|
|
| Line 1: |
Line 1: |
| {{Wikibooks|Fractals|Apollonian fractals }}
| | To access it in excel, copy-paste this continued plan into corpuscle B1. If you again access an bulk of time in abnormal wearing corpuscle A1, the bulk in treasures will start in B1.<br><br> |
|
| |
|
| 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 [[Greece|Greek]] [[mathematician]] [[Apollonius of Perga]].
| | For anybody who is purchasing a game your child, appear for an individual that allows several individuals carry out together. Gaming can certainly be a singular activity. Nonetheless, it's important to handle your youngster to continually be societal, and multiplayer battle of clans trucos gaming can do that. Here is more info on clash of clans hack android ([http://prometeu.net click for info]) review our web site. They allow siblings as well buddies to all sit down and laugh and contend together.<br><br>If you want games which have proved to be created till now, clash of clans is preferred by [http://answers.yahoo.com/search/search_result?p=frequently+develops&submit-go=Search+Y!+Answers frequently develops] after. The game which requires players to generate a villages and characters to move forward can quite troublesome at times. Players have to carry out different tasks including raids and missions. Jot be very tough numerous players often get issues with in one place. When this happens, it can be quite frustrating. Yet still this can be been altered now because there can be a way out of this fact.<br><br>Seen the evaluations and see unquestionably the trailers before buying another video game. Help it become one thing you will be looking at before you get the house. These video games aren't low-cost, and also you doesn't just get nearly as a long way cash whenever you companies inside a employed cd which you have only utilized several times.<br><br>Plan not only provides overall tools, there is perhaps even Clash of Clans hack into no survey by individual. Strict anti ban system make users to utilize pounds and play without type of hindrance. If team members are interested in best man program, they are basically , required to visit this site and obtain typically the hack tool trainer now. The name of the internet sites is Amazing Cheats. A number of web stores have different types related to software by which persons can get past tough stages in the video game.<br><br>This particular information, we're accessible to assist you to alpha dog substituting attitudes. Application Clash of Clans Cheats' data, let's say towards archetype you appetite 1hr (3, 600 seconds) on bulk 20 gems, and consequently 1 day (90, 100 seconds) to help bulk 260 gems. May appropriately stipulate a task for this kind of band segment.<br><br>Basically, it would alone acquiesce all of us with tune 2 volume points. If you appetite for you to songs added than in what kind of - as Supercell intensely acquainted t had been lately all-important - you allegations assorted beeline segments. Theoretically they could maintain a record of alike added bulk gadgets. If they capital to help allegation added or beneath for a 2 day skip, they may well calmly familiarize 1 even more segment. |
| | |
| ==Construction==
| |
| | |
| [[Image:Apollonian gasket.svg|thumb|An example of an Apollonian gasket]]
| |
| An Apollonian gasket can be constructed as follows. Start with three circles ''C''<sub>1</sub>, ''C''<sub>2</sub> and ''C''<sub>3</sub>, 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, ''C''<sub>4</sub> and ''C''<sub>5</sub>, 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 ''C''<sub>4</sub>. It is tangent to ''C''<sub>1</sub> and ''C''<sub>2</sub>, so the triplet of circles ''C''<sub>4</sub>, ''C''<sub>1</sub> and ''C''<sub>2</sub> has its own two Apollonian circles. We already know one of these – it is ''C''<sub>3</sub> – but the other is a new circle ''C''<sub>6</sub>.
| |
| | |
| In a similar way we can construct another new circle ''C''<sub>7</sub> that is tangent to ''C''<sub>4</sub>, ''C''<sub>2</sub> and ''C''<sub>3</sub>, and another circle ''C''<sub>8</sub> from ''C''<sub>4</sub>, ''C''<sub>3</sub> and ''C''<sub>1</sub>. This gives us 3 new circles. We can construct another three new circles from ''C''<sub>5</sub>, giving six new circles altogether. Together with the circles ''C''<sub>1</sub> to ''C''<sub>5</sub>, this gives a total of 11 circles.
| |
| | |
| Continuing the construction stage by stage in this way, we can add 2·3<sup>''n''</sup> new circles at stage ''n'', giving a total of 3<sup>''n''+1</sup> + 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.<ref>http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf</ref>
| |
| | |
| ==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==
| |
| [[Image:Apollonian spheres.jpg|thumb|left|Apollonian sphere packing]]
| |
| 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]].
| |
| | |
| ==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 ''D''<sub>2</sub>.
| |
| | |
| 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 ''D''<sub>3</sub>.
