1936:
5521:
4245:(twice continuously differentiable) embedding of a flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a point of contact must be a tangency. But that would imply that part of the torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction.) On the other hand, according to the
5268:
4360:, yielding a so-called "smooth fractal". The key to obtain the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths". (These infinitely recursive corrugations are used only for embedding into three dimensions; they are not an intrinsic feature of the flat torus.) This is the first time that any such embedding was defined by explicit equations or depicted by computer graphics.
4257:
4555:
4544:
426:
3163:
4650:
403:
380:
3402:
1876:
4579:
51:
448:
75:
3378:
647:
1584:
4957:
59:
5131:
1499:
4478:
two-tori. (And so the torus itself is the surface of genus 1.) To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the boundary circles. (That is, merge the two boundary circles so they become just one circle.) To form the connected
3643:
everywhere. It is flat in the same sense that the surface of a cylinder is flat. In 3 dimensions, one can bend a flat sheet of paper into a cylinder without stretching the paper, but this cylinder cannot be bent into a torus without stretching the paper (unless some regularity and differentiability
2413:
that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal'
2417:
If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or
3357:
whose top and bottom faces are connected with a 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge
462:
2404:
1514:, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.
5248:
5478:
4184:
1871:{\displaystyle {\begin{aligned}A&=4\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)=\pi ^{2}(p+q)(p-q),\\V&=2\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)^{2}={\tfrac {1}{4}}\pi ^{2}(p+q)(p-q)^{2}.\end{aligned}}}
1318:
4824:
4973:, 1), its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); all homotopy equivalences of the torus can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism.
5015:
2612:
4431:
to a) the square torus (total angle = π) and b) the hexagonal torus (total angle = 2π/3). These are the only conformal equivalence classes of flat tori that have any conformal automorphisms other than those generated by translations and negation.
2473:
2418:
through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of a rectangle together, choosing the other two sides instead will cause the same reversal of orientation.
1079:
1289:
921:
341:
An example of a torus can be constructed by taking a rectangular strip of flexible material such as rubber, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare
4419:
along their (identical) boundaries, where each triangle has angles of π/2, π/3, and 0. As a result the area of each triangle can be calculated as π - (π/2 + π/3 + 0) = π/6, so it follows that the compactified moduli space
5236:, the upper bound is tight.) Equivalently, in a torus divided into regions, it is always possible to color the regions using no more than seven colors so that no neighboring regions are the same color. (Contrast with the
3941:
2283:
1589:
1323:
642:{\displaystyle {\begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\sin \theta \\\end{aligned}}}
4372:, one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists a smooth homeomorphism between them that is both angle-preserving and orientation-preserving. The
4402:
by adding one additional point that represents the limiting case as a rectangular torus approaches an aspect ratio of 0 in the limit. The result is that this compactified moduli space is a sphere with
3758:
3267:
5337:
4735:
4392:
may be identified with a punctured sphere that is smooth except for two points that have less angle than 2π (radians) around them: One has total angle = π and the other has total angle = 2π/3.
467:
4030:
3602:
3320:
2659:
2239:
5763:
1911:, and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles".
699:
3381:
In three dimensions, one can bend a rectangle into a torus, but doing this necessarily affects the distances measured along the surface, as seen by the distortion of the checkered pattern.
4952:{\displaystyle \operatorname {MCG} _{\operatorname {Ho} }(\mathbb {T} ^{n})=\operatorname {Aut} (\pi _{1}(X))=\operatorname {Aut} (\mathbb {Z} ^{n})=\operatorname {GL} (n,\mathbb {Z} ).}
5234:
6360:
5126:{\displaystyle 1\to \operatorname {Homeo} _{0}(\mathbb {T} ^{n})\to \operatorname {Homeo} (\mathbb {T} ^{n})\to \operatorname {MCG} _{\operatorname {TOP} }(\mathbb {T} ^{n})\to 1.}
936:
5188:
2533:
282:
5007:
4793:
4764:
4350:
4313:
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3555:
3517:
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3115:
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2783:
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2721:
2688:
2137:
2099:
1193:
822:
5267:
3639:
This metric of the square flat torus can also be realised by specific embeddings of the familiar 2-torus into
Euclidean 4-space or higher dimensions. Its surface has zero
1494:{\displaystyle {\begin{aligned}A&=\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr,\\V&=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}.\end{aligned}}}
4407:
points each having less than 2π total angle around them. (Such points are termed "cusps".) This additional point will have zero total angle around it. Due to symmetry,
1965:
6109:
4356:
as it is constructed by repeatedly corrugating an ordinary torus. Like fractals, it has no defined
Gaussian curvature. However, unlike fractals, it does have defined
4284:
317:
5882:
776:
2262:
813:
745:
722:
118:
with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a
6028:
284:, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into
6353:
3136:
5505:
5251:
This construction shows the torus divided into seven regions, every one of which touches every other, meaning each must be assigned a unique color.
3386:
2453:
on the 2-torus can be represented as a two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the
6140:
4479:
sum of more than two surfaces, successively take the connected sum of two of them at a time until they are all connected. In this sense, a genus
5689:, Author: Kozak Ana Maria, Pompeya Pastorelli Sonia, Verdanega Pedro Emilio, Editorial: McGraw-Hill, Edition 2007, 744 pages, language: Spanish
334:, a torus is any topological space that is homeomorphic to a torus. The surface of a coffee cup and a doughnut are both topological tori with
6346:
5781:
5726:
5686:
4388:" of the torus to contain one point for each conformal equivalence class, with the appropriate topology. It turns out that this moduli space
6299:
6188:
4352:
was found. It is a flat torus in the sense that, as a metric space, it is isometric to a flat square torus. It is similar in structure to a
3812:
2399:{\displaystyle \pi _{1}(\mathbb {T} ^{2})=\pi _{1}(\mathbb {S} ^{1})\times \pi _{1}(\mathbb {S} ^{1})\cong \mathbb {Z} \times \mathbb {Z} .}
5809:
5311:
exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the
6046:
5718:
5703:
2414:
paths commute. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding.
1987:
3653:
4806:, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the
78:
A ring torus with aspect ratio 3, the ratio between the diameters of the larger (magenta) circle and the smaller (red) circle.
62:
As the distance from the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally
3215:
6069:
5473:{\displaystyle {\begin{pmatrix}n+2\\n-1\end{pmatrix}}+{\begin{pmatrix}n\\n-1\end{pmatrix}}={\tfrac {1}{6}}(n^{3}+3n^{2}+8n)}
3416:
with 1 vertex, 2 orthogonal edges, and one square face. It is seen here stereographically projected into 3-space as a torus.
4179:{\displaystyle T=\left\{(x,y,z,w)\in \mathbb {S} ^{3}\mid x^{2}+y^{2}={\frac {1}{2}},\ z^{2}+w^{2}={\frac {1}{2}}\right\}.}
6050:
5590:
5575:
3205:
1903:
measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of
163:). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered
4700:
3647:
A simple 4-dimensional
Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows:
5284:
with 1s as panels and 0s as holes in the mesh – with consistent orientation, every 3×3 matrix appears exactly once
1312:
4963:
3561:
3279:
2620:
1948:
6499:
5929:
Padgett, Adele (2014). "FUNDAMENTAL GROUPS: MOTIVATION, COMPUTATION METHODS, AND APPLICATIONS" REA Program, Uchicago.
4502:
while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called
2160:
1919:
1881:
1958:
1952:
1944:
5835:
4532:
surface is topologically equivalent to either the sphere or the connect sum of some number of tori, disks, and real
654:
4253:
embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding.
3806:-space requires one to stretch it, in which case it looks like a regular torus. For example, in the following map:
3406:
2786:
2048:
by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of
3421:
2147:
701:
representing rotation around the tube and rotation around the torus' axis of revolution, respectively, where the
5969:
5193:
1969:
6535:
5650:
3390:
3365:
in the work of Dmitri
Tymoczko and collaborators (Felipe Posada, Michael Kolinas, et al.), being used to model
2274:
2107:
2024:
63:
6162:
2106:
to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by
6608:
6539:
4639:
111:
58:
31:
5136:
The mapping class group of higher genus surfaces is much more complicated, and an area of active research.
3636:. This particular flat torus (and any uniformly scaled version of it) is known as the "square" flat torus.
2866:
with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.
6338:
5994:
4803:
4522:
4373:
4246:
3176:
2607:{\displaystyle \mathbb {T} ^{n}=\underbrace {\mathbb {S} ^{1}\times \cdots \times \mathbb {S} ^{1}} _{n}.}
324:
42:
6475:
6417:
5645:
5605:
4013:
2072:
1915:
792:
103:
5520:
5162:
2951:
with an integral inverse; these are just the integral matrices with determinant ±1. Making them act on
2261:
247:
6132:
4983:
4769:
4740:
4326:
4289:
4208:
3531:
3493:
3464:
3427:
3091:
3047:
2954:
2914:
2795:
2759:
2726:
2697:
2664:
2113:
2075:
6597:
6572:
5986:
5580:
5308:
5297:
4977:
4737:
of invertible integer matrices, which can be realized as linear maps on the universal covering space
4601:
3346:
3086:
3026:
3019:
5999:
6567:
6561:
5241:
4694:
4686:
4583:
4573:
4459:
4017:
3520:
3135:
3125:
2450:
2249:
1188:
456:
335:
5909:
6462:
6295:
6192:
6020:
5665:
5620:
5550:
5526:
5312:
5237:
4690:
4455:
4381:
3640:
3523:, as well as the structure of an abelian Lie group. Perhaps the simplest example of this is when
2996:
2446:
130:
6304:
5805:
5930:
6633:
6441:
6324:
6289:
6241:
6101:
6012:
5977:
5858:
5777:
5722:
5714:
5699:
5682:
5483:
parts. (This assumes the pieces may not be rearranged but must remain in place for all cuts.)
