalizing the Laplacian in differential geometry
{\displaystyle \partial _{i}:={\frac {\partial }{\partial x^{i}}}} of the tangent bundle T M {\displaystyle TM} and ∧ {\displaystyle \wedge } is the wedge product
Laplace–Beltrami_operator
one in 1988 by Jonathan Rosenberg in Continuous-Trace Algebras from the Bundle Theoretic Point of View. In physics, it has been conjectured to classify
Twisted_K-theory
Study of vector bundles, principal bundles, and fibre bundles
theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused
Gauge_theory_(mathematics)
Concept in mathematics
the group of unit quaternions. The sphere S 4 n + 3 {\displaystyle S^{4n+3}} then becomes a principal Sp(1)-bundle over H P n {\displaystyle \mathbb {HP}
Quaternionic_projective_space
2026 studio album by Illenium
Longhurst, Courtney (August 1, 2025). "Illenium Releases New Two-Single Bundle: 'Refuge' & 'Ur Alive'". EDMTunes. Retrieved January 31, 2026. Talim, Ansh
Odyssey_(Illenium_album)
Branch of algebraic topology
the tautological bundle on S 2 = P 1 ( C ) , {\displaystyle \mathbb {S} ^{2}=\mathbb {P} ^{1}(\mathbb {C} ),} i.e. the Riemann sphere. K ~ n + 2 ( X )
Topological_K-theory
regarding the line bundle K M ⊗ L ⊗ m {\displaystyle K_{M}\otimes L^{\otimes m}} constructed from a positive holomorphic line bundle L {\displaystyle L}
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
large enough (around twice g) this becomes a projective space bundle (the Picard bundle). It has been studied in detail, for example by Kempf and Mukai
Symmetric product of an algebraic curve
Symmetric_product_of_an_algebraic_curve
Concept in mathematics
regular everywhere except at 0 and ∞ {\displaystyle \infty } on the Riemann sphere. If we continue the function, following a loop around the origin, the value
Riemann–Hilbert correspondence
Riemann–Hilbert_correspondence
Disk theorem) Sphere theorem Haken manifold JSJ decomposition Branched surface Lamination Examples 3-sphere Torus bundles Surface bundles over the circle
List of geometric topology topics
List_of_geometric_topology_topics
Differential geometry topic
tangent k-planes in the tangent bundle TM. The target space for the Gauss map N is a Grassmann bundle built on the tangent bundle TM. In the case where M =
Gauss_map
Embedding Exotic structure – See exotic sphere and exotic R 4 {\textstyle \mathbb {R} ^{4}} . Fiber – In a fiber bundle, π : E → B {\displaystyle \pi :E\to
Glossary of differential geometry and topology
Glossary_of_differential_geometry_and_topology
Generalization of affine connections
concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan
Cartan_connection
American ring announcer
13th overall pick of the first round of the 2024 NHL entry draft at the Sphere in Vegas using his trademarked catchphrase, introducing draft pick Jett
Michael_Buffer
SPHERE BUNDLE
SPHERE BUNDLE
Girl/Female
French, German, Hebrew
Little and Womanly; Dear; Man; The Plain
Girl/Female
Indian, Telugu
Veda means Vedham and Shree means Sriman Narayana
Male
Hebrew
(עֵפֶר) Hebrew name EPHER means "calf" or "gazelle." In the bible, this is the name of several characters, including a son of Ezra.
Surname or Lastname
English
English : variant of Spear.
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Boy/Male
British, English
Spear-man
Surname or Lastname
English
English : variant spelling of Shear 1.Indian (Maharashtra); pronounced as two syllables : Hindu (Vani) name, probably from Marathi šera ‘rate’.
Girl/Female
American, Christian, French, German, Hebrew
Darling; Little and Womanly; Beloved; The Plain
Surname or Lastname
English
English : topographic name for someone who lived by the seashore, Middle English schore.English : topographic name for someone who lived on or by a bank or steep slope, Old English scora. There are minor places named with this word in Lancashire and West Yorkshire, and the surname may also be a habitational name from these.Americanized spelling of Ashkenazic Jewish S(c)hor(r) or Szor, variants of Schauer.
Boy/Male
American, British, English
Spear
Female
English
English variant spelling of Greek Phoebe, PHEBE means "shining one."
Girl/Female
French, German, Hebrew
Beloved; A Man; The Plain
Boy/Male
Australian, French, Portuguese
Stern; Severe
Female
English
Variant spelling of English Sherry, SHERI means "darling."
Surname or Lastname
English and Irish (County Limerick; of English origin)
English and Irish (County Limerick; of English origin) : from Old English scīr, Middle English s(c)hire ‘shire’, perhaps a topographic name for someone who lived by the meeting place of a shire.
Female
English
Variant spelling of English Sherry, SHERIE means "darling."
Male
English
Variant spelling of English Ophir, OPHER means "gold" or "reducing to ashes."
Surname or Lastname
English
English : variant of Sherrin.
Female
English
Variant spelling of English Sherry, SHEREE means "darling."
Surname or Lastname
English
English : variant of Shear 1.Jewish (eastern Ashkenazic) : variant spelling of Scher.
SPHERE BUNDLE
SPHERE BUNDLE
Girl/Female
Hindu
Lighting
Boy/Male
Tamil
Dharamvir | தரமவீரÂ
One who gets victory on religion
Girl/Female
Indian, Punjabi, Sikh
The Perfect God
Female
English
Feminine form of English Benjamin, BENJAMINA means "blessed."
Girl/Female
Arabic, Australian, Muslim
Glamorous; The Sunrise; Magical
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Rukhmini; Wife of Krishna
Boy/Male
Arabic
Imposter; Big Liar
Boy/Male
Assamese, Hindu, Indian, Kannada, Marathi, Oriya, Sanskrit, Tamil, Telugu
Master of the Mountains
Boy/Male
Tamil
Thinker
Girl/Female
Indian, Telugu
Name of a Place; Kaashi (Banaras)
SPHERE BUNDLE
SPHERE BUNDLE
SPHERE BUNDLE
SPHERE BUNDLE
SPHERE BUNDLE
a.
Of or pertaining to the heavenly orbs, or to the sphere or spheres in which, according to ancient astronomy and astrology, they were set.
v. t.
To remove, as a planet, from its sphere or orb.
imp. & p. p.
of Sphere
a.
Rounded like a sphere; sphere-shaped; hence, symmetrical; complete; perfect.
a.
Having the form of a sphere; like a sphere; globular; orbicular; as, a spherical body.
a.
Of or pertaining to the spheres.
a.
Of or pertaining to a sphere.
n.
The apparent surface of the heavens, which is assumed to be spherical and everywhere equally distant, in which the heavenly bodies appear to have their places, and on which the various astronomical circles, as of right ascension and declination, the equator, ecliptic, etc., are conceived to be drawn; an ideal geometrical sphere, with the astronomical and geographical circles in their proper positions on it.
n.
A sphere.
v. t.
To form into roundness; to make spherical, or spheral; to perfect.
n.
A sphere or scheme of operation.
n.
A sphere.
superl.
Sharp; afflictive; distressing; violent; extreme; as, severe pain, anguish, fortune; severe cold.
v. t.
To place in, or as in, an orb a sphere. Cf. Ensphere.
v. i.
To form a scheme or schemes.
v. t.
To form into a sphere.
adv.
In this place; in the place where the speaker is; -- opposed to there.
v. t.
To place in a sphere, or among the spheres; to insphere.
v. t.
To place in a sphere; to envelop.
a.
Of or pertaining to a sphere or the spheres.