Hyperbolic geometry kleinian groups and the apollonian gasket a thesis submitted in partial fulfilment of the requirements for the degree of master of science in. Hyperbolic geometry is a subset of a large class of geometries called noneuclidean geometries. The result of this iteration is called the apollonian gasket. The nicest way to resolve these problems is to use the theory of apollonian families. Let p be the center ofthe sphere that represents the hyperplane nx 0.
For a bounded packing p, let n p t be the number of circles in the packing having curvature at most t. A unit circle is any circle in the euclidean plane is a circle with radius one. This is a set of notes from a 5day doityourself or perhaps discoverityourself introduction to hyperbolic geometry. For any p on the circle, the internal and external bisectors of angle apb pass through the fixed points c and d. Now we study some properties of hyperbolic geometry which do not hold in euclidean geometry. It outlines the basic structure of lorentz 3space which allows the rst model of the hyperbolic plane to be derived. All points in the interior of the circle are part of the hyperbolic plane. The newly constructed circles are part of several more triples of tangent circles, so one can iterate this process, producing more and more tangent circles. On the hyperbolic plane, given a line land a point pnot contained by l, there are two parallel lines to lthat contains pand move. We show these groups are hyperbolic coxeter groups. Pdf apollonian circles and hyperbolic geometry matti. Simplicial complexes are the manybody generalization of networks 1,2,3,4,5,6 and they can encode interactions occurring between two or more nodes 7,8,9,10,11,12,14. The hyperbolic re ection r m is the restriction of the inversion c to h2.
The hyperbolic tangents theorem if p lies on the h circle k kq,r, then 1 there is a unique htangent. We first obtain the hyperbolic area and length formula of euclidean disk and a circle represented by its euclidean center and radius. Our formula involves notions from hyperbolic geometry. Show that the geodesics on a sphere are exactly the great circles by. A pictorial overview abstract this article provides a simple pictorial introduction to universal hyperbolic geometry. The circle of apollonius in hyperbolic geometry geometricorum. Graham, ronald l lagarias, jeffrey c mallows, colin l wilks, allan r yan, catherine h 2005. Geometry and arithmetic of crystallographic sphere packings. As its application, we give the hyperbolic area of a. Apollonian circle packings are examples of circle packing obtained in. The apollonian octets and an inversive form of krauses. In euclidean geometry the circle of apollonius is the locus of points in the plane from which two collinear adjacent segments are perceived as. Euclidean and noneuclidean geometry information page, fall 2016 i will be in my office on friday, 12 pm.
Stange, university of colorado, boulder joint work with sneha chaubey, university of illinois at urbanachampaign elena fuchs, university of illinois at urbanachampaign robert hines, university of colorado, boulder height0. The first problem is to compare the hyperbolic and euclidean. Let points e and f be the midpoints of the base and summit, respectively. Counting problems for apollonian circle packings an apollonian circle packing is one of the most of beautiful circle packings whose construction can be described in a very simple manner based on an old theorem of apollonius of perga. Oct, 2010 terms of apollonian circles and give examples for the determination of some natural concepts of hyperbolic geometry such as the midp oint of a geodesic and the base points of a g eodesic. Since the hyperbolic re ection r mtakes hyperbolic lines to hyperbolic lines, the images r m and r m0 must be a. The apollonian circles are defined in two different ways by a line segment denoted cd each circle in the first family the blue circles in the figure is associated with a positive real number r, and is defined as the locus of points x such that the ratio of distances from x to c and to d equals r. We study the relation between the mob2action and the isoq2action, showing that the. In fractal geometry, there are other notions of dimensions which often have di erent values.
Points on the circumference of the circle are not part of the plane itself. Description in this seminar we will explore various topics related to symmetry groups, tilings of euclidean or hyperbolic space, hyperbolic geometry and number theory. These circles form the basis for bipolar coordinates. A conformal model is one for which the metric is a pointbypoint scaling of the euclidean metric.
Albert einsteins special theory of relativity is based on hyperbolic. Hee oh of brown university discusses counting and equidistribution results for circle packings in the plane invariant under a kleinian group at the 50th annual cornell topology festival, may 6, 2012. Aug 21, 2017 circle of appolonius mathematics study material online visit our website for complete lectures study materials notes gu. Hyperbolic axiom 1 let there be a line l and a point p such that p does not lie on l. Play with apollonian gasket of the first dozen circles. Pdf the goal of this paper is to study two basic problems of hyperbolic geometry. Pdf hyperbolic geometry kleinian groups and the apollonian. Since, for the same a and b, each of the apollonian circles corresponds to a different r, no two apollonian circles intersect.
Instead, we will develop hyperbolic geometry in a way that emphasises the similarities and more interestingly. Compacti cation and isometries of hyperbolic space 36 2. Emergent hyperbolic network geometry scientific reports. Apollonian packings and hyperbolic geometry springerlink. Distance is measured in such a way that equal hyperbolic distances are represented by ever smaller euclidean distances toward the bounding circle. Replacing interior angles with vertices coordinates, we also obtain a new hyperbolic area formula of a hyperbolic triangle. For values of r close to zero, the corresponding circle is close.
Three are conformal models associated with the name of henri poincar e. Constructing apollonian circle using kseg interactive geometry software. Hyperbolic geometry, symmetry groups, and more prof. In euclidean geometry the circle of apollonius is the locus of points in the plane from which two collinear adjacent segments are perceived as having the same length. But geometry is concerned about the metric, the way things are measured. The problem of apollonius is to find the distinct circles or lines, or points which can be. Applications of hyperbolic geometry to continued fractions and diophantine approximation thesis directed by prof. Apollonius circle construction problems famous math.