5272:
4807:
4412:
4357:
3799:
3780:
2988:
2944:
2941:
2870:
2454:
2270:
2068:
1296:
1074:{\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right).}
788:
6423:
6283:
6091:
6081:
6004:
5769:
5153:
4811:
4533:
4416:
4323:(continuously differentiable) embedding of a flat torus into 3-dimensional Euclidean space
4256:
3385:
3354:
3078:
2430:
2008:
1137:
1131:
930:
237:
151:
5791:
4554:
4543:
4262:
295:
6525:
6054:
5787:
5540:
5261:
4529:
4445:
4369:
3784:
3605:
3323:
3038:
2790:
2410:
2151:
1899:
As a torus has, effectively, two center points, the centerpoints of the angles are moved;
1300:
1083:
The three classes of standard tori correspond to the three possible aspect ratios between
816:
285:
198:
6487:
3338:
3155:
1284:{\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}<r^{2}}
753:
5990:
3341:, the edge corresponding to the orbifold points where the two coordinates coincide. For
6096:
5615:
5600:
5570:
5555:
5545:
5247:
5157:
5149:
5145:
3772:
3162:
3008:
2897:, which (like tori) are compact connected abelian groups, which are not required to be
2863:
2477:
2422:
2037:
2004:
1292:
916:{\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}=r^{2}.}
798:
730:
707:
425:
320:
5693:
4649:
784:
of the torus. The typical doughnut confectionery has an aspect ratio of about 3 to 2.
6627:
6588:
6544:
6530:
6428:
5660:
5635:
5630:
5595:
5565:
5560:
5289:
4980:
of the mapping class group splits (an identification of the torus as the quotient of
4590:
4526:
4471:
4384:. In the case of a torus, the constant curvature must be zero. Then one defines the "
4377:
3366:
2878:
2852:
2845:
2434:
233:
185:
5732:
3377:
6435:
6272:
6024:
5755:
5585:
4697:(the connected components of the homeomorphism group) is surjective onto the group
4559:
4548:
4515:
4511:
4385:
3401:
2904:
2103:
2053:
1308:
780:
402:
379:
343:
6244:
3139:
The configuration space of 2 not necessarily distinct points on the circle is the
1884:
is sometimes used. In traditional spherical coordinates there are three measures,
1521:
of an outermost point on the surface of the torus to the center, and the distance
6215:
3994:. In particular, for certain very specific choices of a square flat torus in the
3420:
A flat torus is a torus with the metric inherited from its representation as the
6315:
6133:"Mathematicians Produce First-Ever Image of Flat Torus in 3D | Mathematics"
3413:
2859:
2458:
2426:
2245:
2061:
926:
193:
115:
38:
4638:
The term "toroidal polyhedron" is also used for higher-genus polyhedra and for
3183:
segment of the right edge of the repeated parallelogram is identified with the
724:
is the distance from the center of the tube to the center of the torus and the
6308:
6277:
5773:
5655:
5625:
5516:
4815:
4597:
4578:
3350:
181:
149:. If the axis of revolution passes twice through the circle, the surface is a
50:
6268:
4795:(this corresponds to integer coefficients) and thus descend to the quotient.
4600:
with the topological type of a torus are called toroidal polyhedra, and have
6603:
6492:
6249:
6163:"Mathematics: first-ever image of a flat torus in 3D – CNRS Web site – CNRS"
6086:
6008:
5866:
5759:
2855:
2830:
2472:
2045:
447:
289:
177:
6105:
6016:
5943:
5861:
3345:= 3 this quotient may be described as a solid torus with cross-section an
201:, rather than a circle, around an axis. A solid torus is a torus plus the
74:
17:
6480:
5831:
4799:
4016:
solid tori subsets with the aforesaid flat torus surface as their common
3995:
3991:
3776:
3362:
3274:
3140:
2898:
2894:
2882:
2848:
2030:
2000:
1117:
corresponds to the horn torus, which in effect is a torus with no "hole".
331:
229:
218:
214:
83:
133:
does not touch the circle, the surface has a ring shape and is called a
5535:
4353:
3168:
142:
4193:
having this partitioning property include the square tori of the form
3936:{\displaystyle (x,y,z)=((R+P\sin v)\cos u,(R+P\sin v)\sin u,P\cos v).}
3212:
ordered, not necessarily distinct points on the circle. Symbolically,
2885:
which is a torus of the largest possible dimension. Such maximal tori
355:
319:
in the plane with itself. This produces a geometric object called the
6388:
5640:
5610:
5300:
of symbols from an alphabet (often just 0 and 1) that contains every
4499:
4488:
2519:.) Recalling that the torus is the product space of two circles, the
2012:
1304:
241:
210:
202:
169:
164:
107:
67:
2821:(with the action being taken as vector addition). Equivalently, the
1880:
As a torus is the product of two circles, a modified version of the
1129:
describes the self-intersecting spindle torus; its inner shell is a
37:
This article is about the mathematical surface. For the volume, see
6166:
5246:
4577:
4255:
3400:
3384:
3376:
3161:
3134:
2471:
2260:
451:
Poloidal direction (red arrow) and toroidal direction (blue arrow)
446:
359:
is a Latin word for "a round, swelling, elevation, protuberance".
222:
176:
Real-world objects that approximate a torus of revolution include
73:
57:
49:
167:. If the revolved curve is not a circle, the surface is called a
4689:(or the subgroup of diffeomorphisms) of the torus is studied in
2873:. This is due in part to the fact that in any compact Lie group
2457:. In fact, the conformal type of the torus is determined by the
6342:
3397:
can be projected into 3-dimensions and rotated on a fixed axis.
4644:
2862:
is a compact abelian Lie group (when identified with the unit
2036:
of radius √2. This topological torus is also often called the
1929:
5768:. Geometry and Computing. Vol. 9. Springer, Heidelberg.
1888:, the distance from the center of the coordinate system, and
1105:, the surface will be the familiar ring torus or anchor ring.
288:, but another way to do this is the Cartesian product of the
5931:
https://math.uchicago.edu/~may/REU2014/REUPapers/Padgett.pdf
4498:
As examples, a genus zero surface (without boundary) is the
2480:
in four dimensions performing a simple rotation through the
5500:
3954:
in the above flat torus parametrization form a unit vector
3771:
are positive constants determining the aspect ratio. It is
3753:{\displaystyle (x,y,z,w)=(R\cos u,R\sin u,P\cos v,P\sin v)}
1517:
Expressing the surface area and the volume by the distance
5765:
5498:
1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, ... (sequence
2507:
for short. (This is the more typical meaning of the term "
5910:"Applications of the Clifford torus to material textures"
5494:= 0, not covered by the above formulas), are as follows:
4615:= 0. For any number of holes, the formula generalizes to
4427:
The other two cusps occur at the points corresponding in
2911:
are easily constructed from automorphisms of the lattice
2488:
The torus has a generalization to higher dimensions, the
6070:"Doc Madhattan: A flat torus in three dimensional space"
4810:, as these are all naturally isomorphic; also the first
3358:
corresponds to points with all 3 coordinates identical.
2869:
Toroidal groups play an important part in the theory of
1503:
These formulas are the same as for a cylinder of length
205:
inside the torus. Real-world objects that approximate a
6369:
Compact topological surfaces and their immersions in 3D
4661:
3262:{\displaystyle \mathbb {T} ^{n}=(\mathbb {S} ^{1})^{n}}
2889:
have a controlling role to play in theory of connected
2248:
by pasting the opposite edges together, described as a
54:
A ring torus with a selection of circles on its surface
6280:
Fly-through cross-sections of a four-dimensional torus
5418:
5387:
5346:
4012:
above, the torus will partition the 3-sphere into two
2752:
under integral shifts in any coordinate. That is, the
1803:
1198:
827:
5340:
5196:
5165:
5018:
4986:
4827:
4772:
4743:
4703:
4411:
may be constructed by glueing together two congruent
4329:
4292:
4265:
4211:
4033:
3815:
3656:
3564:
3534:
3496:
3467:
3430:
3282:
3273:, not necessarily distinct points is accordingly the
3218:
3094:
3050:
2957:
2917:
2798:
2762:
2729:
2700:
2667:
2661:. The torus discussed above is the standard 2-torus,
2623:
2536:
2286:
2163:
2116:
2078:
1587:
1321:
1196:
939:
825:
801:
756:
733:
710:
657:
465:
298:
250:
2277:
of the fundamental group of the circle with itself:
6581:
6553:
6518:
6509:
6455:
6410:
6381:
6374:
6191:. Math.univ-lyon1.fr. 18 April 2012. Archived from
5009:gives a splitting, via the linear maps, as above):
4730:{\displaystyle \operatorname {GL} (n,\mathbb {Z} )}
5735:Encyclopédie des Formes Mathématiques Remarquables
5472:
5228:
5182:
5125:
5001:
4951:
4787:
4758:
4729:
4344:
4307:
4278:
4226:
4178:
3935:
3752:
3596:
3549:
3511:
3482:
3445:
3314:
3261:
3109:
3065:
2972:
2932:
2813:
2777:
2744:
2715:
2682:
2653:
2606:
2445:The 2-torus double-covers the 2-sphere, with four
2398:
2233:
2131:
2093:
1870:
1493:
1283:
1073:
915:
807:
770:
739:
716:
693:
641:
311:
276:
6286:Visualizing high dimensional data with flat torus
3597:{\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}}
3315:{\displaystyle \mathbb {T} ^{n}/\mathbb {S} _{n}}
2654:{\displaystyle \mathbb {T} ^{1}=\mathbb {S} ^{1}}
5679:Nociones de Geometría Analítica y Álgebra Lineal
4458:there is a more general family of objects, the "
2234:{\displaystyle (x,y)\sim (x+1,y)\sim (x,y+1),\,}
1957:but its sources remain unclear because it lacks
6074:Proceedings of the National Academy of Sciences
5315:, which can be considered a special case where
4249:, which was proven in the 1950s, an isometric
694:{\displaystyle \theta ,\varphi \in [0,2\pi ),}
6354:
3519:. This gives the quotient the structure of a
1525:of an innermost point to the center (so that
1242:
1201:
1167:, the torus degenerates to the circle radius
1151:, the torus degenerates to the sphere radius
871:
830:
8:
4487:doughnuts stuck together side by side, or a
3322:, which is the quotient of the torus by the
2429:to the fundamental group (this follows from
6309:"Topology of a Twisted Torus – Numberphile"
6515:
6378:
6361:
6347:
6339:
5327:A solid torus of revolution can be cut by
5229:{\displaystyle \chi ({\mathsf {K_{7}}})=7}
5148:is seven, meaning every graph that can be
114:one full revolution about an axis that is
6216:"The Tortuous Geometry of the Flat Torus"
6095:
6085:
5998:
5452:
5436:
5417:
5382:
5341:
5339:
5209:
5204:
5203:
5195:
5172:
5167:
5166:
5164:
5108:
5104:
5103:
5090:
5074:
5070:
5069:
5047:
5043:
5042:
5029:
5017:
4993:
4989:
4988:
4985:
4939:
4938:
4911:
4907:
4906:
4875:
4850:
4846:
4845:
4832:
4826:
4779:
4775:
4774:
4771:
4750:
4746:
4745:
4742:
4720:
4719:
4702:
4376:guarantees that every Riemann surface is
4336:
4332:
4331:
4328:
4299:
4295:
4294:
4291:
4270:
4264:
4218:
4214:
4213:
4210:
4158:
4149:
4136:
4116:
4107:
4094:
4081:
4077:
4076:
4032:
3814:
3655:
3588:
3584:
3583:
3577:
3571:
3567:
3566:
3563:
3541:
3537:
3536:
3533:
3503:
3499:
3498:
3495:
3474:
3470:
3469:
3466:
3437:
3433:
3432:
3429:
3306:
3302:
3301:
3295:
3289:
3285:
3284:
3281:
3253:
3243:
3239:
3238:
3225:
3221:
3220:
3217:
3172:is an example of a torus in music theory.
