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3-dimensional object
torus. A solid torus is a torus plus the region inside the torus, with a non-zero volume. Real-world objects that approximate a solid torus include O-rings
Solid_torus
Doughnut-shaped surface of revolution
that approximate a torus of revolution include swim rings, inner tubes and ringette rings. A torus is different from a solid torus, which is formed by
Torus
Cartesian product of 3 circles
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, T 3 = S 1
3-torus
Type of mathematical knot
an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite
Satellite_knot
Open 3-manifold that is contractible but not homeomorphic to R3
three-dimensional sphere. Now find a compact unknotted solid torus T 1 {\displaystyle T_{1}} inside the sphere. (A solid torus is the topological space S 1 × D 2 {\displaystyle
Whitehead_manifold
two solid tori, along a 2-torus: see Clifford torus. Each of the solid tori is then foliated internally, in codimension 1, and the dividing torus surface
Reeb_foliation
Surface of revolution with a hole in the middle
surface of a torus having a topological genus, g, of 1 or greater. The Euler characteristic χ of a g holed toroid is 2(1−g). The torus is an example
Toroid
Class of compact connected topological spaces
embedded solid tori in R3. Fix a sequence of natural numbers {ni}, ni ≥ 2. Let T0 = S1 × D be a solid torus. For each i ≥ 0, choose a solid torus Ti+1 that
Solenoid_(mathematics)
Non-orientable mathematical surface
image of the other, yield a fundamental region of the torus. The universal cover of both the torus and the Klein bottle is the plane R2. The fundamental
Klein_bottle
Simplest non-trivial closed knot with three crossings
3t\end{aligned}}} The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus ( r − 2 ) 2 + z 2 = 1
Trefoil_knot
Operation used to modify three-dimensional topological spaces
\cup T_{k}} , we may glue in one solid torus by a homeomorphism (resp. diffeomorphism) of its boundary to each of the torus boundary components T i {\displaystyle
Dehn_surgery
Geometrical object in four-dimensional space
In differential geometry, the Clifford torus is the standard embedding of the 2-torus as a product of circles in Euclidean space R4 (equivalently C2).
Clifford_torus
is a mapping torus with solid tori glued in so that the core circle of each torus runs parallel to the boundary of the fiber. Each torus in ∂Σφ is fibered
Open_book_decomposition
Operation on a knot producing a link with two components
Bing double of the unknot in the solid torus surrounding it, as shown in the figure, and then twisting that solid torus into the shape of K. This definition
Bing_double
longitude for each boundary torus, i.e. simple closed curves that are generators for the fundamental group of the torus. Let M ( u 1 , u 2 , … , u n
Hyperbolic_Dehn_surgery
Complement of a knot in three-sphere
M is the 3-sphere). Let N be a tubular neighborhood of K; so N is a solid torus. The knot complement is then the complement of N, X K = M − interior
Knot_complement
Topics referred to by the same term
sagittal torus, a structure found in crania Torus, a structure of the xylem Solid torus, a solid whose surface is a torus. Torus knot Algebraic torus Umbilic
Torus_(disambiguation)
Non-orientable surface with one edge
forms a slice through the solid torus swept out by this disk. Because of the one-sidedness of this slice, the sliced torus remains connected. A line or
Möbius_strip
How many times curves wind around each other
geometric argument. The complement of a standard circle is homeomorphic to a solid torus with a point removed (this can be seen by interpreting 3-space as the
Linking_number
Breakfast cereal made by General Mills
United States and Canada, consisting of pulverized oats in the shape of a solid torus. In Europe, Cheerios is marketed by Cereal Partners under the Nestlé
Cheerios
Mapping which preserves all topological properties of a given space
square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations
Homeomorphism
Knot that can't be tied in a string of constant diameter
