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 Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function (or “map”) from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere is mapped to from a distinct great circle of the 3-sphere (Hopf 1931).^[1] Thus the 3-sphere is composed of fibers, where each fiber is a circle — one for each point of the 2-sphere.

This fiber bundle structure is denoted

{\displaystyle S^{1}\hookrightarrow S^{3}{\xrightarrow {\ p\,}}S^{2},}

meaning that the fiber space S¹ (a circle) is embedded in the total space S³ (the 3-sphere), and p : S³ → S² (Hopf’s map) projects S³ onto the base space S² (the ordinary 2-sphere). The Hopf fibration, like any fiber bundle, has the important property that it is locally a product space. However it is not a trivial fiber bundle, i.e., S³ is not globally a product of S² and S¹ although locally it is indistinguishable from it.

This has many implications: for example the existence of this bundle shows that the higher homotopy groups of spheres are not trivial in general. It also provides a basic example of a principal bundle, by identifying the fiber with the circle group.

Stereographic projection of the Hopf fibration induces a remarkable structure on R³, in which space is filled with nested tori made of linking Villarceau circles. Here each fiber projects to a circle in space (one of which is a line, thought of as a “circle through infinity”). Each torus is the stereographic projection of the inverse image of a circle of latitude of the 2-sphere. (Topologically, a torus is the product of two circles.) These tori are illustrated in the images at right. When R³ is compressed to the boundary of a ball, some geometric structure is lost although the topological structure is retained (see Topology and geometry). The loops are homeomorphic to circles, although they are not geometric circles.

There are numerous generalizations of the Hopf fibration. The unit sphere in complex coordinate space Cⁿ⁺¹ fibers naturally over the complex projective space CPⁿ with circles as fibers, and there are also real, quaternionic,^[2] and octonionic versions of these fibrations. In particular, the Hopf fibration belongs to a family of four fiber bundles in which the total space, base space, and fiber space are all spheres:

{\displaystyle S^{0}\hookrightarrow S^{1}\to S^{1},}

{\displaystyle S^{1}\hookrightarrow S^{3}\to S^{2},}

{\displaystyle S^{3}\hookrightarrow S^{7}\to S^{4},}

{\displaystyle S^{7}\hookrightarrow S^{15}\to S^{8}.}

By Adams’s theorem such fibrations can occur only in these dimensions.

The Hopf fibration is important in twistor theory.

Definition and construction

For any natural number n, an n-dimensional sphere, or n-sphere, can be defined as the set of points in an {\displaystyle (n+1)}-dimensional space which are a fixed distance from a central point. For concreteness, the central point can be taken to be the origin, and the distance of the points on the sphere from this origin can be assumed to be a unit length. With this convention, the n-sphere, {\displaystyle S^{n}}, consists of the points {\displaystyle (x_{1},x_{2},\ldots ,x_{n+1})} in {\displaystyle \mathbb {R} ^{n+1}} with x₁² + x₂² + ⋯+ x_{n + 1}² = 1. For example, the 3-sphere consists of the points (x₁, x₂, x₃, x₄) in R⁴ with x₁² + x₂² + x₃² + x₄² = 1.

The Hopf fibration p: S³ → S² of the 3-sphere over the 2-sphere can be defined in several ways.

Direct construction[edit]

Identify R⁴ with C² and R³ with C × R (where C denotes the complex numbers) by writing:

{\displaystyle (x_{1},x_{2},x_{3},x_{4})\leftrightarrow (z_{0},z_{1})=(x_{1}+ix_{2},x_{3}+ix_{4})}

and

{\displaystyle (x_{1},x_{2},x_{3})\leftrightarrow (z,x)=(x_{1}+ix_{2},x_{3})}.

