A tisket, a tasket, an apollonian gasket dana mackenzie i n the spring of 2007 i had the good fortune to spend a semester at the mathematical sciences research institute in berkeley, an institution of higher learning that takes higher to a whole new extreme. This fractal is not particularly well known, perhaps because it is not as straightforward to construct as many other fractals such as the related sierpinski gasket or the menger sponge. The good thing is it can start w any arbitrary configuration of main circles, unlike the standard way. Apollonian gaskets and descartes theorem the math less. Part 1 the construction of the apollonian gasket in order to describe the construction of the apollonian gasket, we will first introduce some important terms. Apollonian gasket 15,32,32,33 by kihyuck hong file apolonian gasket containes about 8140 circles, but it looks like made of about 20 steps. The butterfly plot inherits its genes from these fractals. If we know the curvatures of any three mutually tangent circles in an apollonian gasket then we can use repeated applications of descartes theorem to construct the whole gasket. The apollonian and dionysian is a philosophical and literary concept and dichotomy dialectic, based on apollo and dionysus in greek mythology. Each circle in the apollonian gasket is tangent to the adjacent circles in other words, the circles in the apollonian gasket make contact at infinitely small points. The apollonian gasket is integral if and only if gcd, 1.
This book is devoted to a phenomenon of fractal sets, or simply fractals. More to the point, the were both gods in the ancient greek pantheon, despite representing nearly opposing values and orientations. Localglobal phenomena in thin orbits apollonian group. Consider three mutually tangent circles, and draw their inner soddy circle. Atisket, atasket, an apollonian gasket 3 quarks daily. This is where something called descartes theorem begins to play a starring role.
The result is a multilevel residence with winding passageways and mosaiccovered patios overlooking the mediterranean. But to actually draw an apollonian gasket, we have to start dealing concretely with centers, radii, and so onhow do we actually compute the positions and sizes of the circles involved. But nissims obsession with endless construction, and his prickly personality, has driven away his family. Along with fundamental definitions and properties, some of the key concepts and approaches covered include the laplace operator harmonic.
Membership in the apollonian society is open to all friends of the ucla school of dentistryalumni, faculty members, residency program graduates, professional colleagues, and anyone else who has an interest in advancing oral health education. Apollonian gasket, 978625878, please note that the content of this book primarily consists of articles available from wikipedia or other. Sierpinski gasket, harmonic functions on sierpinski gasket, applications of generalized numerical systems, apollonian gasket, arithmetic properties of apollonian gaskets, geometric and grouptheoretic approach. Friedrich nietzsche used the term in his book the birth of tragedy to describe one of the two opposing tendencies or elements in greek tragedy. The brief history and a general description of the fractal will be. That is, for a fixed integral, primitive apollonian gasket, almost every in the sense of density admissible passing local obstructions integer is the curvature of some circle in the gasket. Apollonian gasket plural apollonian gaskets english wikipedia has an article on. Pearls, mathematician david wright compared the gasket to dr. Named for the greek mathematician apollonius of perga, this type of fractal can. Apollonian and dionysian impact on literature english literature essay. Given a circle with radius, define the curvature of, denoted, by. The mandelbrot set as shown herein may appear to be merely a distant cousin of the butterfly. Abstract this paper provides an introduction to the apollonian fractal, also known by some as the curvilinear sierpinski gasket.
Over the course of the book, they are developed and discussed in various contexts. An introduction to the apollonian fractal sciencedirect. Apollonian and dionysian are terms used by friedrich. Free fractals books download ebooks online textbooks. As its title suggests, this book focuses primarily on two fractals. These open regions fill up quickly leaving a fractal of infinitely many points that are never in any circle. This example uses that method to build an apollonian gasket. According to nietzsche, the apollonian attributes are reason, culture, harmony, and restraint. By slightly changing the circlesschottky group, the limit set can be made to look like a twisting gasket. Apollonian lit has 6 books on goodreads, and recently added insurgent by veronica roth, itch by michelle d.