"thickened", that is, if there exists an extension to an embedding of the solid torus S1×D2 into the 3-sphere. A knot is tame if and only if it can be represented
Wild_knot
Number of "holes" of a surface
to the number of handles on it. For instance: A ball has genus 0. A solid torus D2 × S1 has genus 1. The genus of a graph is the minimal integer n such
Genus_(mathematics)
Mathematical theory
For example the Clifford torus construction in the 3-sphere shows that the complement of a solid torus is another solid torus; which will be open if the
Alexander_duality
Surface of genus g Torus Double torus 3-sphere, S3 3-torus, T3 Poincaré homology sphere SO(3) ≅ RP3 Solid Klein bottle Solid torus Whitehead manifold
List_of_manifolds
Embedding of Cantor set in 3-dimensional Euclidean space
solid torus A0 (iteration 0). Next, construct a "necklace" of smaller, linked tori that lie inside A0. This necklace is A1 (iteration 1). Each torus composing
Antoine's_necklace
Space which has no holes through it
convex subset of R n {\displaystyle \mathbb {R} ^{n}} is simply connected. A torus, the (elliptic) cylinder, the Möbius strip, the projective plane and the
Simply_connected_space
Describes how distinct surgery presentations of a given 3-manifold are related
along two disjoint 3-balls. A 2-handle is attached along a solid torus; since this solid torus is embedded in a 3-manifold, there is a relation between
Kirby_calculus
Three-dimensional geometric shape
4 π r 2 {\displaystyle 4\pi r^{2}} . Spherical pressure vessel Ball Solid torus Bubble Sphere Focaloid Weisstein, Eric W. "Spherical Shell". mathworld
Spherical_shell
Embedding of the circle in three dimensional Euclidean space
A framed knot is the extension of a tame knot to an embedding of the solid torus D2 × S1 in S3. The framing of the knot is the linking number of the image
Knot_(mathematics)
Decomposition of a manifold into standard pieces
circle) and is called a solid torus. All other handlebodies may be obtained by taking the boundary-connected sum of a collection of solid tori. Handle decomposition
Handlebody
Manifold of dimension 3 equipped with a hyperbolic metric
obtained is a manifold with a torus boundary and under some (not generic) conditions it is possible to glue a hyperbolic solid torus on each boundary component
Hyperbolic_3-manifold
Knot invariant
⊂ S 3 {\displaystyle S^{1}\times D^{2}\subset S^{3}} is an unknotted solid torus containing K ′ {\displaystyle K'} ), then Δ K ( t ) = Δ f ( S 1 × { 0
Alexander_polynomial
4-dimensional object
that is the boundary between the two bounding (solid) torus cells. It is in the shape of a Clifford torus, which is the Cartesian product of two circles
Duocylinder
Topological space
fiber has a tubular neighborhood that forms a standard fibered torus. A standard fibered torus corresponding to a pair of coprime integers ( a , b ) {\displaystyle
Seifert_fiber_space
Knot defined by parametric equations defining Lissajous curves
studied in other domains, for instance in a cylinder or in a (flat) solid torus (Lissajous-toric knot). Because a knot cannot be self-intersecting, the
Lissajous_knot
q {\displaystyle p=q} the knot is a torus knot. In braid form these knots can be defined in a square solid torus (i.e. the cube [ − 1 , 1 ] 3 {\displaystyle
Lissajous-toric_knot
List of concrete topologies and topological spaces
analytic manifold that is not paracompact. Real projective line Torus 3-torus Solid torus Unknot Whitehead manifold − An open 3-manifold that is contractible
List_of_topologies
Pathological embedding of the sphere in 3D space
standard torus: Remove a radial slice of the torus. Connect a standard punctured torus to each side of the cut, interlinked with the torus on the other
Alexander_horned_sphere
Sport
physics, and biomechanics. Gyroscopic Stability: A tennikoit ring is a solid torus. If thrown without spin, aerodynamic forces will cause it to tumble unpredictably
Tennikoit
Branch of topology
M is the 3-sphere). Let N be a tubular neighborhood of K; so N is a solid torus. The knot complement is then the complement of N, X K = M − interior
Low-dimensional_topology