3101:
3097:
3096:
3093:
3057:
3053:
3052:
3049:
2964:
2960:
2959:
2956:
2924:
2920:
2919:
2916:
2840:-torus in this sense is an example of an
2805:
2801:
2800:
2797:
2769:
2765:
2764:
2761:
2736:
2732:
2731:
2728:
2707:
2703:
2702:
2699:
2674:
2670:
2669:
2666:
2645:
2641:
2640:
2630:
2626:
2625:
2622:
2617:The standard 1-torus is just the circle:
2595:
2583:
2579:
2578:
2562:
2558:
2557:
2553:
2543:
2539:
2538:
2535:
2389:
2388:
2381:
2380:
2368:
2364:
2363:
2353:
2337:
2333:
2332:
2322:
2306:
2302:
2301:
2291:
2285:
2244:or, equivalently, as the quotient of the
2230:
2162:
2123:
2119:
2118:
2115:
2085:
2081:
2080:
2077:
1988:Learn how and when to remove this message
1896:, angles measured from the center point.
1855:
1818:
1802:
1793:
1771:
1744:
1734:
1674:
1645:
1619:
1609:
1588:
1586:
1478:
1465:
1425:
1384:
1322:
1320:
1275:
1262:
1247:
1241:
1240:
1225:
1212:
1206:
1200:
1199:
1197:
1195:
1057:
1044:
1029:
1013:
1002:
989:
976:
963:
950:
938:
904:
891:
876:
870:
869:
854:
841:
835:
829:
828:
826:
824:
800:
795:for a torus radially symmetric about the
760:
755:
732:
709:
656:
589:
525:
466:
464:
303:
297:
268:
255:
249:
5265:
4538:
3330:letters (by permuting the coordinates).
2409:Intuitively speaking, this means that a
6068:Filippelli, Gianluigi (27 April 2012).
5747:
5486:The first 11 numbers of parts, for 0 ≤
4286:isometric embedding of a flat torus in
3405:The simplest tiling of a flat torus is
3361:These orbifolds have found significant
3015:-torus is a free abelian group of rank
2833:by gluing the opposite faces together.
2067:The surface described above, given the
1311:of its torus are easily computed using
5210:
5206:
5173:
5169:
4205:is a rotation of 4-dimensional space
3117:whose generators are the duals of the
2980:in the usual way, one has the typical
2858:. This follows from the fact that the
191:A torus should not be confused with a
5838:from the original on 13 December 2014
4364:Conformal classification of flat tori
3990:/2 parameterize the unit 3-sphere as
3644:conditions are given up, see below).
3604:, which can also be described as the
3175:The Tonnetz is only truly a torus if
2851:. It is also an example of a compact
2523:-dimensional torus is the product of
2146:The torus can also be described as a
27:Doughnut-shaped surface of revolution
7:
5698:. Cambridge University Press, 2002.
4238:is a member of the Lie group SO(4).
2265:Turning a punctured torus inside-out
6323:Anders Sandberg (4 February 2014).
4766:that preserve the standard lattice
4398:may be turned into a compact space
91:
6290:Polydoes, doughnut-shaped polygons
6131:Enrico de Lazaro (18 April 2012).
6034:from the original on 25 July 2011.
5917:Journal of Applied Crystallography
5812:from the original on 29 April 2012
5806:"Equations for the Standard Torus"
5709:V. V. Nikulin, I. R. Shafarevich.
5190:can be embedded on the torus, and
3412:, constructed on the surface of a
2893:. Toroidal groups are examples of
2723:can be described as a quotient of
2690:. And similar to the 2-torus, the
1918:are more commonly used to discuss
25:
6112:from the original on 25 June 2012
5183:{\displaystyle {\mathsf {K_{7}}}}
4483:surface resembles the surface of
4241:It is known that there exists no
3200:-fold product of the circle, the
2023:. This can be viewed as lying in
440:: self-intersecting spindle torus
277:{\displaystyle S^{1}\times S^{1}}
6143:from the original on 1 June 2012
5970:"The Geometry of Musical Chords"
5968:Tymoczko, Dmitri (7 July 2006).
5887:Oxford English Dictionary Online
5519:
5002:{\displaystyle \mathbb {R} ^{n}}
4788:{\displaystyle \mathbb {Z} ^{n}}
4759:{\displaystyle \mathbb {R} ^{n}}
4648:
4553:
4542:
4345:{\displaystyle \mathbb {R} ^{3}}
4308:{\displaystyle \mathbb {R} ^{3}}
4227:{\displaystyle \mathbb {R} ^{4}}
3550:{\displaystyle \mathbb {Z} ^{2}}
3512:{\displaystyle \mathbb {Z} ^{2}}
3483:{\displaystyle \mathbb {R} ^{2}}
3446:{\displaystyle \mathbb {R} ^{2}}
3110:{\displaystyle \mathbb {Z} ^{n}}
3066:{\displaystyle \mathbb {T} ^{n}}
2973:{\displaystyle \mathbb {R} ^{n}}
2933:{\displaystyle \mathbb {Z} ^{n}}
2814:{\displaystyle \mathbb {Z} ^{n}}
2778:{\displaystyle \mathbb {R} ^{n}}
2745:{\displaystyle \mathbb {R} ^{n}}
2716:{\displaystyle \mathbb {T} ^{n}}
2683:{\displaystyle \mathbb {T} ^{2}}
2511:-torus", the other referring to
2476:A stereographic projection of a
2132:{\displaystyle \mathbb {R} ^{3}}
2094:{\displaystyle \mathbb {R} ^{3}}
1934:
1295:(and, hence, homeomorphic) to a
424:
401:
378:
197:, which is formed by rotating a
145:to the circle, the surface is a
6189:"Flat tori finally visualized!"
6045:Phillips, Tony (October 2006).
5908:De Graef, Marc (7 March 2024).
5733:"Tore (notion géométrique)" at
4525:for surfaces states that every
2433:since the fundamental group is
141:. If the axis of revolution is
5808:. Geom.uiuc.edu. 6 July 1995.
5467:
5429:
5217:
5200:
5117:
5114:
5099:
5083:
5080:
5065:
5056:
5053:
5038:
5022:
4943:
4929:
4917:
4902:
4890:
4887:
4881:
4868:
4856:
4841:
4724:
4710:
4069:
4045:
3927:
3900:
3879:
3864:
3843:
3840:
3834:
3816:
3747:
3687:
3681:
3657:
3250:
3234:
2374:
2359:
2343:
2328:
2312:
2297:
2224:
2206:
2200:
2182:
2176:
2164:
1852:
1839:
1836:
1824:
1707:
1695:
1692:
1680:
925:Algebraically eliminating the
685:
670:
613:
601:
580:
559:
549:
537:
516:
495:
485:
473:
1:
6051:American Mathematical Society
5576:Irrational winding of a torus
5331:(> 0) planes into at most
5156:of at most seven. (Since the
3269:. The configuration space of
3077:) can be identified with the
6284:"Relational Perspective Map"
5490:≤ 10 (including the case of
4540:
4518:are also occasionally used.
3363:applications to music theory
2825:-torus is obtained from the
2110:the topological torus into
2108:stereographically projecting
1307:of this solid torus and the
6047:"Take on Math in the Media"
5319:is 1 (one dimension).
4319:In April 2012, an explicit
4020:. One example is the torus
3775:to a regular torus but not
1920:magnetic confinement fusion
1882:spherical coordinate system
747:is the radius of the tube.
394:: ring torus or anchor ring
6650:
5591:Loewner's torus inequality
5259:
4571:
4443:
3798:) into Euclidean 3-space.
3608:under the identifications
3461:is a discrete subgroup of
3123:
2940:, which are classified by
2154:under the identifications
1907:is moved to the center of
1135:and its outer shell is an
651:using angular coordinates
36:
29:
6214:Hoang, Lê Nguyên (2016).
