Clifford torus

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A stereographic projection of a Clifford torus performing a simple rotation
Topologically a rectangle is the fundamental polygon of a torus, with opposite edges sewn together.

In geometric topology, the Clifford torus is a special kind of torus sitting inside the unit 3-sphere S3 in R4, the Euclidean space of four dimensions. Or equivalently, it can be seen as a torus sitting inside C2 since C2 is topologically equivalent to R4. It is specifically the torus in S3 that is geometrically the cartesian product of two circles, each of radius sqrt(1/2).

The Clifford torus is an example of a square torus, because it is isometric to a square 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 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. The Clifford torus cannot exist in Euclidean three-dimensional space.[1]

Formal definition[edit]

The unit circle S1 in R2 can be parameterized by an angle coordinate:

S^1 = \{ ( \cos{\theta}, \sin{\theta} ) \, | \, 0 \leq \theta < 2\pi \}.

In another copy of R2, take another copy of the unit circle

S^1 = \{ ( \cos{\phi}, \sin{\phi} ) \, | \, 0 \leq \phi < 2\pi \}.

Then the Clifford torus is

(\sqrt{1/2})S^1 \times (\sqrt{1/2})S^1 = \{ \sqrt{1/2}( \cos{\theta}, \sin{\theta}, \cos{\phi}, \sin{\phi} ) \, | \, 0 \leq \theta < 2\pi, 0 \leq \phi < 2\pi \}.

Since each copy of S1 is an embedded submanifold of R2, the Clifford torus is an embedded torus in R2 × R2 = R4.

If R4 is given by coordinates (x1, y1, x2, y2), then the Clifford torus is given by

x_1^2 + y_1^2 = 1/2 = x_2^2 + y_2^2. \, .

This shows that in R4 the Clifford torus is a submanifold of the unit 3-sphere S3.

It is easy to verify that the Clifford torus is in fact a minimal surface in S3.

Alternative derivation using complex numbers[edit]

It is also common to consider the Clifford torus as an embedded torus in C2. In two copies of C, we have the following unit circles (still parametrized by an angle coordinate):

S^1 = \{ e^{i\theta} \, | \, 0 \leq \theta < 2\pi \}


S^1 = \{ e^{i\phi} \, | \, 0 \leq \phi < 2\pi \}.

Now the Clifford torus appears as

(\sqrt{1/2})S^1 \times (\sqrt{1/2})S^1 = \{ \sqrt{1/2} ( e^{i\theta}, e^{i\phi} ) \, | \, 0 \leq \theta < 2\pi, 0 \leq \phi < 2\pi \}.

As before, this is an embedded submanifold, in the unit sphere S3 in C2.

If C2 is given by coordinates (z1, z2), then the Clifford torus is given by

\left| z_{1} \right|^{2} = 1/2 = \left| z_{2} \right|^{2}.

In the Clifford torus as defined above, the distance of any point of the Clifford torus to the origin of C2 is

\sqrt{ \left| e^{i\theta} \right|^2 + \left| e^{i\phi} \right|^2 } = 1.

The set of all points at a distance of 1 from the origin of C2 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.)[2])

Since O(4) acts on R4 by orthogonal transformations, 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) 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.[2] 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[edit]

The flat tori in the unit 3-sphere S3 that are the product of circles of radius = r in one 2-plane R2 and radius = √(1-r2 in another 2-plane R2 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 0 ≤ θ ≤ π/2 (where we include the degenerate cases θ = 0 and θ = π/2).

The union for 0 ≤ θ ≤ π/2 of all of these tori of form

   Tθ  =  S(cos(θ)) × S(sin(θ))

(where S(r) denotes the circle in the plane R2 defined by having center = (0,0) and radius = r)

is the 3-sphere S3. (Note that we must include the two degenerate cases θ = 0 and θ = π/2, each of which corresponds to a great circle of S3, and which together constitute a pair of polar great circles.)

This torus Tθ is readily seen to have area(Tθ) = 4π2cos(θ)sin(θ) = 2π2sin(2θ), so only the torus Tπ/4 has the maximum possible area of 2π2. This torus Tπ/4 is the torus Tθ that is most commonly called the "Clifford torus" — and it is also the only one of the Tθ that is a minimal surface in S3.

Still more general definition of Clifford tori in higher dimensions[edit]

Any unit sphere S2n-1 in an even-dimensional euclidean space R2n = ℂn may be expressed in terms of the complex coordinates as follows:

                          S2n-1  =  {(z1,...,zn) ∈ ℂn |  |z1|2 + ... + |zn|2  =  1}

Then for any non-negative numbers r1,...,rn such that

                           r12 + ... + rn2  =  1

we may define a generalized Clifford torus as follows:

                          Tr1,...,rn  =  {(z1,...,zn) ∈ ℂn |  |zk| = rk, 1 ≤ k ≤ n}

These generalized Clifford tori are all disjoint from one another. We may once again conclude that the union of each one of these tori Tr1,...,rn is the unit (2n-1)-sphere S2n-1 (where we must again include the degenerate cases where at least one of the radii rk = 0).


  • 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[edit]

In symplectic geometry, the Clifford torus gives an example of an embedded Lagrangian submanifold of C2 with the standard symplectic structure. (Of course, any product of embedded circles in C gives a Lagrangian torus of C2, 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.

See also[edit]


  1. ^ Luminet, Jean-Pierre; Starkman, Glenn D; Weeks, Jeffrey R. (April 1999). "Is Space Finite?" (PDF). Scientific American. Retrieved 6 Sep 2014. 
  2. ^ a b Norbs, P (September 2005). "The 12th problem" (PDF). The Australian Mathematical Society Gazette 32 (4): 244–246.