Determining whether a knot is the unknot
of them transforms the complement into a standard triangulation of a solid torus. The time for this method would be triply exponential; however, experimental
Unknotting_problem
Method of describing higher-order polyhedra
square torus, {4,4}1,0 A regular 4x4 square torus, {4,4}4,0 tQ24×12 projected to torus taQ24×12 projected to torus actQ24×8 projected to torus tH24×12
Conway_polyhedron_notation
In mathematics, a partition of a manifold into submanifolds
irrational number, the torus R 2 / Z 2 {\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}} is foliated by the set of straight lines in the torus of slope m. Each
Foliation
Result about foliation of compact 3-manifolds
foliation of the 3-sphere S3 has a compact leaf. The leaf is a torus T2 bounding a solid torus with the Reeb foliation. The theorem was proved by Sergei Novikov
Novikov's compact leaf theorem
Novikov's_compact_leaf_theorem
Loop seen as a trivial knot
infinite cyclic group, and its knot complement is homeomorphic to a solid torus. If a diagram lies on the surface of a sphere rather than a plane, unknotting
Unknot
Mathematics textbook
W. Alexander states that at least one side of any torus in Euclidean space must be a solid torus. However, for more complicated manifolds, cutting along
Introduction_to_3-Manifolds
For instance, if K is a trefoil knot embedded in the boundary of a solid torus V and S is the closure of a small annular neighborhood of K in ∂ V {\displaystyle
Boundary-incompressible surface
Boundary-incompressible_surface
American mathematician
879-910, with Xuwen Chen. Linear stability analysis of a hot plasma in a solid torus, Arch. Rat. Mech. Anal. 211 (2014), 619-672, with T. Nguyen. Stability
Walter_Alexander_Strauss
Solid with six equal square faces
A cube is a three-dimensional solid object in geometry. It has eight vertices and twelve straight edges of the same length, so that these edges form six
Cube
Generalized manifold
faces identified with a 120° twist (a 1/3 twist) – equivalently, as a solid torus in 3 dimensions with a cross-section an equilateral triangle and such
Orbifold
Algebraic structure associated with a topological space
The torus is defined as a product of two circles T 2 = S 1 × S 1 {\displaystyle T^{2}=S^{1}\times S^{1}} . The torus has a single path-connected
Homology_(mathematics)
Moment of inertia of diff geometric shapes
solids, the ratio of moments of inertia is based on the dimension. More precisely, if the 'hollow' version is generated from a scaling of the solid (more
List_of_moments_of_inertia
Four-dimensional analog of the dodecahedron
60-cell solid torus. One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint
120-cell
Linear accelerator
McLean, H. S. (14 January 1991). "Quasistatic compression of a compact torus". Physical Review Letters. 66 (2): 165–168. Bibcode:1991PhRvL..66..165M
Plasma_railgun
2006 film directed by Kenji Kamiyama
Ghost in the Shell: Stand Alone Complex – Solid State Society (Japanese: 攻殻機動隊 STAND ALONE COMPLEX Solid State Society, Hepburn: Kōkaku Kidōtai Sutando
Ghost in the Shell: Stand Alone Complex – Solid State Society
Ghost_in_the_Shell:_Stand_Alone_Complex_–_Solid_State_Society
Symmetric tessellation of a closed surface
a plane as a chessboard to a cylinder section to a torus. The projection from a cylinder to a torus distorts the geometry in 3 dimensions, but can be done
Regular_map_(graph_theory)
Geometric shape
The apple and lemon together make up a spindle torus (or self-crossing torus or self-intersecting torus). The lemon forms the boundary of a convex set
Lemon_(geometry)
2002 video game
a 2002 run and gun video game published by Activision and developed by Torus Games for the Game Boy Advance. The game, played from a side-scrolling perspective
The Invincible Iron Man (video game)
The_Invincible_Iron_Man_(video_game)
Surface created by rotating a curve about an axis
intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus). The sections of the surface of revolution made
Surface_of_revolution
Two-dimensional manifold
arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions
Surface_(topology)
Partition of a toroidal surface into polygons
a toroidal polyhedron is a polyhedron which is also a toroid (a g-holed torus), having a topological genus (g) of 1 or greater. Notable examples include
Toroidal_polyhedron
Three dimensional analogue of uniformization conjecture
the mapping torus of an Anosov map of a torus has a finite volume solv structure, but its JSJ decomposition cuts it open along one torus to produce a
Geometrization_conjecture
Simple curve of Euclidean geometry
Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv:2010
Circle
Innermost Galilean moon of Jupiter
occasionally provides sodium ions in the plasma torus with an electron, removing those new "fast" neutrals from the torus. These particles retain their velocity
Io_(moon)
Hypothesized entity in outer space
result is an inner region rotating at a single rate with a loosely connected torus orbiting beyond it. Synestias also have differences in the mantles, both
Synestia
Electrical resonant transformer circuit invented by Nikola Tesla
capacitive electrode (top load) (E) in the form of a smooth metal sphere or torus attached to the secondary terminal of the coil. Its large surface area suppresses
Tesla_coil
Electricity generation by nuclear fusion
variations, including the Levitated Dipole Experiment (LDX), use a solid superconducting torus that is magnetically levitated inside the reactor chamber. Magnetic
Fusion_power
Planar maps require at most four colors
the torus has Euler characteristic χ = 0 (and genus g = 1) and thus p = 7, so no more than seven colors are required to color any map on a torus. This
Four_color_theorem
Ring of cosmic dust orbiting an astronomical object
A ring system is a disc or torus orbiting an astronomical object that is composed of numerous solid chunks of ice, rock, and dust. In the case of giant
Ring_system
Property of a planar object which maps onto itself after rotation by any angle
or duocylindrical symmetry. For example, the duocylinder and Clifford torus have circular symmetry in two orthogonal axes. A spherinder has spherical
Circular_symmetry
Topics referred to by the same term
band Celestiial Ashen (2004 video game), a game for N-Gage developed by Torus Ashen (2018 video game) Stuart Ashen (born 1976), also known as Ashens,
Ashen
Theorem in geometric topology
ball (which is known in mathematics as the two-dimensional sphere) or of a torus, are two-dimensional. The surface of a ball has trivial fundamental group
Poincaré_conjecture
Topological invariant in mathematics
surfaces of toroidal polyhedra all have Euler characteristic 0, like the torus. The Euler characteristic can be defined for connected plane graphs by the
Euler_characteristic
cubic Monkey saddle (saddle-like surface for 3 legs.) Torus Dupin cyclide (inversion of a torus) Whitney umbrella Right conoid (a ruled surface) Apollonian
List_of_mathematical_shapes
File format for 3D printing and scanning
begins with the line: solid name where name is an optional string (though if name is omitted there must still be a space after solid, for compatibility with
STL_(file_format)
Flat-sided three-dimensional shape
vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface. The terms solid polyhedron and polyhedral surface are commonly
Polyhedron
3D model
curves or curves that double back on themselves (such as the aforementioned torus), whereas a surface of revolution could not because such curves cannot be
Lathe_(graphics)
Supernova remnant in the constellation Taurus
the Crab Nebula is a helium-rich torus which is visible as an east–west band crossing the pulsar region. The torus composes about 25% of the visible
Crab_Nebula
Compression of an electrically conducting filament by magnetic forces
cylinder around into a torus. Unfortunately this breaks θ symmetry, as paths on the inner portion (inboard side) of the torus are shorter than similar
Pinch_(plasma_physics)
Results on the surface areas and volumes of surfaces and solids of revolution
C: A = s d . {\displaystyle A=sd.} For example, the surface area of the torus with minor radius r and major radius R is A = ( 2 π r ) ( 2 π R ) = 4 π