Thus S³ is identified with the subset of all (z₀, z₁) in C² such that |z₀|² + |z₁|² = 1, and S² is identified with the subset of all (z, x) in C×R such that |z|² + x² = 1. (Here, for a complex number z = x + iy, |z|² = z z^∗ = x² + y², where the star denotes the complex conjugate.) Then the Hopf fibration p is defined by

{\displaystyle p(z_{0},z_{1})=(2z_{0}z_{1}^{\ast },\left|z_{0}\right|^{2}-\left|z_{1}\right|^{2}).}

The first component is a complex number, whereas the second component is real. Any point on the 3-sphere must have the property that |z₀|² + |z₁|² = 1. If that is so, then p(z₀, z₁) lies on the unit 2-sphere in C × R, as may be shown by squaring the complex and real components of p

{\displaystyle 2z_{0}z_{1}^{\ast }\cdot 2z_{0}^{\ast }z_{1}+\left(\left|z_{0}\right|^{2}-\left|z_{1}\right|^{2}\right)^{2}=4\left|z_{0}\right|^{2}\left|z_{1}\right|^{2}+\left|z_{0}\right|^{4}-2\left|z_{0}\right|^{2}\left|z_{1}\right|^{2}+\left|z_{1}\right|^{4}=\left(\left|z_{0}\right|^{2}+\left|z_{1}\right|^{2}\right)^{2}=1}

Furthermore, if two points on the 3-sphere map to the same point on the 2-sphere, i.e., if p(z₀, z₁) = p(w₀, w₁), then (w₀, w₁) must equal (λ z₀, λ z₁) for some complex number λ with |λ|² = 1. The converse is also true; any two points on the 3-sphere that differ by a common complex factor λ map to the same point on the 2-sphere. These conclusions follow, because the complex factor λ cancels with its complex conjugate λ^∗ in both parts of p: in the complex 2z₀z₁^∗ component and in the real component |z₀|² − |z₁|².

Since the set of complex numbers λ with |λ|² = 1 form the unit circle in the complex plane, it follows that for each point m in S², the inverse image p⁻¹(m) is a circle, i.e., p⁻¹m ≅ S¹. Thus the 3-sphere is realized as a disjoint union of these circular fibers.

A direct parametrization of the 3-sphere employing the Hopf map is as follows.^[3]

{\displaystyle z_{0}=e^{i\,{\frac {\xi _{1}+\xi _{2}}{2}}}\sin \eta }

{\displaystyle z_{1}=e^{i\,{\frac {\xi _{2}-\xi _{1}}{2}}}\cos \eta .}

or in Euclidean R⁴

{\displaystyle x_{1}=\cos \left({\frac {\xi _{1}+\xi _{2}}{2}}\right)\sin \eta }

{\displaystyle x_{2}=\sin \left({\frac {\xi _{1}+\xi _{2}}{2}}\right)\sin \eta }

{\displaystyle x_{3}=\cos \left({\frac {\xi _{2}-\xi _{1}}{2}}\right)\cos \eta }

{\displaystyle x_{4}=\sin \left({\frac {\xi _{2}-\xi _{1}}{2}}\right)\cos \eta }

Where η runs over the range 0 to π/2, ξ₁ runs over the range 0 and 2π and ξ₂ can take any values between 0 and 4π. Every value of η, except 0 and π/2 which specify circles, specifies a separate flat torus in the 3-sphere, and one round trip (0 to 4π) of either ξ₁ or ξ₂ causes you to make one full circle of both limbs of the torus.

A mapping of the above parametrization to the 2-sphere is as follows, with points on the circles parametrized by ξ₂.

{\displaystyle z=\cos(2\eta )}

{\displaystyle x=\sin(2\eta )\cos \xi _{1}}

{\displaystyle y=\sin(2\eta )\sin \xi _{1}}

Geometric interpretation using the complex projective line

A geometric interpretation of the fibration may be obtained using the complex projective line, CP¹, which is defined to be the set of all complex one-dimensional subspaces of C². Equivalently, CP¹ is the quotient of C²\{0} by the equivalence relation which identifies (z₀, z₁) with (λ z₀, λ z₁) for any nonzero complex number λ. On any complex line in C² there is a circle of unit norm, and so the restriction of the quotient map to the points of unit norm is a fibration of S³ over CP¹.

CP¹ is diffeomorphic to a 2-sphere: indeed it can be identified with the Riemann sphere C_∞ = C ∪ {∞}, which is the one point compactification of C (obtained by adding a point at infinity). The formula given for p above defines an explicit diffeomorphism between the complex projective line and the ordinary 2-sphere in 3-dimensional space. Alternatively, the point (z₀, z₁) can be mapped to the ratio z₁/z₀ in the Riemann sphere C_∞.

