Apollonian gasket
Encyclopedia
In mathematics
, an Apollonian gasket or Apollonian net is a fractal
generated from triples of circles, where each circle is tangent
to the other two. It is named after Greek
mathematician
Apollonius of Perga
.
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 can be any size, as long as they have common tangents). 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 (see Descartes' theorem
). 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 C1, 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 Apollonian gasket has a Hausdorff dimension
of about 1.3057 http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf.
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 circles that are tangent to one of the two straight lines form a family of Ford circle
s.
The three-dimensional equivalent of the Apollonian gasket is the Apollonian sphere packing
.
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.
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, which can be thought of as a tessellation
of the hyperbolic plane by circles and hyperbolic triangles.
The Apollonian gasket is the limit set of a group of Möbius transformations known as a Kleinian group
.
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
. As this ratio is not rational, no integral Apollonian circle packings possess this D3 symmetry, although many packings come close.
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 factorizations demonstrate an interesting chain of factors throughout the sequence of gaskets, and the multiplier appears to be converging toward
(−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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an Apollonian gasket or Apollonian net is a fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
generated from triples of circles, where each circle is tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
to the other two. It is named after Greek
Greece
Greece , officially the Hellenic Republic , and historically Hellas or the Republic of Greece in English, is a country in southeastern Europe....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Apollonius of Perga
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
.
Construction
Gasket
thumb|sright|250px|Some seals and gaskets1. [[o-ring]]2. fiber [[Washer |washer]]3. paper gaskets4. [[cylinder head]] [[head gasket|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 can be any size, as long as they have common tangents). 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 (see Descartes' theorem
Descartes' theorem
In geometry, Descartes' theorem, named after René Descartes, establishes a relationship between four kissing, or mutually tangent, circles. The theorem can be used to construct a fourth circle tangent to three given, mutually tangent circles.-History:...
). 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 C1, 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 Apollonian gasket has a Hausdorff dimension
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
of about 1.3057 http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf.
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
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 circles that are tangent to one of the two straight lines form a family of Ford circle
Ford circle
In mathematics, a Ford circle is a circle with centre at and radius 1/, where p/q is an irreducible fraction, i.e. p and q are coprime integers...
s.
The three-dimensional equivalent of the Apollonian gasket is the Apollonian sphere packing
Apollonian sphere packing
Apollonian sphere packing is the three dimensional equivalent of the Apollonian gasket. The principle of construction is very similar: with any four spheres that are cotangent to each other, it is then possible to construct two more spheres that are cotangent to four of them.The fractal dimension...
.
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
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, so in hyperbolic geometry all Apollonian gaskets are congruent. In a sense, there is therefore only one Apollonian gasket, which can be thought of as a tessellation
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
of the hyperbolic plane by circles and hyperbolic triangles.
The Apollonian gasket is the limit set of a group of Möbius transformations known as a Kleinian group
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
.
Integral Apollonian circle packings
If any four mutually tangent circles in an Apollonian gasket all have integer curvature then all circles in the gasket will have integer curvature. 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.| Beginning curvatures | Symmetry |
|---|---|
| −1, 2, 2, 3, 3 | D2 |
| −2, 3, 6, 7, 7 | D1 |
| −3, 4, 12, 13, 13 | D1 |
| −3, 5, 8, 8, 12 | D1 |
| −4, 5, 20, 21, 21 | D1 |
| −4, 8, 9, 9, 17 | D1 |
| −5, 6, 30, 31, 31 | D1 |
| −5, 7, 18, 18, 22 | D1 |
| −6, 7, 42, 43, 43 | D1 |
| −6, 10, 15, 19, 19 | D1 |
| −6, 11, 14, 15, 23 | C1 |
| −7, 8, 56, 57, 57 | D1 |
| −7, 9, 32, 32, 36 | D1 |
| −7, 12, 17, 20, 24 | C1 |
| −8, 9, 72, 73, 73 | D1 |
| −8, 12, 25, 25, 33 | D1 |
| −8, 13, 21, 24, 28 | C1 |
| −9, 10, 90, 91, 91 | D1 |
| −9, 11, 50, 50, 54 | D1 |
| −9, 14, 26, 27, 35 | C1 |
| −9, 18, 19, 22, 34 | C1 |
| −10, 11, 110, 111, 111 | D1 |
| −10, 14, 35, 39, 39 | D1 |
| −10, 18, 23, 27, 35 | C1 |
| Beginning curvatures | Symmetry |
|---|---|
| −11, 12, 132, 133, 133 | D1 |
| −11, 13, 72, 72, 76 | D1 |
| −11, 16, 36, 37, 45 | C1 |
| −11, 21, 24, 28, 40 | C1 |
| −12, 13, 156, 157, 157 | D1 |
| −12, 16, 49, 49, 57 | D1 |
| −12, 17, 41, 44, 48 | C1 |
| −12, 21, 28, 37, 37 | D1 |
| −12, 21, 29, 32, 44 | C1 |
| −12, 25, 25, 28, 48 | D1 |
| −13, 14, 182, 183, 183 | D1 |
| −13, 15, 98, 98, 102 | D1 |
| −13, 18, 47, 50, 54 | C1 |
| −13, 23, 30, 38, 42 | C1 |
| −14, 15, 210, 211, 211 | D1 |
| −14, 18, 63, 67, 67 | D1 |
| −14, 19, 54, 55, 63 | C1 |
| −14, 22, 39, 43, 51 | C1 |
| −14, 27, 31, 34, 54 | C1 |
| −15, 16, 240, 241, 241 | D1 |
| −15, 17, 128, 128, 132 | D1 |
| −15, 24, 40, 49, 49 | D1 |
| −15, 24, 41, 44, 56 | C1 |
| −15, 28, 33, 40, 52 | C1 |
| −15, 32, 32, 33, 65 | D1 |
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
. 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 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 factorizations demonstrate an interesting chain of factors throughout the sequence of gaskets, and the multiplier appears to be converging toward
| 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
- Sierpiński triangleSierpinski triangleThe Sierpinski triangle , also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral...
- Apollonian networkApollonian networkIn combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable...
, a graph derived from finite subsets of the Apollonian gasket
External links
- Alexander Bogomolny, Apollonian Gasket, cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- An interactive Apollonian gasket running on pure HTML5 (the link is dead) 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.. Newspaper story about an artwork in the form of a partial Apollonian gasket, with an outer circumference of nine miles.