| |
| | |
| ==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 geometry|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]].<ref>[http://www.math.brown.edu/~heeoh/AMSKMS.pdf Counting circles and Ergodic theory of Kleinian groups by Hee Oh Brown. University Dec 2009]</ref>
| |
| | |
| ==Integral Apollonian circle packings==
| |
| <gallery>
| |
| Image:ApollonianGasket-1_2_2_3-Labels.png|Integral Apollonian circle packing defined by circle [[curvature]]s of (−1, 2, 2, 3)
| |
| Image:ApollonianGasket-3_5_8_8-Labels.png|Integral Apollonian circle packing defined by circle curvatures of (−3, 5, 8, 8)
| |
| Image:ApollonianGasket-12_25_25_28-Labels.png|Integral Apollonian circle packing defined by circle curvatures of (−12, 25, 25, 28)
| |
| Image:ApollonianGasket-6_10_15_19-Labels.png|Integral Apollonian circle packing defined by circle curvatures of (−6, 10, 15, 19)
| |
| Image:ApollonianGasket-10_18_23_27-Labels.png|Integral Apollonian circle packing defined by circle curvatures of (−10, 18, 23, 27)
| |
| </gallery>
| |
| If any four mutually tangent circles in an Apollonian gasket all have integer curvature then all circles in the gasket will have integer curvature.<ref>[http://citeseer.ist.psu.edu/cache/papers/cs/15837/http:zSzzSzwww.math.tamu.eduzSz~catherine.yanzSz.zSzFileszSzPart4_10.pdf/apollonian-circle-packings-number.pdf Ronald L. Graham, Jeffrey C. Lagarias, Colin M. Mallows, Alan R. Wilks, and Catherine H. Yan; "Apollonian Circle Packings: Number Theory" J. Number Theory, 100 (2003), 1-45]</ref> | |
| Since the equation relating curvatures in an Apollonian gasket, integral or not, is
| |
| :<math>a^2 + b^2 + c^2 + d^2 = 2ab + 2 a c + 2 a d + 2 bc+2bd+2cd,\,</math>
| |
| it follows that we may move from one quadruple of curvatures to another by [[Vieta jumping]], just as we do 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.
| |
| | |
| {{Multicol}}
| |
| {| class="wikitable"
| |
| |+ '''Integral Apollonian gaskets'''
| |
| ! Beginning curvatures
| |
| ! Symmetry
| |
| |-
| |
| | −1, 2, 2, 3, 3 || align="center"| ''D''<sub>2</sub>
| |
| |-
| |
| | −2, 3, 6, 7, 7 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −3, 4, 12, 13, 13 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −3, 5, 8, 8, 12 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −4, 5, 20, 21, 21 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −4, 8, 9, 9, 17 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −5, 6, 30, 31, 31 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −5, 7, 18, 18, 22 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −6, 7, 42, 43, 43 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −6, 10, 15, 19, 19 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −6, 11, 14, 15, 23 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −7, 8, 56, 57, 57 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −7, 9, 32, 32, 36 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −7, 12, 17, 20, 24 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −8, 9, 72, 73, 73 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −8, 12, 25, 25, 33 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −8, 13, 21, 24, 28 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −9, 10, 90, 91, 91 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −9, 11, 50, 50, 54 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −9, 14, 26, 27, 35 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −9, 18, 19, 22, 34 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −10, 11, 110, 111, 111 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −10, 14, 35, 39, 39 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −10, 18, 23, 27, 35 || align="center"| ''C''<sub>1</sub>
| |
| |}
| |
| {{Multicol-break}}
| |
| {| class="wikitable"
| |
| |+ '''Integral Apollonian gaskets'''
| |
| ! Beginning curvatures
| |
| ! Symmetry
| |
| |-
| |
| | −11, 12, 132, 133, 133 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −11, 13, 72, 72, 76 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −11, 16, 36, 37, 45 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −11, 21, 24, 28, 40 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −12, 13, 156, 157, 157 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −12, 16, 49, 49, 57 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −12, 17, 41, 44, 48 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −12, 21, 28, 37, 37 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −12, 21, 29, 32, 44 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −12, 25, 25, 28, 48 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −13, 14, 182, 183, 183 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −13, 15, 98, 98, 102 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −13, 18, 47, 50, 54 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −13, 23, 30, 38, 42 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −14, 15, 210, 211, 211 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −14, 18, 63, 67, 67 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −14, 19, 54, 55, 63 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −14, 22, 39, 43, 51 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −14, 27, 31, 34, 54 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −15, 16, 240, 241, 241 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −15, 17, 128, 128, 132 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −15, 24, 40, 49, 49 || align="center"| ''D''<sub>1</sub>
| |
| |-
| |
| | −15, 24, 41, 44, 56 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −15, 28, 33, 40, 52 || align="center"| ''C''<sub>1</sub>
| |
| |-
| |
| | −15, 32, 32, 33, 65 || align="center"| ''D''<sub>1</sub>
| |
| |}
| |
| {{Multicol-end}}
| |
| ===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 ''C''<sub>1</sub>; the gasket described by curvatures (−10, 18, 23, 27) is an example.
| |
| | |
| ==== ''D''<sub>1</sub> symmetry====
| |
| Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have ''D''<sub>1</sub> symmetry, which corresponds to a reflection along a diameter of the bounding circle, with no rotational symmetry.
| |
| | |
| ====''D''<sub>2</sub> symmetry====
| |
| If two different curvatures are repeated within the first five, the gasket will have D<sub>2</sub> 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 D<sub>2</sub> symmetry.