Fiber bundle structure

The Hopf fibration defines a fiber bundle, with bundle projection p. This means that it has a “local product structure”, in the sense that every point of the 2-sphere has some neighborhood U whose inverse image in the 3-sphere can be identified with the product of U and a circle: p⁻¹(U) ≅ U × S¹. Such a fibration is said to be locally trivial.

For the Hopf fibration, it is enough to remove a single point m from S² and the corresponding circle p⁻¹(m) from S³; thus one can take U = S²\{m}, and any point in S² has a neighborhood of this form.

Geometric interpretation using rotations

Another geometric interpretation of the Hopf fibration can be obtained by considering rotations of the 2-sphere in ordinary 3-dimensional space. The rotation group SO(3) has a double cover, the spin group Spin(3), diffeomorphic to the 3-sphere. The spin group acts transitively on S² by rotations. The stabilizer of a point is isomorphic to the circle group. It follows easily that the 3-sphere is a principal circle bundle over the 2-sphere, and this is the Hopf fibration.

To make this more explicit, there are two approaches: the group Spin(3) can either be identified with the group Sp(1) of unit quaternions, or with the special unitary group SU(2).

In the first approach, a vector (x₁, x₂, x₃, x₄) in R⁴ is interpreted as a quaternion q ∈ H by writing

{\displaystyle q=x_{1}+\mathbf {i} x_{2}+\mathbf {j} x_{3}+\mathbf {k} x_{4}.\,\!}

The 3-sphere is then identified with the versors, the quaternions of unit norm, those q ∈ H for which |q|² = 1, where |q|² = q q^∗, which is equal to x₁² + x₂² + x₃² + x₄² for q as above.

On the other hand, a vector (y₁, y₂, y₃) in R³ can be interpreted as an imaginary quaternion

{\displaystyle p=\mathbf {i} y_{1}+\mathbf {j} y_{2}+\mathbf {k} y_{3}.\,\!}

Then, as is well-known since Cayley (1845), the mapping

{\displaystyle p\mapsto qpq^{*}\,\!}

is a rotation in R³: indeed it is clearly an isometry, since |q p q^∗|² = q p q^∗ q p^∗ q^∗ = q p p^∗ q^∗ = |p|², and it is not hard to check that it preserves orientation.

In fact, this identifies the group of versors with the group of rotations of R³, modulo the fact that the versors q and −q determine the same rotation. As noted above, the rotations act transitively on S², and the set of versors q which fix a given right versor p have the form q = u + v p, where u and v are real numbers with u² + v² = 1. This is a circle subgroup. For concreteness, one can take p = k, and then the Hopf fibration can be defined as the map sending a versor ω to ω k ω^∗. All the quaternions ωq, where q is one of the circle of versors that fix k, get mapped to the same thing (which happens to be one of the two 180° rotations rotating k to the same place as ω does).

Another way to look at this fibration is that every versor ω moves the plane spanned by {1, k} to a new plane spanned by {ω, ωk}. Any quaternion ωq, where q is one of the circle of versors that fix k, will have the same effect. We put all these into one fibre, and the fibres can be mapped one-to-one to the 2-sphere of 180° rotations which is the range of ωkω^*.

This approach is related to the direct construction by identifying a quaternion q = x₁ + i x₂ + j x₃ + k x₄ with the 2×2 matrix:

{\displaystyle {\begin{bmatrix}x_{1}+\mathbf {i} x_{2}&x_{3}+\mathbf {i} x_{4}\\-x_{3}+\mathbf {i} x_{4}&x_{1}-\mathbf {i} x_{2}\end{bmatrix}}.\,\!}

This identifies the group of versors with SU(2), and the imaginary quaternions with the skew-hermitian 2×2 matrices (isomorphic to C × R).