5889:. Oxford University Press
5774:10.1007/978-3-642-34364-3
5275:model of de Bruijn torus
4506:-holed tori (or, rarely,
4380:to one that has constant
3337:= 2, the quotient is the
3187:segment of the left edge.
3143:quotient of the 2-torus,
2273:of the torus is just the
1313:Pappus's centroid theorem
173:, as in a square toroid.
106:generated by revolving a
5651:Torus-based cryptography
4635:is the number of holes.
3391:stereographic projection
3179:is assumed, so that the
1943:This section includes a
30:Not to be confused with
6500:Sphere with three holes
6087:10.1073/pnas.1118478109
6009:10.1126/science.1126287
4964:Eilenberg–MacLane space
4642:of toroidal polyhedra.
4510:-fold tori). The terms
4424:has area equal to π/3.
2139:from the north pole of
2029:and is a subset of the
1972:more precise citations.
372:vertical cross-sections
161:self-intersecting torus
112:three-dimensional space
5474:
5285:
5252:
5230:
5184:
5127:
5003:
4962:Since the torus is an
4953:
4789:
4760:
4731:
4594:
4523:classification theorem
4378:conformally equivalent
4374:Uniformization theorem
4346:
4316:
4309:
4280:
4228:
4180:
3937:
3754:
3598:
3551:
3513:
3484:
3447:
3417:
3398:
3382:
3316:
3263:
3189:
3177:enharmonic equivalence
3159:
3111:
3067:
3025:. It follows that the
2974:
2934:
2877:one can always find a
2815:
2779:
2746:
2717:
2684:
2655:
2608:
2485:
2400:
2266:
2235:
2133:
2095:
1872:
1495:
1285:
1075:
917:
809:
772:
741:
718:
695:
643:
452:
313:
278:
79:
71:
66:into a double-covered
55:
43:Torus (disambiguation)
41:. For other uses, see
6418:Real projective plane
6403:Pretzel (genus 3) ...
5948:mathworld.wolfram.com
5711:Geometries and Groups
5646:Toroidal and poloidal
5606:Real projective plane
5475:
5271:
5250:
5231:
5185:
5150:embedded on the torus
5128:
5004:
4954:
4790:
4761:
4732:
4581:
4572:Further information:
4347:
4310:
4281:
4279:{\displaystyle C^{1}}
4259:
4229:
4181:
3938:
3755:
3599:
3552:
3514:
3485:
3448:
3404:
3388:
3380:
3353:; equivalently, as a
3317:
3264:
3165:
3138:
3112:
3068:
2975:
2935:
2816:
2780:
2747:
2718:
2685:
2656:
2609:
2475:
2401:
2264:
2236:
2134:
2096:
1916:toroidal and poloidal
1873:
1496:
1286:
1076:
918:
810:
793:Cartesian coordinates
773:
742:
719:
696:
644:
450:
314:
312:{\displaystyle S^{1}}
279:
104:surface of revolution
77:
61:
53:
6573:Euler characteristic
5581:Joint European Torus
5338:
5194:
5163:
5016:
4984:
4978:short exact sequence
4825:
4770:
4741:
4701:
4602:Euler characteristic
4327:
4290:
4263:
4234:, or in other words
4209:
4031:
3813:
3654:
3562:
3532:
3494:
3465:
3428:
3347:equilateral triangle
3280:
3216:
3092:
3048:
3033:-torus is 0 for all
3027:Euler characteristic
2955:
2915:
2881:; that is, a closed
2796:
2760:
2727:
2698:
2665:
2621:
2534:
2461:of the four points.
2284:
2161:
2114:
2076:
1585:
1319:
1194:
937:
823:
799:
754:
731:
708:
655:
463:
296:
248:
6307:(27 January 2014).
6269:Creation of a torus
5991:2006Sci...313...72T
5942:Weisstein, Eric W.
4695:mapping class group
4687:homeomorphism group
4584:toroidal polyhedron
4574:Toroidal polyhedron
4315:, with corrugations
4247:Nash-Kuiper theorem
3521:Riemannian manifold
3206:configuration space
3131:Configuration space
3126:Quasitoric manifold
3121:nontrivial cycles.
2496:, often called the
2492:n-dimensional torus
2451:conformal structure
2447:ramification points
2250:fundamental polygon
1301:Euclidean open disk
771:{\displaystyle R/r}
157:self-crossing torus
135:torus of revolution
6400:Number 8 (genus 2)
6242:Weisstein, Eric W.
6057:on 5 October 2008.
5859:Weisstein, Eric W.
5713:. Springer, 1987.
5695:Algebraic Topology
5666:Villarceau circles
5621:Surface (topology)
5551:Annulus (geometry)
5527:Mathematics portal
5470:
5427:
5408:
5373:
5313:De Bruijn sequence
5286:
5253:
5238:four color theorem
5226:
5180:
5123:
4999:
4949:
4785:
4756:
4727:
4691:geometric topology
4660:. You can help by
4595:
4568:Toroidal polyhedra
4495:handles attached.
4466:surfaces. A genus
4413:geodesic triangles
4382:Gaussian curvature
4342:
4317:
4305:
4276:
4224:
4176:
3933:
3750:
3641:Gaussian curvature
3594:
3547:
3509:
3480:
3443:
3418:
3399:
3383:
3312:
3259:
3190:
3160:
3107:
3063:
2997:free abelian group
2982:toral automorphism
2970:
2930:
2871:compact Lie groups
2811:
2775:
2742:
2713:
2680:
2651:
2604:
2600:
2593:
2527:circles. That is:
2515:holes or of genus
2486:
2468:-dimensional torus
2455:Weierstrass points
2396:
2267:
2231:
2129:
2091:
1945:list of references
1868:
1866:
1812:
1491:
1489:
1303:and a circle. The
1281:
1253:
1071:
913:
882:
805:
768:
737:
714:
691:
639:
637:
453:
309:
274:
232:, a ring torus is
137:, also known as a
131:axis of revolution
80:
72:
56:
6621:
6620:
6617:
6616:
6451:
6450:
6080:(19): 7218–7223.
5783:978-3-642-34363-6
5727:978-3-540-15281-1
5687:978-970-10-6596-9
5426:
4808:fundamental group
4678:
4677:
4565:
4564:
4534:projective planes
4454:In the theory of
4166:
4131:
4124:
3188:
2989:fundamental group
2984:on the quotient.
2945:integral matrices
2554:
2552:
2441:Two-sheeted cover
2271:fundamental group
2069:relative topology
1998:
1997:
1990:
1811:
1787:
1760:
1661:
1635:
1291:of this torus is
1231:
860:
808:{\displaystyle z}
740:{\displaystyle r}
717:{\displaystyle R}
370:Bottom-halves and
238:Cartesian product
213:, non-inflatable
16:(Redirected from
6641:
6536:Triangulatedness
6516:
6379:
6375:Without boundary
6363:
6356:
6349:
6340:
6335:
6333:
6331:
6319:
6313:
6256:
6255:
6254:
6237:
6231:
6230:
6228:
6226:
6211:
6205:
6204:
6202:
6200:
6185:
6179:
6178:
6176:
6174:
6165:. Archived from
6159:
6153:
6152:
6150:
6148:
6128:
6122:
6121:
6119:
6117:
6099:
6089:
6065:
6059:
6058:
6053:. Archived from
6042:
6036:
6035:
6033:
6002:
5974:
5965:
5959:
5958:
5956:
5954:
5939:
5933:
5927:
5921:
5920:
5914:
5905:
5899:
5898:
5896:
5894:
5879:
5873:
5872:
5871:
5854:
5848:
5847:
5845:
5843:
5834:. Spatial Corp.