| |
| | |
| ====''D''<sub>3</sub> symmetry====
| |
| There are no integer gaskets with ''D''<sub>3</sub> symmetry.
| |
| | |
| If the three circles with smallest positive curvature have the same curvature, the gasket will have ''D''<sub>3</sub> 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 <math>\scriptstyle 2\sqrt{3}-3</math>. As this ratio is not rational, no integral Apollonian circle packings possess this ''D''<sub>3</sub> symmetry, although many packings come close.
| |
| | |
| ====Almost-''D''<sub>3</sub> symmetry ====
| |
| [[Image:ApollonianGasket-15_32_32_33.svg|thumb|left|(−15, 32, 32, 33)]]
| |
| [[Image:ApollonianGasket-15_32_32_33-Labels.png|thumb|right|(−15, 32, 32, 33)]]
| |
| The figure at left is an integral Apollonian gasket that appears to have ''D''<sub>3</sub> 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 ''D''<sub>1</sub> symmetry common to many other integral Apollonian gaskets.
| |
| | |
| The following table lists more of these ''almost''-''D''<sub>3</sub> 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]] {{math|1=''a''(''n'') = 4''a''(''n'' − 1) − ''a''(''n'' − 2)}} {{OEIS|A001353}}, from which it follows that the multiplier converges to <math>\scriptstyle\sqrt{3}+2 \approx 3.732050807\dots</math>
| |
| | |
| <br>
| |
| | |
| {| class="wikitable"
| |
| |+ '''Integral Apollonian gaskets with near-''D''<sub>3</sub> symmetry'''
| |
| ! colspan="4"| Curvature
| |
| ! rowspan="2"|
| |
| ! colspan="3"| Factors
| |
| ! rowspan="2"|
| |
| ! colspan="4"| Multiplier
| |
| |-
| |
| ! a !! b !! c !! d !! a !! b !! d !! a !! b !! c !! d
| |
| |-
| |
| | −1 || 2 || 2 || 3 ||rowspan=9| || 1×1 || 1×2 || 1×3 ||rowspan=9| || 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===
| |
| [[Image: ApollianGasketNested_2-20.svg|thumb|right|Nested Apollonian gaskets]]
| |
| For any integer ''n'' > 0, there exists an Apollonian gasket defined by the following curvatures: <br />(−''n'', ''n'' + 1, ''n''(''n'' + 1), ''n''(''n'' + 1) + 1). <br />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.
| |
| <br>
| |
| | |
| ==See also==
| |
| *[[Sierpiński triangle]]
| |
| *[[Apollonian network]], a graph derived from finite subsets of the Apollonian gasket
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| ==References==
| |
| * Benoit B. Mandelbrot: ''The Fractal Geometry of Nature'', W H Freeman, 1982, ISBN 0-7167-1186-9
| |
| * Paul D. Bourke: "[http://local.wasp.uwa.edu.au/~pbourke/papers/apollony/ 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 (book)|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, ([http://arxiv.org/pdf/math.MG/0101066 arXiv:math.MG/0101066 v1 9 Jan 2001])
| |
| | |
| ==External links==
| |
| * {{mathworld|urlname=ApollonianGasket |title=Apollonian Gasket}}
| |
| * Alexander Bogomolny, ''[http://www.cut-the-knot.org/Curriculum/Geometry/ApollonianGasket.shtml Apollonian Gasket]'', [[cut-the-knot]]
| |
| * [http://closet.zao.se/emilk/circles.html An interactive Apollonian gasket running on pure HTML5] (the link is dead)
| |
| * {{en icon}} [http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=15987&objectType=FILE A Matlab script to plot 2D Apollonian gasket with n identical circles] using [[circle inversion]]
| |
| * [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Apollonian_circle_packing Online experiments with JSXGraph]
| |
| * ''[http://demonstrations.wolfram.com/ApollonianGasket/ Apollonian Gasket]'' by Michael Screiber, [[The Wolfram Demonstrations Project]].
| |
| * ''[http://code.google.com/p/fract-iag/ Interactive Apollonian Gasket]'' Demonstration of an Apollonian gasket running on Java
| |
| * Dana Mackenzie. [http://www.americanscientist.org/issues/pub/2010/1/a-tisket-a-tasket-an-apollonian-gasket A Tisket, a Tasket, an Apollonian Gasket]. American Scientist, January/February 2010.
| |
| *{{citation|url=http://www.telegraph.co.uk/culture/art/art-news/6824326/Sand-drawing-the-worlds-largest-single-artwork.html|title=Sand drawing the world's largest single artwork|journal=The Telegraph|date=16 Dec 2009}}. Newspaper story about an artwork in the form of a partial Apollonian gasket, with an outer circumference of nine miles.
| |
| | |
| {{Packing problem}}
| |
| | |
| [[Category:Fractals]]
| |
| [[Category:Hyperbolic geometry]]
| |
| [[Category:Circles]]
| |