Explicit formulae

The rotation induced by a unit quaternion q = w + i x + j y + k z is given explicitly by the orthogonal matrix

{\displaystyle {\begin{bmatrix}1-2(y^{2}+z^{2})&2(xy-wz)&2(xz+wy)\\2(xy+wz)&1-2(x^{2}+z^{2})&2(yz-wx)\\2(xz-wy)&2(yz+wx)&1-2(x^{2}+y^{2})\end{bmatrix}}.}

Here we find an explicit real formula for the bundle projection by noting that the fixed unit vector along the z axis, (0,0,1), rotates to another unit vector,

{\displaystyle {\Big (}2(xz+wy),2(yz-wx),1-2(x^{2}+y^{2}){\Big )},\,\!}

which is a continuous function of (w, x, y, z). That is, the image of q is the point on the 2-sphere where it sends the unit vector along the z axis. The fiber for a given point on S² consists of all those unit quaternions that send the unit vector there.

We can also write an explicit formula for the fiber over a point (a, b, c) in S². Multiplication of unit quaternions produces composition of rotations, and

{\displaystyle q_{\theta }=\cos \theta +\mathbf {k} \sin \theta }

is a rotation by 2θ around the z axis. As θ varies, this sweeps out a great circle of S³, our prototypical fiber. So long as the base point, (a, b, c), is not the antipode, (0, 0, −1), the quaternion

{\displaystyle q_{(a,b,c)}={\frac {1}{\sqrt {2(1+c)}}}(1+c-\mathbf {i} b+\mathbf {j} a)}

will send (0, 0, 1) to (a, b, c). Thus the fiber of (a, b, c) is given by quaternions of the form q_(a, b, c)q_θ, which are the S³ points

{\displaystyle {\frac {1}{\sqrt {2(1+c)}}}{\Big (}(1+c)\cos(\theta ),a\sin(\theta )-b\cos(\theta ),a\cos(\theta )+b\sin(\theta ),(1+c)\sin(\theta ){\Big )}.\,\!}

Since multiplication by q_(a,b,c) acts as a rotation of quaternion space, the fiber is not merely a topological circle, it is a geometric circle.

The final fiber, for (0, 0, −1), can be given by defining q_(0,0,−1) to equal i, producing

{\displaystyle {\Big (}0,\cos(\theta ),-\sin(\theta ),0{\Big )},}

which completes the bundle. But note that this one-to-one mapping between S³ and S²×S¹ is not continuous on this circle, reflecting the fact that S³ is not topologically equivalent to S²×S¹.

Thus, a simple way of visualizing the Hopf fibration is as follows. Any point on the 3-sphere is equivalent to a quaternion, which in turn is equivalent to a particular rotation of a Cartesian coordinate frame in three dimensions. The set of all possible quaternions produces the set of all possible rotations, which moves the tip of one unit vector of such a coordinate frame (say, the z vector) to all possible points on a unit 2-sphere. However, fixing the tip of the z vector does not specify the rotation fully; a further rotation is possible about the z–axis. Thus, the 3-sphere is mapped onto the 2-sphere, plus a single rotation.

The rotation can be represented using the Euler angles θ, φ, and ψ. The Hopf mapping maps the rotation to the point on the 2-sphere given by θ and φ, and the associated circle is parametrized by ψ. Note that when θ = π the Euler angles φ and ψ are not well defined individually, so we do not have a one-to-one mapping (or a one-to-two mapping) between the 3-torus of (θ, φ, ψ) and S³.

Fluid mechanics

If the Hopf fibration is treated as a vector field in 3 dimensional space then there is a solution to the (compressible, non-viscous) Navier-Stokes equations of fluid dynamics in which the fluid flows along the circles of the projection of the Hopf fibration in 3 dimensional space. The size of the velocities, the density and the pressure can be chosen at each point to satisfy the equations. All these quantities fall to zero going away from the centre. If a is the distance to the inner ring, the velocities, pressure and density fields are given by:

{\displaystyle \mathbf {v} (x,y,z)=A\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-2}\left(2(-ay+xz),2(ax+yz),a^{2}-x^{2}-y^{2}+z^{2}\right)}

{\displaystyle p(x,y,z)=-A^{2}B\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-3},}

{\displaystyle \rho (x,y,z)=3B\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-1}}

for arbitrary constants A and B. Similar patterns of fields are found as soliton solutions of magnetohydrodynamics:^[4]

Generalizations

The Hopf construction, viewed as a fiber bundle p: S³ → CP¹, admits several generalizations, which are also often known as Hopf fibrations. First, one can replace the projective line by an n-dimensional projective space. Second, one can replace the complex numbers by any (real) division algebra, including (for n = 1) the octonions.

