In
geometric topology, the Clifford torus is the simplest and most symmetric
flat embedding of the
cartesian product of two
circles
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after
William Kingdon Clifford. It resides in R
4, as opposed to in R
3. To see why R
4 is necessary, note that if ''S'' and ''S'' each exists in its own independent embedding space R and R, the resulting product space will be R
4 rather than R
3. The historically popular view that the cartesian product of two circles is an R
3 torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
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 R
3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R
4. 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 ''S'' and ''S'' each has a radius of
, their Clifford torus product will fit perfectly within the unit
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
''S''
3, which is a 3-dimensional submanifold of R
4. When mathematically convenient, the Clifford torus can be viewed as residing inside the
complex coordinate space
In mathematics, the ''n''-dimensional complex coordinate space (or complex ''n''-space) is the set of all ordered ''n''-tuples of complex numbers. It is denoted \Complex^n, and is the ''n''-fold Cartesian product of the complex plane \Complex wi ...
C
2, since C
2 is topologically equivalent to R
4.
The Clifford torus is an example of a square torus, because it is
isometric to a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
with opposite sides identified. It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
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
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedding, embedded into some Euclidean space. Isometry, Isometric means preserving the length of every ...
; one possible embedding modifies the standard torus by a
fractal set of ripples running in two perpendicular directions along the surface.
[.]
Formal definition
The
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
''S''
1 in R
2 can be parameterized by an angle coordinate:
:
In another copy of R
2, take another copy of the unit circle
:
Then the Clifford torus is
:
Since each copy of ''S''
1 is an embedded
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
of R
2, the Clifford torus is an embedded torus in = R
4.
If R
4 is given by coordinates (''x''
1, ''y''
1, ''x''
2, ''y''
2), then the Clifford torus is given by
:
This shows that in R
4 the Clifford torus is a submanifold of the unit 3-sphere ''S''
3.
It is easy to verify that the Clifford torus is a minimal surface in ''S''
3.
Alternative derivation using complex numbers
It is also common to consider the Clifford torus as an
embedded torus in C
2. In two copies of C, we have the following unit circles (still parametrized by an angle coordinate):
:
and
:
Now the Clifford torus appears as
:
As before, this is an embedded submanifold, in the unit sphere ''S''
3 in C
2.
If C
2 is given by coordinates (''z''
1, ''z''
2), then the Clifford torus is given by
:
In the Clifford torus as defined above, the distance of any point of the Clifford torus to the origin of C
2 is
:
The set of all points at a distance of 1 from the origin of C
2 is the unit 3-sphere, and so the Clifford torus sits inside this 3-sphere. In fact, the Clifford torus divides this 3-sphere into two congruent
solid tori (see
Heegaard splitting
In the mathematical field of geometric topology, a Heegaard splitting () is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
Definitions
Let ''V'' and ''W'' be handlebodies of genus ''g'', an ...
[
]).
Since
O(4) acts on R
4 by
orthogonal transformation In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we h ...
s, we can move the "standard" Clifford torus defined above to other equivalent tori via rigid rotations. These are all called "Clifford tori". The six-dimensional group O(4) acts transitively on the space of all such Clifford tori sitting inside the 3-sphere. However, this action has a two-dimensional stabilizer (see
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
) since rotation in the meridional and longitudinal directions of a torus preserves the torus (as opposed to moving it to a different torus). Hence, there is actually a four-dimensional space of Clifford tori.
In fact, there is a one-to-one correspondence between Clifford tori in the unit 3-sphere and pairs of polar great circles (i.e., great circles that are maximally separated). Given a Clifford torus, the associated polar great circles are the core circles of each of the two complementary regions. Conversely, given any pair of polar great circles, the associated Clifford torus is the locus of points of the 3-sphere that are equidistant from the two circles.
More general definition of Clifford tori
The flat tori in the unit 3-sphere ''S''
3 that are the product of circles of radius ''r'' in one 2-plane R
2 and radius in another 2-plane R
2 are sometimes also called "Clifford tori".
The same circles may be thought of as having radii that are cos(''θ'') and sin(''θ'') for some angle ''θ'' in the range (where we include the degenerate cases and ).
The union for of all of these tori of form
:
(where ''S''(''r'') denotes the circle in the plane R
2 defined by having center and radius ''r'') is the 3-sphere ''S''
3. (Note that we must include the two degenerate cases and , each of which corresponds to a great circle of ''S''
3, and which together constitute a pair of polar great circles.)
This torus ''T''
''θ'' is readily seen to have area
:
so only the torus ''T''
/4 has the maximum possible area of 2
2. This torus ''T''
/4 is the torus ''T''
''θ'' that is most commonly called the "Clifford torus" – and it is also the only one of the ''T''
''θ'' that is a minimal surface in ''S''
3.
Still more general definition of Clifford tori in higher dimensions
Any unit sphere S
2''n''−1 in an even-dimensional euclidean space may be expressed in terms of the complex coordinates as follows:
:
Then, for any non-negative numbers ''r''
1, ..., ''r''
''n'' such that ''r''
12 + ... + ''r''
''n''2 = 1, we may define a generalized Clifford torus as follows:
:
These generalized Clifford tori are all disjoint from one another. We may once again conclude that the union of each one of these tori T
''r''1, ..., ''r''''n'' is the unit (2''n'' − 1)-sphere ''S''
2''n''−1 (where we must again include the degenerate cases where at least one of the radii ''r''
''k'' = 0).
Properties
* The Clifford torus is "flat"; it can be flattened out to a plane without stretching, unlike the standard torus of revolution.
* The Clifford torus divides the 3-sphere into two congruent solid tori. (In a
stereographic projection, the Clifford torus appears as a standard torus of revolution. The fact that it divides the 3-sphere equally means that the interior of the projected torus is equivalent to the exterior, which is not easily visualized).
Uses in mathematics
In
symplectic geometry, the Clifford torus gives an example of an embedded
Lagrangian submanifold of C
2 with the standard symplectic structure. (Of course, any product of embedded circles in C gives a Lagrangian torus of C
2, so these need not be Clifford tori.)
The
Lawson conjecture states that every
minimally embedded torus in the 3-sphere with the
round metric must be a Clifford torus. This conjecture was proved by
Simon Brendle in 2012.
Clifford tori and their images under conformal transformations are the global minimizers of the
Willmore functional.
See also
*
Duocylinder
The duocylinder, also called the double cylinder or the bidisc, is a geometric object embedded in 4- dimensional Euclidean space, defined as the Cartesian product of two disks of respective radii ''r''1 and ''r''2:
:D = \left\
It is analogo ...
*
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz H ...
*
Clifford parallel In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in s ...
and Clifford surface
*
William Kingdom Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...
References
Geometric topology
Four-dimensional geometry