5828:
5822:
5821:
5819:
5817:
5802:
5796:
5795:
5752:
5529:
5524:
5523:
5503:
5479:
5477:
5476:
5471:
5457:
5456:
5441:
5440:
5428:
5419:
5413:
5412:
5378:
5377:
5282:
5270:
5235:
5233:
5232:
5227:
5216:
5215:
5214:
5213:
5189:
5187:
5186:
5181:
5179:
5178:
5177:
5176:
5154:chromatic number
5140:Coloring a torus
5132:
5130:
5129:
5124:
5113:
5112:
5107:
5095:
5094:
5079:
5078:
5073:
5052:
5051:
5046:
5034:
5033:
5008:
5006:
5005:
5000:
4998:
4997:
4992:
4958:
4956:
4955:
4950:
4942:
4916:
4915:
4910:
4880:
4879:
4855:
4854:
4849:
4837:
4836:
4812:cohomology group
4798:At the level of
4794:
4792:
4791:
4786:
4784:
4783:
4778:
4765:
4763:
4762:
4757:
4755:
4754:
4749:
4736:
4734:
4733:
4728:
4723:
4673:
4670:
4652:
4645:
4589:
4557:
4546:
4539:
4417:hyperbolic plane
4370:Riemann surfaces
4368:In the study of
4351:
4349:
4348:
4343:
4341:
4340:
4335:
4314:
4312:
4311:
4306:
4304:
4303:
4298:
4285:
4283:
4282:
4277:
4275:
4274:
4233:
4231:
4230:
4225:
4223:
4222:
4217:
4185:
4183:
4182:
4177:
4172:
4168:
4167:
4159:
4154:
4153:
4141:
4140:
4129:
4125:
4117:
4112:
4111:
4099:
4098:
4086:
4085:
4080:
4011:
4009:
3992:Hopf coordinates
3989:
3973:
3942:
3940:
3939:
3934:
3797:
3779:. It can not be
3759:
3757:
3756:
3751:
3635:
3603:
3601:
3600:
3595:
3593:
3592:
3587:
3581:
3576:
3575:
3570:
3557:
3556:
3554:
3553:
3548:
3546:
3545:
3540:
3518:
3516:
3515:
3510:
3508:
3507:
3502:
3489:
3487:
3486:
3481:
3479:
3478:
3473:
3452:
3450:
3449:
3444:
3442:
3441:
3436:
3355:triangular prism
3321:
3319:
3318:
3313:
3311:
3310:
3305:
3299:
3294:
3293:
3288:
3268:
3266:
3265:
3260:
3258:
3257:
3248:
3247:
3242:
3230:
3229:
3224:
3186:
3182:
3174:
3116:
3114:
3113:
3108:
3106:
3105:
3100:
3079:exterior algebra
3072:
3070:
3069:
3064:
3062:
3061:
3056:
2979:
2977:
2976:
2971:
2969:
2968:
2963:
2939:
2937:
2936:
2931:
2929:
2928:
2923:
2820:
2818:
2817:
2812:
2810:
2809:
2804:
2784:
2782:
2781:
2776:
2774:
2773:
2768:
2751:
2749:
2748:
2743:
2741:
2740:
2735:
2722:
2720:
2719:
2714:
2712:
2711:
2706:
2689:
2687:
2686:
2681:
2679:
2678:
2673:
2660:
2658:
2657:
2652:
2650:
2649:
2644:
2635:
2634:
2629:
2613:
2611:
2610:
2605:
2599:
2594:
2589:
2588:
2587:
2582:
2567:
2566:
2561:
2548:
2547:
2542:
2494:
2493:
2431:Hurewicz theorem
2425:of the torus is
2405:
2403:
2402:
2397:
2392:
2384:
2373:
2372:
2367:
2358:
2357:
2342:
2341:
2336:
2327:
2326:
2311:
2310:
2305:
2296:
2295:
2240:
2238:
2237:
2232:
2138:
2136:
2135:
2130:
2128:
2127:
2122:
2100:
2098:
2097:
2092:
2090:
2089:
2084:
1993:
1986:
1982:
1979:
1973:
1968:this section by
1959:inline citations
1938:
1937:
1930:
1910:
1906:
1902:
1895:
1891:
1887:
1877:
1875:
1874:
1869:
1867:
1860:
1859:
1823:
1822:
1813:
1804:
1798:
1797:
1792:
1788:
1783:
1772:
1765:
1761:
1756:
1745:
1739:
1738:
1679:
1678:
1666:
1662:
1657:
1646:
1640:
1636:
1631:
1620:
1614:
1613:
1580:
1579:
1577:
1576:
1573:
1570:
1552:
1551:
1549:
1548:
1545:
1542:
1524:
1520:
1513:
1509:
1500:
1498:
1497:
1492:
1490:
1483:
1482:
1470:
1469:
1454:
1450:
1435:
1431:
1430:
1429:
1389:
1388:
1373:
1369:
1354:
1350:
1290:
1288:
1287:
1282:
1280:
1279:
1267:
1266:
1254:
1252:
1251:
1246:
1245:
1232:
1230:
1229:
1217:
1216:
1207:
1205:
1204:
1186:
1172:
1166:
1156:
1150:
1128:
1116:
1104:
1090:
1086:
1080:
1078:
1077:
1072:
1067:
1063:
1062:
1061:
1049:
1048:
1034:
1033:
1018:
1017:
1012:
1008:
1007:
1006:
994:
993:
981:
980:
968:
967:
955:
954:
931:quartic equation
922:
920:
919:
914:
909:
908:
896:
895:
883:
881:
880:
875:
874:
861:
859:
858:
846:
845:
836:
834:
833:
814:
812:
811:
806:
777:
775:
774:
769:
764:
746:
744:
743:
738:
723:
721:
720:
715:
700:
698:
697:
692:
648:
646:
645:
640:
638:
593:
529:
439:
428:
416:
405:
393:
382:
330:In the field of
318:
316:
315:
310:
308:
307:
283:
281:
280:
275:
273:
272:
260:
259:
93:
21:
6649:
6648:
6644:
6643:
6642:
6640:
6639:
6638:
6624:
6623:
6622:
6613:
6577:
6554:Characteristics
6549:
6511:
6505:
6447:
6406:
6370:
6367:
6329:
6327:
6322:
6311:
6305:Séquin, Carlo H
6303:
6300:Wayback Machine
6265:
6260:
6259:
6245:"Torus Cutting"
6240:
6239:
6238:
6234:
6224:
6222:
6213:
6212:
6208:
6198:
6196:
6195:on 18 June 2012
6187:
6186:
6182:
6172:
6170:
6161:
6160:
6156:
6146:
6144:
6130:
6129:
6125:
6115:
6113:
6067:
6066:
6062:
6044:
6043:
6039:
6031:
6000:10.1.1.215.7449
5985:(5783): 72–74.
5972:
5967:
5966:
5962:
5952:
5950:
5941:
5940:
5936:
5928:
5924:
5912:
5907:
5906:
5902:
5892:
5890:
5881:
5880:
5876:
5857:
5856:
5855:
5851:
5841:
5839:
5830:
5829:
5825:
5815:
5813:
5804:
5803:
5799:
5784:
5754:
5753:
5749:
5744:
5692:Allen Hatcher.
5675:
5670:
5541:Algebraic torus
5525:
5518:
5515:
5499:
5448:
5432:
5407:
5406:
5394:
5393:
5383:
5372:
5371:
5359:
5358:
5342:
5336:
5335:
5325:
5323:Cutting a torus
5294:de Bruijn torus
5292:mathematics, a
5281:
5277:
5266:
5264:
5262:de Bruijn torus
5258:
5256:de Bruijn torus
5205:
5192:
5191:
5168:
5161:
5160:
5142:
5102:
5086:
5068:
5041:
5025:
5014:
5013:
4987:
4982:
4981:
4905:
4871:
4844:
4828:
4823:
4822:
4773:
4768:
4767:
4744:
4739:
4738:
4699:
4698:
4683:
4674:
4668:
4665:
4658:needs expansion
4587:
4576:
4570:
4558:
4547:
4470:surface is the
4452:
4442:
4366:
4358:surface normals
4330:
4325:
4324:
4293:
4288:
4287:
4266:
4261:
4260:
4212:
4207:
4206:
4145:
4132:
4103:
4090:
4075:
4044:
4040:
4029:
4028:
4007:
4002:
3987:
3955:
3811:
3810:
3788:
3652:
3651:
3609:
3606:Cartesian plane
3582:
3565:
3560:
3559:
3535:
3530:
3529:
3524:
3497:
3492:
3491:
3468:
3463:
3462:
3431:
3426:
3425:
3410:
3375:
3324:symmetric group
3300:
3283:
3278:
3277:
3249:
3237:
3219:
3214:
3213:
3184:
3180:
3173:
3154:, which is the
3153:
3133:
3128:
3095:
3090:
3089:
3051:
3046:
3045:
3039:cohomology ring
2958:
2953:
2952:
2918:
2913:
2912:
2864:complex numbers
2799:
2794:
2793:
2789:of the integer
2763:
2758:
2757:
2730:
2725:
2724:
2701:
2696:
2695:
2668:
2663:
2662:
2639:
2624:
2619:
2618:
2577:
2556:
2555:
2537:
2532:
2531:
2491:
2490:
2470:
2443:
2362:
2349:
2331:
2318:
2300:
2287:
2282:
2281:
2159:
2158:
2152:Cartesian plane
2117:
2112:
2111:
2079:
2074:
2073:
2007:defined as the
2003:, a torus is a
1994:
1983:
1977:
1974:
1963:
1949:related reading
1939:
1935:
1928:
1914:In modern use,
1908:
1904:
1900:
1893:
1889:
1885:
1865:
1864:
1851:
1814:
1773:
1767:
1766:
1746:
1740:
1730:
1720:
1714:
1713:
1670:
1647:
1641:
1621:
1615:
1605:
1595:
1583:
1582:
1574:
1571:
1562:
1561:
1559:
1554:
1546:
1543:
1534:
1533:
1531:
1526:
1522:
1518:
1511:
1504:
1488:
1487:
1474:
1461:
1440:
1436:
1421:
1417:
1413:
1406:
1400:
1399:
1380:
1359:
1355:
1340:
1336:
1329:
1317:
1316:
1271:
1258:
1239:
1221:
1208:
1192:
1191:
1178:
1168:
1161:
1152:
1145:
1120:
1108:
1096:
1088:
1084:
1053:
1040:
1039:
1035:
1025:
998:
985:
972:
959:
946:
945:
941:
940:
935:
934:
900:
887:
868:
850:
837:
821:
820:
797:
796:
752:
751:
729:
728:
706:
705:
653:
652:
636:
635:
616:
595:
594:
552:
531:
530:
488:
461:
460:
455:A torus can be
445:
444:
443:
442:
441:
431:
429:
420:
419:
418:
408:
406:
397:
396:
395:
385:
383:
374:
373:
371:
365:
352:
323:, a surface in
299:
294:
293:
286:Euclidean space
264:
251:
246:
245:
46:
35:
28:
23:
22:
15:
12:
11:
5:
6647:
6645:
6637:
6636:
6626:
6625:
6619:
6618:
6615:
6614:
6612:
6611:
6606:
6600:
6594:
6591:
6585:
6583:
6579:
6578:
6576:
6575:
6570:
6565:
6557:
6555:
6551:
6550:
6548:
6547:
6542:
6533:
6528:
6522:
6520:
6513:
6507:
6506:
6504:
6503:
6497:
6496:
6495:
6485:
6484:
6483:
6478:
6470:
6469:
6468:
6459:
6457:
6453:
6452:
6449:
6448:
6446:
6445:
6442:Dyck's surface
6439:
6433:
6432:
6431:
6426:
6414:
6412:
6411:Non-orientable
6408:
6407:
6405:
6404:
6401:
6398:
6392:
6385:
6383:
6376:
6372:
6371:
6368:
6366:
6365:
6358:
6351:
6343:
6337:
6336:
6320:
6292:
6287:
6281:
6275:
6264:
6263:External links
6261:
6258:
6257:
6232:
6206:
6180:
6169:on 5 July 2012
6154:
6123:
6060:
6037:
5960:
5934:
5922:
5900:
5874:
5849:
5823:
5797:
5782:
5746:
5745:
5743:
5740:
5739:
5738:
5730:
5707:
5690:
5674:
5671:
5669:
5668:
5663:
5658:
5653:
5648:
5643:
5638:
5633:
5628:
5623:
5618:
5616:Spiric section
5613:
5608:
5603:
5601:Period lattice
5598:
5593:
5588:
5583:
5578:
5573:
5571:Elliptic curve
5568:
5563:
5558:
5556:Clifford torus
5553:
5548:
5546:Angenent torus
5543:
5538:
5532:
5531:
5530:
5514:
5511:
5510:
5509:
5481:
5480:
5469:
5466:
5463:
5460:
5455:
5451:
5447:
5444:
5439:
5435:
5431:
5425:
5422:
5416:
5411:
5405:
5402:
5399:
5396:
5395:
5392:
5389:
5388:
5386:
5381:
5376:
5370:
5367:
5364:
5361:
5360:
5357:
5354:
5351:
5348:
5347:
5345:
5324:
5321:
5279:
5260:Main article:
5257:
5254:
5225:
5222:
5219:
5212:
5208:
5202:
5199:
5175:
5171:
5158:complete graph
5146:Heawood number
5141:
5138:
5134:
5133:
5122:
5119:
5116:
5111:
5106:
5101:
5098:
5093:
5089:
5085:
5082:
5077:
5072:
5067:
5064:
5061:
5058:
5055:
5050:
5045:
5040:
5037:
5032:
5028:
5024:
5021:
4996:
4991:
4960:
4959:
4948:
4945:
4941:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4914:
4909:
4904:
4901:
4898:
4895:
4892:
4889:
4886:
4883:
4878:
4874:
4870:
4867:
4864:
4861:
4858:
4853:
4848:
4843:
4840:
4835:
4831:
4814:generates the
4782:
4777:
4753:
4748:
4726:
4722:
4718:
4715:
4712:
4709:
4706:
4682:
4679:
4676:
4675:
4655:
4653:
4569:
4566:
4563:
4562:
4551:
4444:Main article:
4441:
4434:
4365:
4362:
4339:
4334:
4302:
4297:
4273:
4269:
4221:
4216:
4189:Other tori in
4187:
4186:
4175:
4171:
4165:
4162:
4157:
4152:
4148:
4144:
4139:
4135:
4128:
4123:
4120:
4115:
4110:
4106:
4102:
4097:
4093:
4089:
4084:
4079:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4043:
4039:
4036:
3944:
3943:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3761:
3760:
3749:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3591:
3586:
3580:
3574:
3569:
3544:
3539:
3506:
3501:
3490:isomorphic to
3477:
3472:
3440:
3435:
3408:
3374:
3371:
3367:musical triads
3309:
3304:
3298:
3292:
3287:
3256:
3252:
3246:
3241:
3236:
3233:
3228:
3223:
3204:-torus is the
3196:-torus is the
3151:
3132:
3129:
3104:
3099:
3060:
3055:
3009:homology group
2967:
2962:
2927:
2922:
2808:
2803:
2772:
2767:
2739:
2734:
2710:
2705:
2677:
2672:
2648:
2643:
2638:
2633:
2628:
2615:
2614:
2603:
2598:
2592:
2586:
2581:
2576:
2573:
2570:
2565:
2560:
2551:
2546:
2541:
2506:
2502:
2495:
2478:Clifford torus
2469:
2463:
2442:
2439:
2423:homology group
2407:
2406:
2395:
2391:
2387:
2383:
2379:
2376:
2371:
2366:
2361:
2356:
2352:
2348:
2345:
2340:
2335:
2330:
2325:
2321:
2317:
2314:
2309:
2304:
2299:
2294:
2290:
2275:direct product
2242:
2241:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2126:
2121:
2088:
2083:
2038:Clifford torus
2005:closed surface
1996:
1995:
1953:external links
1942:
1940:
1933:
1927:
1924:
1863:
1858:
1854:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1821:
1817:
1810:
1807:
1801:
1796:
1791:
1786:
1782:
1779:
1776:
1770:
1764:
1759:
1755:
1752:
1749:
1743:
1737:
1733:
1729:
1726:
1723:
1721:
1719:
1716:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1677:
1673:
1669:
1665:
1660:
1656:
1653:
1650:
1644:
1639:
1634:
1630:
1627:
1624:
1618:
1612:
1608:
1604:
1601:
1598:
1596:
1594:
1591:
1590:
1486:
1481:
1477:
1473:
1468:
1464:
1460:
1457:
1453:
1449:
1446:
1443:
1439:
1434:
1428:
1424:
1420:
1416:
1412:
1409:
1407:
1405:
1402:
1401:
1398:
1395:
1392:
1387:
1383:
1379:
1376:
1372:
1368:
1365:
1362:
1358:
1353:
1349:
1346:
1343:
1339:
1335:
1332:
1330:
1328:
1325:
1324:
1278:
1274:
1270:
1265:
1261:
1257:
1250:
1244:
1238:
1235:
1228:
1224:
1220:
1215:
1211:
1203:
1175:
1174:
1158:
1142:
1118:
1106:
1070:
1066:
1060:
1056:
1052:
1047:
1043:
1038:
1032:
1028:
1024:
1021:
1016:
1011:
1005:
1001:
997:
992:
988:
984:
979:
975:
971:
966:
962:
958:
953:
949:
944:
912:
907:
903:
899:
894:
890:
886:
879:
873:
867:
864:
857:
853:
849:
844:
840:
832:
804:
778:is called the
767:
763:
759:
736:
713:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
634:
631:
628:
625:
622:
619:
617:
615:
612:
609:
606:
603:
600:
597:
596:
592:
588:
585:
582:
579:
576:
573:
570:
567:
564:
561:
558:
555:
553:
551:
548:
545:
542:
539:
536:
533:
532:
528:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
489:
487:
484:
481:
478:
475:
472:
469:
468:
430:
423:
422:
421:
407:
400:
399:
398:
384:
377:
376:
375:
369:
368:
367:
366:
364:
361:
351:
348:
321:Clifford torus
306:
302:
271:
267:
263:
258:
254:
186:ringette rings
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6646:
6635:
6632:
6631:
6629:
6610:
6607:
6605:
6601:
6599:
6595:
6593:Making a hole
6592:
6590:
6589:Connected sum
6587:
6586:
6584:
6580:
6574:
6571:
6569:
6566:
6563:
6559:
6558:
6556:
6552:
6546:
6545:Orientability
6543:
6541:
6537:
6534:
6532:
6529:
6527:
6526:Connectedness
6524:
6523:
6521:
6517:
6514:
6508:
6501:
6498:
6494:
6491:
6490:
6489:
6486:
6482:
6479:
6477:
6474:
6473:
6471:
6466:
6465:
6464:
6461:
6460:
6458:
6456:With boundary
6454:
6444:(genus 3) ...