Real Hopf fibrations

A real version of the Hopf fibration is obtained by regarding the circle S¹ as a subset of R² in the usual way and by identifying antipodal points. This gives a fiber bundle S¹ → RP¹ over the real projective line with fiber S⁰ = {1, −1}. Just as CP¹ is diffeomorphic to a sphere, RP¹ is diffeomorphic to a circle.

More generally, the n-sphere Sⁿ fibers over real projective space RPⁿ with fiber S⁰.

Complex Hopf fibrations

The Hopf construction gives circle bundles p : S²ⁿ⁺¹ → CPⁿ over complex projective space. This is actually the restriction of the tautological line bundle over CPⁿ to the unit sphere in Cⁿ⁺¹.

Quaternionic Hopf fibrations

Similarly, one can regard S⁴ⁿ⁺³ as lying in Hⁿ⁺¹ (quaternionic n-space) and factor out by unit quaternion (= S³) multiplication to get the quaternionic projective space HPⁿ. In particular, since S⁴ = HP¹, there is a bundle S⁷ → S⁴ with fiber S³.

Octonionic Hopf fibrations

A similar construction with the octonions yields a bundle S¹⁵ → S⁸ with fiber S⁷. But the sphere S³¹ does not fiber over S¹⁶ with fiber S¹⁵. One can regard S⁸ as the octonionic projective line OP¹. Although one can also define an octonionic projective plane OP², the sphere S²³ does not fiber over OP² with fiber S⁷.^[5][6]

Fibrations between spheres

Sometimes the term “Hopf fibration” is restricted to the fibrations between spheres obtained above, which are

• S¹ → S¹ with fiber S⁰
• S³ → S² with fiber S¹
• S⁷ → S⁴ with fiber S³
• S¹⁵ → S⁸ with fiber S⁷

As a consequence of Adams’s theorem, fiber bundles with spheres as total space, base space, and fiber can occur only in these dimensions. Fiber bundles with similar properties, but different from the Hopf fibrations, were used by John Milnor to construct exotic spheres.

Geometry and applications

The Hopf fibration has many implications, some purely attractive, others deeper. For example, stereographic projection S³ → R³ induces a remarkable structure in R³, which in turn illuminates the topology of the bundle (Lyons 2003). Stereographic projection preserves circles and maps the Hopf fibers to geometrically perfect circles in R³ which fill space. Here there is one exception: the Hopf circle containing the projection point maps to a straight line in R³ — a “circle through infinity”.

The fibers over a circle of latitude on S² form a torus in S³ (topologically, a torus is the product of two circles) and these project to nested toruses in R³ which also fill space. The individual fibers map to linking Villarceau circles on these tori, with the exception of the circle through the projection point and the one through its opposite point: the former maps to a straight line, the latter to a unit circle perpendicular to, and centered on, this line, which may be viewed as a degenerate torus whose minor radius has shrunken to zero. Every other fiber image encircles the line as well, and so, by symmetry, each circle is linked through every circle, both in R³ and in S³. Two such linking circles form a Hopf link in R³

Hopf proved that the Hopf map has Hopf invariant 1, and therefore is not null-homotopic. In fact it generates the homotopy group π₃(S²) and has infinite order.

In quantum mechanics, the Riemann sphere is known as the Bloch sphere, and the Hopf fibration describes the topological structure of a quantum mechanical two-level system or qubit. Similarly, the topology of a pair of entangled two-level systems is given by the Hopf fibration

{\displaystyle S^{3}\hookrightarrow S^{7}\to S^{4}.}

(Mosseri & Dandoloff 2001).

