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A stereographic projection of a Clifford torus performing a simple rotationTopologically a rectangle is the fundamental polygon of a torus, with opposite edges sewn together.
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles S1 a and S1 b (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1 a and S1 b each exists in its own independent embedding space R2 a and R2 b, the resulting product space will be R4 rather than R3. The historically popular view that the Cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.
Stated another way, a torus embedded in R3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R4. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lower-dimensional image that accurately captures the connectivity of the cube edges, but also requires the arbitrary selection and removal of one of the three fully symmetric and interchangeable axes of the cube.
If S1 a and S1 b each has a radius of 1/√2, their Clifford torus product will fit perfectly within the unit 3-sphere S3, which is a 3-dimensional submanifold of R4. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C2, since C2 is topologically equivalent to R4.
The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified. (Some video games, including Asteroids, are played on a square torus; anything that moves off one edge of the screen reappears on the opposite edge with the same orientation.) It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometry[clarification needed] as if it were flat, whereas the surface of a common "doughnut"-shaped torus is positively curved on the outer rim and negatively curved on the inner. Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a fractal set of ripples running in two perpendicular directions along the surface.[1]
^Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. (April 2012), "Flat tori in three-dimensional space and convex integration", Proceedings of the National Academy of Sciences, 109 (19): 7218–7223, doi:10.1073/pnas.1118478109, PMC 3358891, PMID 22523238.
submanifold of R2, the Cliffordtorus is an embedded torus in R2 × R2 = R4. If R4 is given by coordinates (x1, y1, x2, y2), then the Cliffordtorus is given by x...
called the Cliffordtorus, a surface in 4-space. In the field of topology, a torus is any topological space that is homeomorphic to a torus. The surface...
planes, or duocylindrical symmetry. For example, the duocylinder and Cliffordtorus have circular symmetry in two orthogonal axes. A spherinder has spherical...
correspond to the five partitions of 4, the number of dimensions. Cliffordtorus Duoprism Flat torus Hopf fibration Manifold The Fourth Dimension Simply Explained...
itself is orthogonal to that axis. The word "torus" originates from the Latin word "protuberance." Torus fractures are low risk and may cause acute pain...
tori, along a 2-torus: see Cliffordtorus. Each of the solid tori is then foliated internally, in codimension 1, and the dividing torus surface forms one...
operators Clifford module, a mathematical representation Clifford theory, dealing with representations, named after Alfred H. CliffordCliffordtorus, a figure...
mapping class group of the two-torus that only lens spaces have splittings of genus one. Three-torus Recall that the three-torus T 3 {\displaystyle T^{3}}...
William Kingdon Clifford FRS (4 May 1845 – 3 March 1879) was a British mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced...
one-to-one correspondence between Clifford parallelisms and planes external to the Klein quadric. Cliffordtorus 24-cell § Clifford parallel polytopes Regular...
representation in four dimensions as the Cartesian product of two circles: see Cliffordtorus. Formally, in mathematics, a developable surface is a surface with zero...
family of uniform duoprisms, which are products of two regular polygons. Cliffordtorus The Fourth Dimension Simply Explained, Henry P. Manning, Munn & Company...
dimensions, a regular skew polygon can have vertices on a Cliffordtorus and related by a Clifford displacement. Unlike zig-zag skew polygons, skew polygons...
examples. For example the Cliffordtorus construction in the 3-sphere shows that the complement of a solid torus is another solid torus; which will be open...
the geometric objects that are being modeled to new positions. The Cliffordtorus on the surface of the 3-sphere is the simplest and most symmetric flat...
laid out in a 2-dimensional arrangement topologically equivalent to the 2-torus and the ridge of the duocylinder. These can be further divided into three...
distinct planar polygons. They have vertices lying on a Cliffordtorus and related by a Clifford displacement. Unlike 3-dimensional polygons, skew polygons...
rectangle, conic sections, Leonardo da Vinci's Claw, the Koch curve, the Cliffordtorus, San Gaku, Mascheroni's cardioid, Pythagorean triples, spidrons, and...
faces of the duoprisms, with the n-gonal faces as holes and represent a cliffordtorus, and an approximation of a duocylinder {4,4|6} has 36 square faces,...
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