Pappus's_centroid_theorem
2015 novel by N. K. Jemisin
earthquakes, and when angered can unintentionally kill living things in their "torus", or area of influence, by extracting the heat from their bodies to use
The_Fifth_Season_(novel)
Geometric inversion of a torus, cylinder or double cone
cyclides can be defined as inversions of the torus (or the cylinder, or the double cone). Since a standard torus is the orbit of a point under a two dimensional
Dupin_cyclide
Overview of and topical guide to geometry
Quadric Hypersphere, sphere Spheroid Ellipsoid Hyperboloid Paraboloid Cone Torus Root system Similarity Zonotope Projective geometry Arc (projective geometry)
Outline_of_geometry
Continuous deformation between two continuous functions
embeddings, f and g, of the torus into R3. X is the torus, Y is R3, f is some continuous function from the torus to R3 that takes the torus to the embedded surface-of-a-doughnut
Homotopy
Mathematical model of the physical space
geometric objects that are being modeled to new positions. The Clifford torus on the surface of the 3-sphere is the simplest and most symmetric flat embedding
Euclidean_geometry
the Euler characteristic, χ. Uniform tilings on the plane correspond to a torus topology, with Euler characteristic of zero. Density: the Density (polytope)
List_of_uniform_polyhedra
Branch of mathematics
a topologist cannot distinguish a coffee mug from a doughnut. A pliable torus (shaped like a doughnut) can be reshaped to a coffee mug by creating a dimple
Topology
Imprint of mouth tissue
chewing, the denture will tend to pivot around incompressible areas (e.g. torus palatinus) and dig into compressible areas. Mucocompressive means that the
Dental_impression
2001 film
good performance as the exacting heroine, while Nakamura and Matsuoka lend solid support as the two wildly contrasted guys." "Parallel section and other
Unloved
Magnetic field plasma confinement device
Symmetric Torus, use a close-fitting shell as a magnetic coil. Driving current through the shell itself is attractive for reactor design. A solid copper
Reversed_field_pinch
Basic shapes represented in vector graphics
had. In constructive solid geometry, primitives are simple geometric shapes such as a cube, cylinder, sphere, cone, pyramid, torus. Modern 2D computer
Geometric_primitive
Austenitic nickel-chromium superalloys
increasingly used in the boilers of waste incinerators. The Joint European Torus and DIII-D tokamaks' vacuum vessels are made of Inconel. Inconel 718 is
Inconel
2024 video game
McKay, Zach (October 7, 2024). "Review: DRAGON BALL: Sparking! ZERO is a Solid Arena Fighting Game". Hardcore Gamer. Retrieved January 5, 2026. Quesada
Dragon_Ball:_Sparking!_Zero
1997 video game
Blood, set five years after its events. It was directed and produced by Toru Hagihara, with Koji Igarashi acting as assistant director. It features Dracula's
Castlevania: Symphony of the Night
Castlevania:_Symphony_of_the_Night
Vibration control platform
2012. "The Gaia torus is complete". European Space Agency. July 28, 2009. Retrieved Jan 4, 2014. "Schematic diagram of the Gaia torus". European Space
Optical_table
Geometric object
uniform duoprisms, which are products of two regular polygons. Clifford torus The Fourth Dimension Simply Explained, Henry P. Manning, Munn & Company
Spherinder
Method of representing objects in 3D computer graphics
travel forever. Surfaces closed in both directions include a sphere and a torus. Moving in any direction on such surfaces will cause the observer to travel
Computer representation of surfaces
Computer_representation_of_surfaces
Production of voltage by a varying magnetic field
29, 1831, he wrapped two wires around opposite sides of an iron ring or "torus" (an arrangement similar to a modern toroidal transformer).[citation needed]
Electromagnetic_induction
Mechanical, toroid gasket that seals an interface
as a packing or a toric joint, is a mechanical gasket in the shape of a torus; it is a loop of elastomer with a round cross-section, designed to be seated
O-ring
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