The Hopf fibration is equivalent to the fiber bundle structure of the Dirac monopole.^[7]

Notes

References

• Cayley, Arthur (1845), “On certain results relating to quaternions”, Philosophical Magazine, 26: 141–145, doi:10.1080/14786444508562684; reprinted as article 20 in Cayley, Arthur (1889), The collected mathematical papers of Arthur Cayley, I, (1841–1853), Cambridge University Press, pp. 123–126
• Hopf, Heinz (1931), “Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche”, Mathematische Annalen, Berlin: Springer, 104 (1): 637–665, doi:10.1007/BF01457962, ISSN 0025-5831
• Hopf, Heinz (1935), “Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension”, Fundamenta Mathematicae, Warsaw: Polish Acad. Sci., 25: 427–440, ISSN 0016-2736
• Lyons, David W. (April 2003), “An Elementary Introduction to the Hopf Fibration” (PDF), Mathematics Magazine, 76 (2): 87–98, doi:10.2307/3219300, ISSN 0025-570X, JSTOR 3219300
• Mosseri, R.; Dandoloff, R. (2001), “Geometry of entangled states, Bloch spheres and Hopf fibrations”, Journal of Physics A: Mathematical and Theoretical, 34 (47): 10243–10252, arXiv:quant-ph/0108137, Bibcode:2001JPhA…3410243M, doi:10.1088/0305-4470/34/47/324.
• Steenrod, Norman (1951), The Topology of Fibre Bundles, PMS 14, Princeton University Press (published 1999), ISBN 978-0-691-00548-5
• Urbantke, H.K. (2003), “The Hopf fibration-seven times in physics”, Journal of Geometry and Physics, 46 (2): 125–150, Bibcode:2003JGP….46..125U, doi:10.1016/S0393-0440(02)00121-3.

External links[edit]

• Hazewinkel, Michiel, ed. (2001) [1994], “Hopf fibration”, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Dimensions Math Chapters 7 and 8 illustrate the Hopf fibration with animated computer graphics.
• An Elementary Introduction to the Hopf Fibration by David W. Lyons (PDF)
• YouTube animation showing dynamic mapping of points on the 2-sphere to circles in the 3-sphere, by Professor Niles Johnson.
• YouTube animation of the construction of the 120-cell By Gian Marco Todesco shows the Hopf fibration of the 120-cell.
• Video of one 30-cell ring of the 600-cell from http://page.math.tu-berlin.de/~gunn/.
• Interactive visualization of the mapping of points on the 2-sphere to circles in the 3-spher
  e

In mathematics, an Apollonian gasket or Apollonian net is a fractal generated starting from a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mathematician Apollonius of Perga.^[1]

Construction

An Apollonian gasket can be constructed as follows. Start with three circles C₁, C₂ and C₃, each one of which is tangent to the other two (in the general construction, these three circles have to be different sizes, and they must have a common tangent). Apollonius discovered that there are two other non-intersecting circles, C₄ and C₅, which have the property that they are tangent to all three of the original circles – these are called Apollonian circles. Adding the two Apollonian circles to the original three, we now have five circles.

Take one of the two Apollonian circles – say C₄. It is tangent to C₁ and C₂, so the triplet of circles C₄, C₁ and C₂ has its own two Apollonian circles. We already know one of these – it is C₃ – but the other is a new circle C₆.

In a similar way we can construct another new circle C₇ that is tangent to C₄, C₂ and C₃, and another circle C₈ from C₄, C₃ and C₁. This gives us 3 new circles. We can construct another three new circles from C₅, giving six new circles altogether. Together with the circles C₁ to C₅, this gives a total of 11 circles.

Continuing the construction stage by stage in this way, we can add 2·3ⁿ new circles at stage n, giving a total of 3ⁿ⁺¹ + 2 circles after n stages. In the limit, this set of circles is an Apollonian gasket.

The sizes of the new circles are determined by Descartes’ theorem. Let k_i (for i = 1, …, 4) denote the curvatures of four mutually tangent circles. Then Descartes’ Theorem states

{\displaystyle (k_{1}+k_{2}+k_{3}+k_{4})^{2}=2\,(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}+k_{4}^{2}).}

(1)

The Apollonian gasket has a Hausdorff dimension of about 1.3057.^[2]

Curvature

The curvature of a circle (bend) is defined to be the inverse of its radius.

• Negative curvature indicates that all other circles are internally tangent to that circle. This is bounding circle.
• Zero curvature gives a line (circle with infinite radius).
• Positive curvature indicates that all other circles are externally tangent to that circle. This circle is in the interior of circle with negative curvature.