6443:
6440:
6437:
6434:
6430:
6429:Roman surface
6427:
6425:
6424:Boy's surface
6421:
6420:
6419:
6416:
6415:
6413:
6409:
6402:
6399:
6396:
6393:
6390:
6387:
6386:
6384:
6380:
6377:
6373:
6364:
6359:
6357:
6352:
6350:
6345:
6344:
6341:
6326:
6325:"Torus Earth"
6321:
6317:
6310:
6306:
6301:
6297:
6293:
6291:
6288:
6285:
6282:
6279:
6276:
6274:
6270:
6267:
6266:
6262:
6252:
6251:
6246:
6243:
6236:
6233:
6221:
6217:
6210:
6207:
6194:
6190:
6184:
6181:
6168:
6164:
6158:
6155:
6142:
6138:
6134:
6127:
6124:
6111:
6107:
6103:
6098:
6093:
6088:
6083:
6079:
6075:
6071:
6064:
6061:
6056:
6052:
6048:
6041:
6038:
6030:
6026:
6022:
6018:
6014:
6010:
6006:
6001:
5996:
5992:
5988:
5984:
5980:
5979:
5971:
5964:
5961:
5949:
5945:
5938:
5935:
5932:
5926:
5923:
5918:
5911:
5904:
5901:
5888:
5884:
5878:
5875:
5869:
5868:
5863:
5860:
5853:
5850:
5837:
5833:
5827:
5824:
5811:
5807:
5801:
5798:
5793:
5789:
5785:
5779:
5775:
5771:
5767:
5766:
5761:
5757:
5756:Gallier, Jean
5751:
5748:
5741:
5737:
5736:
5731:
5728:
5724:
5720:
5719:3-540-15281-4
5716:
5712:
5708:
5705:
5704:0-521-79540-0
5701:
5697:
5696:
5691:
5688:
5684:
5680:
5677:
5676:
5672:
5667:
5664:
5662:
5661:Umbilic torus
5659:
5657:
5654:
5652:
5649:
5647:
5644:
5642:
5639:
5637:
5636:Toric variety
5634:
5632:
5631:Toric section
5629:
5627:
5624:
5622:
5619:
5617:
5614:
5612:
5609:
5607:
5604:
5602:
5599:
5597:
5596:Maximal torus
5594:
5592:
5589:
5587:
5584:
5582:
5579:
5577:
5574:
5572:
5569:
5567:
5566:Dupin cyclide
5564:
5562:
5561:Complex torus
5559:
5557:
5554:
5552:
5549:
5547:
5544:
5542:
5539:
5537:
5534:
5533:
5528:
5522:
5517:
5512:
5507:
5502:
5497:
5496:
5495:
5493:
5489:
5484:
5464:
5461:
5458:
5453:
5449:
5445:
5442:
5437:
5433:
5423:
5420:
5414:
5409:
5403:
5400:
5397:
5390:
5384:
5379:
5374:
5368:
5365:
5362:
5355:
5352:
5349:
5343:
5334:
5333:
5332:
5330:
5322:
5320:
5318:
5314:
5310:
5307:
5303:
5299:
5295:
5291:
5290:combinatorial
5283:
5274:
5269:
5263:
5255:
5249:
5245:
5243:
5239:
5223:
5220:
5197:
5159:
5155:
5151:
5147:
5139:
5137:
5120:
5109:
5096:
5091:
5087:
5075:
5062:
5059:
5048:
5035:
5030:
5026:
5019:
5012:
5011:
5010:
4994:
4979:
4974:
4972:
4968:
4965:
4946:
4935:
4932:
4926:
4923:
4920:
4912:
4899:
4896:
4893:
4884:
4876:
4872:
4865:
4862:
4859:
4851:
4838:
4833:
4829:
4821:
4820:
4819:
4817:
4813:
4809:
4805:
4801:
4796:
4780:
4751:
4716:
4713:
4707:
4704:
4696:
4692:
4688:
4681:Automorphisms
4680:
4672:
4663:
4659:
4656:This section
4654:
4651:
4647:
4646:
4643:
4641:
4636:
4634:
4630:
4626:
4622:
4618:
4614:
4610:
4606:
4603:
4599:
4592:
4591:quadrilateral
4585:
4580:
4575:
4567:
4561:
4556:
4552:
4550:
4545:
4541:
4537:
4535:
4531:
4528:
4524:
4519:
4517:
4513:
4509:
4505:
4501:
4496:
4494:
4490:
4486:
4482:
4477:
4473:
4472:connected sum
4469:
4465:
4461:
4457:
4451:
4449:
4439:
4435:
4433:
4430:
4425:
4423:
4418:
4414:
4410:
4406:
4401:
4397:
4393:
4391:
4387:
4383:
4379:
4375:
4371:
4363:
4361:
4359:
4355:
4337:
4322:
4300:
4271:
4267:
4258:
4254:
4252:
4248:
4244:
4239:
4237:
4219:
4204:
4200:
4196:
4192:
4173:
4169:
4163:
4160:
4155:
4150:
4146:
4142:
4137:
4133:
4126:
4121:
4118:
4113:
4108:
4104:
4100:
4095:
4091:
4087:
4082:
4072:
4066:
4063:
4060:
4057:
4054:
4051:
4048:
4041:
4037:
4034:
4027:
4026:
4025:
4023:
4019:
4015:
4005:
4000:
3997:
3993:
3985:
3982:, and 0 <
3981:
3977:
3971:
3967:
3963:
3959:
3953:
3949:
3930:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3897:
3894:
3891:
3888:
3885:
3882:
3876:
3873:
3870:
3867:
3861:
3858:
3855:
3852:
3849:
3846:
3837:
3831:
3828:
3825:
3822:
3819:
3809:
3808:
3807:
3805:
3801:
3795:
3791:
3786:
3782:
3778:
3774:
3773:diffeomorphic
3770:
3766:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3684:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3650:
3649:
3648:
3645:
3642:
3637:
3633:
3629:
3625:
3621:
3617:
3613:
3607:
3589:
3578:
3572:
3542:
3527:
3522:
3504:
3475:
3460:
3456:
3438:
3423:
3415:
3411:
3403:
3396:
3392:
3387:
3379:
3372:
3370:
3368:
3364:
3359:
3356:
3352:
3348:
3344:
3340:
3336:
3331:
3329:
3325:
3307:
3296:
3290:
3276:
3272:
3254:
3244:
3231:
3226:
3211:
3207:
3203:
3199:
3195:
3178:
3171:
3170:
3164:
3157:
3150:
3146:
3142:
3137:
3130:
3127:
3122:
3120:
3102:
3088:
3084:
3080:
3076:
3058:
3043:
3040:
3036:
3032:
3028:
3024:
3021:
3018:
3014:
3010:
3006:
3002:
2998:
2994:
2990:
2985:
2983:
2965:
2950:
2946:
2943:
2925:
2910:
2906:
2905:Automorphisms
2902:
2900:
2896:
2892:
2888:
2884:
2880:
2879:maximal torus
2876:
2872:
2867:
2865:
2861:
2857:
2854:
2850:
2847:
2843:
2839:
2834:
2832:
2829:-dimensional
2828:
2824:
2806:
2792:
2788:
2770:
2755:
2737:
2708:
2693:
2675:
2646:
2636:
2631:
2601:
2596:
2590:
2584:
2574:
2571:
2568:
2563:
2549:
2544:
2530:
2529:
2528:
2526:
2522:
2518:
2514:
2510:
2504:
2500:
2497:
2489:
2483:
2479:
2474:
2467:
2464:
2462:
2460:
2456:
2452:
2448:
2440:
2438:
2436:
2432:
2428:
2424:
2419:
2415:
2412:
2393:
2385:
2377:
2369:
2354:
2350:
2346:
2338:
2323:
2319:
2315:
2307:
2292:
2288:
2280:
2279:
2278:
2276:
2272:
2263:
2259:
2257:
2254:
2251:
2247:
2227:
2221:
2218:
2215:
2212:
2209:
2203:
2197:
2194:
2191:
2188:
2185:
2179:
2173:
2170:
2167:
2157:
2156:
2155:
2153:
2149:
2144:
2142:
2124:
2109:
2105:
2101:
2086:
2070:
2065:
2063:
2059:
2055:
2051:
2047:
2043:
2039:
2035:
2032:
2028:
2027:
2022:
2019: ×
2018:
2014:
2010:
2006:
2002:
2001:Topologically
1992:
1989:
1981:
1978:November 2015
1971:
1967:
1961:
1960:
1954:
1950:
1946:
1941:
1932:
1931:
1925:
1923:
1921:
1917:
1912:
1897:
1883:
1878:
1861:
1856:
1848:
1845:
1842:
1833:
1830:
1827:
1819:
1815:
1808:
1805:
1799:
1794:
1789:
1784:
1780:
1777:
1774:
1768:
1762:
1757:
1753:
1750:
1747:
1741:
1735:
1731:
1727:
1724:
1722:
1717:
1710:
1704:
1701:
1698:
1689:
1686:
1683:
1675:
1671:
1667:
1663:
1658:
1654:
1651:
1648:
1642:
1637:
1632:
1628:
1625:
1622:
1616:
1610:
1606:
1602:
1599:
1597:
1592:
1569:
1565:
1557:
1541:
1537:
1529:
1515:
1508:
1501:
1484:
1479:
1475:
1471:
1466:
1462:
1458:
1455:
1451:
1447:
1444:
1441:
1437:
1432:
1426:
1422:
1418:
1414:
1410:
1408:
1403:
1396:
1393:
1390:
1385:
1381:
1377:
1374:
1370:
1366:
1363:
1360:
1356:
1351:
1347:
1344:
1341:
1337:
1333:
1331:
1326:
1314:
1310:
1306:
1302:
1298:
1294:
1293:diffeomorphic
1276:
1272:
1268:
1263:
1259:
1255:
1248:
1236:
1233:
1226:
1222:
1218:
1213:
1209:
1190:
1185:
1181:
1171:
1164:
1159:
1155:
1148:
1143:
1140:
1139:
1134:
1133:
1127:
1123:
1119:
1115:
1111:
1107:
1103:
1099:
1094:
1093:
1092:
1081:
1068:
1064:
1058:
1054:
1050:
1045:
1041:
1036:
1030:
1026:
1022:
1019:
1014:
1009:
1003:
999:
995:
990:
986:
982:
977:
973:
969:
964:
960:
956:
951:
947:
942:
932:
928:
923:
910:
905:
901:
897:
892:
888:
884:
877:
865:
862:
855:
851:
847:
842:
838:
818:
802:
794:
790:
785:
783:
782:
765:
761:
757:
748:
734:
727:
711:
704:
688:
682:
679:
676:
673:
667:
664:
661:
658:
649:
632:
629:
626:
623:
620:
618:
610:
607:
604:
598:
590:
586:
583:
577:
574:
571:
568:
565:
562:
556:
554:
546:
543:
540:
534:
526:
522:
519:
513:
510:
507:
504:
501:
498:
492:
490:
482:
479:
476:
470:
458:
449:
438:
434:
427:
415:
411:
404:
392:
388:
381:
362:
360:
358:
357:
349:
347:
345:
339:
337:
333:
328:
326:
322:
304:
300:
291:
287:
269:
265:
261:
256:
252:
243:
239:
235:
231:
226:
224:
220:
216:
212:
208:
204:
200:
196:
195:
189:
187:
183:
179:
174:
172:
171:
166:
162:
158:
154:
153:
152:spindle torus
148:
144:
140:
136:
132:
127:
125:
121:
117:
113:
109:
105:
101:
97:
89:
85:
76:
69:
65:
60:
52:
48:
44:
40:
33:
19:
6488:Möbius strip
6436:Klein bottle
6394:
6328:. Retrieved
6296:Ghostarchive
6294:Archived at
6273:cut-the-knot
6248:
6235:
6223:. Retrieved
6219:
6209:
6197:. Retrieved
6193:the original