A tisket, a tasket, an apollonian gasket trinity college dublin. Introduction an apollonian circle packing is a packing of circles arising by repeatedly. We prove that a set of density one satisfies the localglobal conjecture for integral apollonian gaskets. An introduction to the apollonian fractal paul bourke. Download mathematica notebook apolloniangasketarray. This paper provides an introduction to the apollonian fractal, also known by some as the curvilinear sierpinski gasket. On the localglobal conjecture for integral apollonian gaskets. Apollo and dionysus were gods in ancient creek religion. This file is licensed under the creative commons attributionshare alike 4. Provided to youtube by routenote apollonian gasket math dealer growing numbers in jars of vermiculite.
Introduction to apollonian circle packings tangent. Wikipedia mathematics a fractal generated from triples of circles, where each circle is tangent. This is not an example of the work produced by our essay writing service. At each step inscribe new tangent disks into each open curvilinear triangle. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. A problem fit for a princess math teachers circles. So it should have about 10 460 353 205 circles apollonian gasketin haskell and diagrams by brent yorgey apollonian. On a diophantine equation that generates all integral. It simply randomly finds 3 tangent circles at each iteration and tries to add new circles. It is named after greek mathematician apollonius of perga. Apollonius of perga 262190 bc was a well known and prolific geometer in. In mathematics, an apollonian gasket or apollonian net is a fractal generated starting from a. The brief history and a general description of the fractal will be given, including the.
Figure 2 locates the curvatures of the theorem in the apollonian gasket. Files are available under licenses specified on their description page. This page was last edited on 27 october 2018, at 04. An introduction to the apollonian fractal semantic scholar. An introduction to the apollonian fractal paul bourke email.
A copy of the license is included in the section entitled gnu free documentation license. Generalized apollonian gasket fractal python recipes. Perhaps the most astounding fact about the apollonian gasket is that if the first four circles have integer bends, then every other circle in the packing does too. And in a book about fractals with the lovely title indras. All structured data from the file and property namespaces is available under the creative commons cc0 license. We show these groups are hyperbolic coxeter groups.
An apollonian gasket can be constructed as follows. And in a book about fractals with the lovely title indras pearls, mathematician david wright compared the gasket. To understand how to make an apollonian gasket as a limit set, read the book. Butterfly in the quantum world, story of a most fascinating quantum fractal, in dubala i. This is not my area of research at all, but i think its super cool. If you are given the first three circles of an apollonian gasket, the bend of the fourth is found as explained above by solving a quadratic equation. It draws apollonian gasket fractal for any n using descartes theorem. Apollonian group which is the group generated by the apollonian and dual apollonian groups together. Apollonian story full frame documentary film festival. Newspaper story about an artwork in the form of a partial apollonian gasket, with an outer.
Fractalsapollonian fractals wikibooks, open books for. The next three arguments are bends of three circles. If you think of each circle as cutting a hole in the enclosing circle so you get a lacelike figure, its an. An apollonian gasket is a type of fractal image that is formed from a collection of evershrinking circles contained within a single large circle. And if any four mutually tangent circles in the gasket have integer curvatures, then all circles in the gasket. 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. Drag the slider to make a twisted apollonian gasket. In addition to fractals, other related concepts include soddy circles and ford circles. Some western philosophical and literary figures have invoked this dichotomy in critical and creative works, most notably friedrich nietzsche and later followers. Geometry apollonian gasket malin christerssons math site. 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. One of several names for a series of circles, nested inside one large circle, and tangent to.
Apollonian definition, etymology and usage, examples and. This kind of a shape is a fractal pattern called an apollonian gasket, honoring apollonius and his tangencies. Apollonian and dionysian impact on literature english. Apollo was the god of light, reason, harmony, balance and prophesy, while. A tisket, a tasket, an apollonian gasket american scientist. Apollonian, of, relating to, or resembling the god apollo. When making a geometrical construction of an apollonian gasket, new circles. An apollonian gasket can also be generated as a socalled limit set of a socalled schottky group made by pairing tangent circles. The lunar crater apollonius was so named to honor his work on the history of the. The first time i heard of apollonian circle packings was at that conference i went to in marseille last month, during the first lecture of hee ohs minicourse. Nissim kahlon has spent the last forty years digging a home out of a limestone cliff in israel. So heres a quick write up background of the first third of. Prominent number theorists have studied some of the properties of the numbers that appear as curvatures in a given gasket pattern. This book tells the tale of the marriage of the apollonian gaskets kiss precise.
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