Variations

An Apollonian gasket can also be constructed by replacing one of the generating circles by a straight line, which can be regarded as a circle passing through the point at infinity.

Alternatively, two of the generating circles may be replaced by parallel straight lines, which can be regarded as being tangent to one another at infinity. In this construction, the additional circles form a family of Ford circles.

The three-dimensional equivalent of the Apollonian gasket is the Apollonian sphere packing.

Symmetries

If two of the original generating circles have the same radius and the third circle has a radius that is two-thirds of this, then the Apollonian gasket has two lines of reflective symmetry; one line is the line joining the centres of the equal circles; the other is their mutual tangent, which passes through the centre of the third circle. These lines are perpendicular to one another, so the Apollonian gasket also has rotational symmetry of degree 2; the symmetry group of this gasket is D₂.

If all three of the original generating circles have the same radius then the Apollonian gasket has three lines of reflective symmetry; these lines are the mutual tangents of each pair of circles. Each mutual tangent also passes through the centre of the third circle and the common centre of the first two Apollonian circles. These lines of symmetry are at angles of 60 degrees to one another, so the Apollonian gasket also has rotational symmetry of degree 3; the symmetry group of this gasket is D₃.

Links with hyperbolic geometry

The three generating circles, and hence the entire construction, are determined by the location of the three points where they are tangent to one another. Since there is a Möbius transformation which maps any three given points in the plane to any other three points, and since Möbius transformations preserve circles, then there is a Möbius transformation which maps any two Apollonian gaskets to one another.

Möbius transformations are also isometries of the hyperbolic plane, so in hyperbolic geometry all Apollonian gaskets are congruent. In a sense, there is therefore only one Apollonian gasket, up to (hyperbolic) isometry.

The Apollonian gasket is the limit set of a group of Möbius transformations known as a Kleinian group.^[3]

Integral Apollonian circle packings

• 

  Integral Apollonian circle packing defined by circle curvatures of (−1, 2, 2, 3)

• 

  Integral Apollonian circle packing defined by circle curvatures of (−3, 5, 8, 8)

• 

  Integral Apollonian circle packing defined by circle curvatures of (−12, 25, 25, 28)

• 

  Integral Apollonian circle packing defined by circle curvatures of (−6, 10, 15, 19)

• 

  Integral Apollonian circle packing defined by circle curvatures of (−10, 18, 23, 27)

If any four mutually tangent circles in an Apollonian gasket all have integer curvature then all circles in the gasket will have integer curvature.^[4] Since the equation relating curvatures in an Apollonian gasket, integral or not, is

{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=2ab+2ac+2ad+2bc+2bd+2cd,\,}

it follows that one may move from one quadruple of curvatures to another by Vieta jumping, just as when finding a new Markov number. The first few of these integral Apollonian gaskets are listed in the following table. The table lists the curvatures of the largest circles in the gasket. Only the first three curvatures (of the five displayed in the table) are needed to completely describe each gasket – all other curvatures can be derived from these three.

Symmetry of integral Apollonian circle packings

No symmetry

If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group C₁; the gasket described by curvatures (−10, 18, 23, 27) is an example.

D₁ symmetry

Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have D₁ symmetry, which corresponds to a reflection along a diameter of the bounding circle, with no rotational symmetry.

D₂ symmetry

If two different curvatures are repeated within the first five, the gasket will have D₂ symmetry; such a symmetry consists of two reflections (perpendicular to each other) along diameters of the bounding circle, with a two-fold rotational symmetry of 180°. The gasket described by curvatures (−1, 2, 2, 3) is the only Apollonian gasket (up to a scaling factor) to possess D₂ symmetry.

D₃ symmetry

There are no integer gaskets with D₃ symmetry.

If the three circles with smallest positive curvature have the same curvature, the gasket will have D₃ symmetry, which corresponds to three reflections along diameters of the bounding circle (spaced 120° apart), along with three-fold rotational symmetry of 120°. In this case the ratio of the curvature of the bounding circle to the three inner circles is 2√3 − 3. As this ratio is not rational, no integral Apollonian circle packings possess this D₃ symmetry, although many packings come close.