6183:
6171:. Retrieved
6167:the original
6157:
6145:. Retrieved
6137:Sci-News.com
6136:
6126:
6114:. Retrieved
6077:
6073:
6063:
6055:the original
6040:
5982:
5976:
5963:
5951:. Retrieved
5947:
5937:
5925:
5916:
5903:
5891:. Retrieved
5886:
5877:
5865:
5852:
5840:. Retrieved
5826:
5814:. Retrieved
5800:
5764:
5750:
5734:
5710:
5694:
5678:
5586:Klein bottle
5491:
5487:
5485:
5482:
5328:
5326:
5316:
5305:
5301:
5293:
5287:
5276:
5144:The torus's
5143:
5135:
4975:
4970:
4966:
4961:
4797:
4684:
4666:
4662:adding to it
4657:
4637:
4632:
4628:
4624:
4620:
4616:
4612:
4608:
4604:
4596:
4520:
4516:triple torus
4512:double torus
4507:
4503:
4497:
4492:
4484:
4480:
4475:
4467:
4463:
4453:
4447:
4437:
4428:
4426:
4421:
4408:
4404:
4399:
4395:
4394:
4389:
4386:moduli space
4367:
4320:
4318:
4250:
4242:
4240:
4235:
4202:
4198:
4194:
4190:
4188:
4021:
4003:
3998:
3983:
3979:
3975:
3969:
3965:
3961:
3957:
3951:
3947:
3945:
3803:
3793:
3789:
3781:analytically
3768:
3764:
3762:
3646:
3638:
3631:
3627:
3623:
3619:
3615:
3611:
3525:
3458:
3454:
3419:
3394:
3360:
3342:
3339:Möbius strip
3334:
3332:
3327:
3270:
3209:
3201:
3197:
3193:
3191:
3167:
3156:Möbius strip
3148:
3144:
3118:
3082:
3074:
3041:
3034:
3030:
3022:
3016:
3012:
3004:
3000:
2995:-torus is a
2992:
2986:
2981:
2948:
2908:
2903:
2890:
2886:
2874:
2868:
2844:dimensional
2841:
2837:
2835:
2826:
2822:
2753:
2691:
2616:
2524:
2520:
2516:
2512:
2508:
2498:
2487:
2481:
2465:
2444:
2420:
2416:
2408:
2268:
2255:
2252:
2243:
2145:
2140:
2104:homeomorphic
2066:
2057:
2054:fiber bundle
2049:
2041:
2033:
2025:
2020:
2016:
1999:
1984:
1975:
1964:Please help
1956:
1913:
1898:
1879:
1567:
1563:
1555:
1539:
1535:
1527:
1516:
1506:
1502:
1309:surface area
1183:
1179:
1176:
1169:
1162:
1153:
1146:
1136:
1130:
1125:
1121:
1113:
1109:
1101:
1097:
1082:
924:
791:equation in
786:
781:aspect ratio
779:
749:
726:minor radius
725:
703:major radius
702:
650:
457:parametrized
454:
436:
432:
417:: horn torus
413:
409:
390:
386:
354:
353:
344:Klein bottle
340:
329:
234:homeomorphic
227:
206:
192:
190:
175:
168:
160:
156:
150:
146:
138:
134:
128:
123:
119:
99:
95:
87:
81:
47:
6531:Compactness
6316:Brady Haran
6220:Science4All
5842:16 November
5278:(16,32;3,3)
4560:genus three
4024:defined by
3414:duocylinder
2860:unit circle
2785:modulo the
2459:cross-ratio
2411:closed path
2246:unit square
2062:Hopf bundle
2040:. In fact,
1970:introducing
1510:and radius
927:square root
207:solid torus
194:solid torus
182:inner tubes
64:degenerates
39:Solid torus
6582:Operations
6564:components
6560:Number of
6540:smoothness
6519:Properties
6467:Semisphere
6382:Orientable
6278:"4D torus"
6225:1 November
5919:: 638–648.
5883:"poloidal"
5760:Xu, Dianna
5742:References
5656:Torus knot
5626:Toric lens
4816:cohomology
4669:April 2010
4640:immersions
4588:6 × 4 = 24
4500:two-sphere
3783:embedded (
3395:flat torus
3373:Flat torus
3124:See also:
2942:invertible
2756:-torus is
2505:hypertorus
2427:isomorphic
2421:The first
2046:filled out
1581:), yields
1315:, giving:
750:The ratio
178:swim rings
147:horn torus
139:ring torus
18:Flat torus
6609:Immersion
6604:cross-cap
6602:Gluing a
6596:Gluing a
6493:Cross-cap
6438:(genus 2)
6422:genus 1;
6397:(genus 1)
6391:(genus 0)
6250:MathWorld
5995:CiteSeerX
5893:10 August
5867:MathWorld
5401:−
5366:−
5198:χ
5118:→
5097:
5084:→
5063:
5057:→
5036:
5023:→
4976:Thus the
4927:
4900:
4873:π
4866:
4839:
4818:algebra:
4708:
4598:Polyhedra
4549:genus two
4530:connected
4088:∣
4073:∈
4014:congruent
3964:) = (cos(
3922:
3907:
3895:
3871:
3859:
3787:of class
3777:isometric
3742:
3727:
3712:
3697:
3349:, with a
3271:unordered
3081:over the
2899:manifolds
2856:Lie group
2831:hypercube
2591:⏟
2575:×
2572:⋯
2569:×
2386:×
2378:≅
2351:π
2347:×
2320:π
2289:π
2204:∼
2180:∼
1922:devices.
1846:−
1816:π
1778:−
1732:π
1702:−
1672:π
1652:−
1607:π
1463:π
1445:π
1419:π
1382:π
1364:π
1345:π
1234:−
996:−
863:−
683:π
668:∈
665:φ
659:θ
633:θ
630:
611:φ
605:θ
591:φ
587:
578:θ
575:
547:φ
541:θ
527:φ
523:
514:θ
511:
483:φ
477:θ
350:Etymology
290:embedding
262:×
219:doughnuts
215:lifebuoys
6634:Surfaces
6628:Category
6562:boundary
6481:Cylinder
6298:and the
6141:Archived
6110:Archived
6106:22523238
6029:Archived
6017:16825563
5836:Archived
5810:Archived
5762:(2013).
5513:See also
5240:for the
4804:homology
4800:homotopy
4631:, where
4489:2-sphere
4456:surfaces
4201:, where
4018:boundary
4001:, where
3996:3-sphere
3802:it into
3457:, where
3422:quotient
3389:Seen in
3275:orbifold
3141:orbifold
2999:of rank
2947:of size
2883:subgroup
2849:manifold
2694:-torus,
2449:. Every
2148:quotient
2031:3-sphere
1926:Topology
1189:interior
929:gives a
789:implicit
363:Geometry
332:topology
230:topology
209:include
124:doughnut
116:coplanar
84:geometry
6512:notions
6510:Related
6476:Annulus
6472:Ribbon
6330:24 July
6312:(video)
6199:21 July
6173:21 July
6147:21 July
6116:21 July
6097:3358891
6025:2877171
5987:Bibcode
5978:Science
5953:27 July
5944:"Torus"
5862:"Torus"
5832:"Torus"
5816:21 July
5792:3026641
5536:3-torus
5504:in the
5501:A003600
4627:= 2 − 2
4527:compact
4450:surface
4440:surface
4415:in the
4354:fractal
3968:), sin(
3800:Mapping
3393:, a 4D
3192:As the
3185:(G♭-B♭)
3181:(F♯-A♯)
3169:Tonnetz
3073:,
3029:of the
2895:protori
2853:abelian
2846:compact
2791:lattice
2435:abelian
2150:of the
2013:circles
2011:of two
2009:product
1966:improve
1578:
1560:
1550:
1532:
1297:product
325:4-space
242:circles
240:of two
236:to the
217:, ring
211:O-rings
143:tangent
129:If the
102:) is a
100:toruses
6598:handle
6389:Sphere
6104:
6094:
6023:
6015:
5997:
5790:
5780:
5725:
5717:
5702:
5685:
5641:Toroid
5611:Sphere
5309:matrix
5296:is an
5152:has a
4693:. Its
4446:Genus
4436:Genus
4130:
3792:, 2 ≤
3785:smooth
3763:where
3087:module
3037:. The
3020:choose
3011:of an
3003:. The
2991:of an
2787:action
2501:-torus
2484:-plane
1305:volume
1187:, the
223:bagels
221:, and
203:volume
170:toroid
165:sphere
108:circle
68:sphere
32:Taurus
6568:Genus
6395:Torus
6032:(PDF)
6021:S2CID
5973:(PDF)
5913:(PDF)
5673:Notes
5298:array
5242:plane
5060:Homeo
5027:Homeo
4593:faces
4586:with
4491:with
4460:genus
4405:three
3986:<
3974:then
3626:) ~ (
3622:+ 1,
3618:) ~ (
3407:{4,4}
3351:twist
2102:, is
2071:from
2060:(the
2056:over
2052:as a
1951:, or
1299:of a
1177:When
1160:When
1144:When
1138:apple
1132:lemon
1124:<
1100:>
1095:When
435:<
389:>
356:Torus
338:one.
336:genus
120:donut
88:torus
6463:Disk
6332:2019
6227:2022
6201:2012
6175:2012
6149:2012
6118:2012
6102:PMID
6013:PMID
5955:2021
5895:2007
5844:2014
5818:2012
5778:ISBN
5723:ISBN
5715:ISBN
5700:ISBN
5683:ISBN
5506:OEIS
5304:-by-
4802:and
4685:The
4521:The
4514:and
3950:and
3767:and
3634:+ 1)
3333:For
3166:The
3007:-th
2987:The
2269:The
1892:and
1553:and
1269:<
1087:and
817:axis
459:as:
199:disk
184:and
155:(or
96:tori
86:, a
6538:or
6502:...
6271:at
6092:PMC
6082:doi
6078:109
6005:doi
5983:313
5770:doi
5288:In
5273:STL
5244:.)
5092:TOP
5088:MCG
4897:Aut
4863:Aut
4830:MCG
4664:.
4474:of
3946:If
3919:cos
3904:sin
3892:sin
3868:cos
3856:sin
3796:≤ ∞
3739:sin
3724:cos
3709:sin
3694:cos
3528:=
3424:,
3409:1,0
3326:on
3208:of
2907:of
2836:An
2503:or
2437:).
2253:ABA
2064:).
2044:is
1165:= 0
1149:= 0
819:is
787:An
627:sin
584:sin
572:cos
520:cos
508:cos
346:).
292:of
244::
228:In
188:.
159:or
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110:in
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6630::
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2015::
1955:,
1947:,
1566:−
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4038:=
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4008:π
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