Almost-D₃ symmetry

The figure at left is an integral Apollonian gasket that appears to have D₃ symmetry. The same figure is displayed at right, with labels indicating the curvatures of the interior circles, illustrating that the gasket actually possesses only the D₁ symmetry common to many other integral Apollonian gaskets.

The following table lists more of these almost–D₃ integral Apollonian gaskets. The sequence has some interesting properties, and the table lists a factorization of the curvatures, along with the multiplier needed to go from the previous set to the current one. The absolute values of the curvatures of the “a” disks obey the recurrence relation a(n) = 4a(n − 1) − a(n − 2) (sequence A001353 in the OEIS), from which it follows that the multiplier converges to √3 + 2 ≈ 3.732050807.

Integral Apollonian gaskets with near-D₃ symmetry

Curvature					Factors				Multiplier
a	b	c	d		a	b	d		a	b	c	d
−1	2	2	3		1×1	1×2	1×3		N/A	N/A	N/A	N/A
−4	8	9	9		2×2	2×4	3×3		4.000000000	4.000000000	4.500000000	3.000000000
−15	32	32	33		3×5	4×8	3×11		3.750000000	4.000000000	3.555555556	3.666666667
−56	120	121	121		8×7	8×15	11×11		3.733333333	3.750000000	3.781250000	3.666666667
−209	450	450	451		11×19	15×30	11×41		3.732142857	3.750000000	3.719008264	3.727272727
−780	1680	1681	1681		30×26	30×56	41×41		3.732057416	3.733333333	3.735555556	3.727272727
−2911	6272	6272	6273		41×71	56×112	41×153		3.732051282	3.733333333	3.731112433	3.731707317
−10864	23408	23409	23409		112×97	112×209	153×153		3.732050842	3.732142857	3.732302296	3.731707317
−40545	87362	87362	87363		153×265	209×418	153×571		3.732050810	3.732142857	3.731983425	3.732026144

Sequential curvatures

For any integer n > 0, there exists an Apollonian gasket defined by the following curvatures:
(−n, n + 1, n(n + 1), n(n + 1) + 1).
For example, the gaskets defined by (−2, 3, 6, 7), (−3, 4, 12, 13), (−8, 9, 72, 73), and (−9, 10, 90, 91) all follow this pattern. Because every interior circle that is defined by n + 1 can become the bounding circle (defined by −n) in another gasket, these gaskets can be nested. This is demonstrated in the figure at right, which contains these sequential gaskets with n running from 2 through 20.

See also

• Descartes’ theorem, for curvatures of mutually tangent circles
• Ford circle, the special case of integral Apollonian gasket (0,0,1,1)
• Sierpiński triangle
• Apollonian network, a graph derived from finite subsets of the Apollonian gasket

Notes

References

• Benoit B. Mandelbrot: The Fractal Geometry of Nature, W H Freeman, 1982, ISBN 0-7167-1186-9
• Paul D. Bourke: “An Introduction to the Apollony Fractal“. Computers and Graphics, Vol 30, Issue 1, January 2006, pages 134–136.
• David Mumford, Caroline Series, David Wright: Indra’s Pearls: The Vision of Felix Klein, Cambridge University Press, 2002, ISBN 0-521-35253-3
• Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks: Beyond the Descartes Circle Theorem, The American Mathematical Monthly, Vol. 109, No. 4 (Apr., 2002), pp. 338–361, (arXiv:math.MG/0101066 v1 9 Jan 2001)

External links

• Weisstein, Eric W. “Apollonian Gasket”. MathWorld.
• Alexander Bogomolny, Apollonian Gasket, cut-the-knot
• An interactive Apollonian gasket running on pure HTML5 (the link is dead)
• (in English) A Matlab script to plot 2D Apollonian gasket with n identical circles using circle inversion
• Online experiments with JSXGraph
• Apollonian Gasket by Michael Screiber, The Wolfram Demonstrations Project.
• Interactive Apollonian Gasket Demonstration of an Apollonian gasket running on Java
• Dana Mackenzie. A Tisket, a Tasket, an Apollonian Gasket. American Scientist, January/February 2010.
• “Sand drawing the world’s largest single artwork”, The Telegraph, 16 Dec 2009. Newspaper story about an artwork in the form of a partial Apollonian gasket, with an outer circumference of nine miles.