Publication date 1996 topics geometry, hyperbolic history sources. In this session, well examine the implications of breaking the 5th postulate by constructing and exploring hyperbolic geometry. Hyperbolic circles as apollonian circles recall that x. If the line l is orthogonal to the xaxis, an inversion about l restricting on h 2 is the same as a hyperbolic re. Points, lines, and triangles in hyperbolic geometry. In 10 it was observed that there exist notions of integer apollonian packings in spherical and hyperbolic geometry. The reason why perpendicular generalized arcs are used as hyperbolic lines to model hyperbolic geometry, is because of the properties of perpendicular circles when doing circle inversion. Given 3 mutually tangent circles in the plane, there exist exactly two circles tangent to all three. Number theory and the circle packings of apollonius peter sarnak. Note also that the lines of the construction form three triples that meet in common points. Diy hyperbolic geometry kathryn mann written for mathcamp 2015 abstract and guide to the reader. The beltramiklein model o r klein model for studying hyperbolic geometry in this model, a circle is fixed with center o and fixed radius. Download a new look at geometry ebook for free in pdf and epub format. If four mutally tangent circles with integral curvature are chosen, further choices of circles tangent to three others an apollonian packing will also have integral curvature.
Euclidean geometry is usually the most convenient to describe the physical world surrounding us. Wilker department of mathematics university of toronto toronto, ontario, canada mss ial submitted by chandler davis abstract suppose we are given three disjoint circles in the euclidean plane with the property that none of them contains the other two. The apollonian group is a hyperbolic coxeter group. Circle of appolonius mathematics study material online youtube. Hyperbolic geometry 63 we shall consider in this exposition ve of the most famous of the analytic models of hyperbolic geometry. In hyperbolic geometry, the analog of this locus is an alge braic curve of degree four which can be bounded or unbounded.
In the proof of the lemma on hcircles, we observed that k. Apollonian circles are two families of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. 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. Determining the possible distributions of curvatures is a di. The second problem is to find hyperbolic counterparts of some basic. The circles defined by the apollonian pursuit problem for the same two points a and b, but with varying ratios of the two speeds, are disjoint from each other and form a continuous family that cover the entire plane. The goal of this paper is to study two basic problems of hyperbolic geometry. Effective bisector estimate with application to apollonian. In hyperbolic geometry, the analog of this locus is an alge. This is a model of h2, the hyperbolic plane, in which the geodesics run along euclidean circles orthogo. In each step, find the interior tangent circle to a triple and generate three new triples, each containing the new circle and a pair of circles from the previous triple. These are based on analogues of the descartes equation valid in these geometries, as 2. In particular, read the one about apollonian circles theorem, which is related to problem 39. Apollonian circles and hyperbolic geometry nasaads.
This particular fractal is known as the apollonian gasket and consists of a complicated arrangement of tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral apollonian circle packing. Apollonian group which is the group generated by the apollonian and dual apollonian groups together. Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In a saccheri quadrilateral, the summit is longer than the base andthe segment joiningtheir midpoints is shorter than each arm. In fact, besides hyperbolic geometry, there is a second noneuclidean geometry that can be characterized by the behavior of parallel lines. Number theory and the circle packings of apollonius. Ford circles are important objects for studying the geometry. For a periodic packing, n p t is the number of such circles in one period. Fuchs, arithmetic properties of apollonian circle packings, ph. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an \em integral apollonian circle packing.
Oct, 2010 the goal of this paper is to study two basic problems of hyperbolic geometry. Then p is a linear combination of nand e, and satis. The foundations of hyperbolic geometry are based on one axiom that replaces euclids fth postulate, known as the hyperbolic axiom. Stange this dissertation explores relations between hyperbolic geometry and diophantine approximation, with an emphasis on continued fractions over the euclidean imaginary quadratic elds, qp. Template of the apollonian window pdf for printing. Moreover, we systematically generalize the construction of packings in terms of the coxeter group theory, and propose a computational algorithm to draw the pictures efficiently.
Mar 11, 20 a particular instance of this question is the apollonian circle packing problem. A new hyperbolic area formula of a hyperbolic triangle and. Basic fact 2 leaves the possibility that an hline and h circle meet once. We also mentioned in the beginning of the course about euclids fifth postulate. Chapter 15 hyperbolic geometry math 4520, spring 2015 so far we have talked mostly about the incidence structure of points, lines and circles. Apollonian circles and hyperbolic metric for x 2b2 n0, the hyperbolic circle centered at x is an apollonian circle with the base points x,x hyperbolic geodesic hyperbolic geodesics are arcs of circles, which are orthogonal to the boundary of the domain. Recent advances in modular forms, ergodic theory, hyperbolic geometry, and additive combinatorics. Hyperbolic geometry is an imaginative challenge that lacks important. At bottom, an approximation of the nished packing, with. Applications of hyperbolic geometry to continued fractions. The apollonian circle packing and ample cones for k3 surfaces 3 is a euclidean metric on. It is named after greek mathematician apollonius of perga.
In euclidean space it is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral apollonian circle packing. From apollonian circle packings to fibonacci numbers. Draw your own apollonian window recipe plus the data for the first seven hundred circles. But for the residual set of an apollonian circle packing, the hausdor dimension, the packing dimension and the box dimension are all equal to each other 55. Here is another place where many interesting results related to inversion are discussed. Since and 0are circles through q, the center of c, the inversions c and c0 are contained in straight lines as opposed to circles. The apollonian fractal is created by repeatedly placing inner soddy circles into the gaps between. The apollonian fractal is named after apollonius of perga circa 262bc to 190bc who. An apollonian circle packing is one of the most of beautiful circle packings. We explain how to understand the subject using only elementary projective geometry, augmented by a distinguished circle. The second problem is to find hyperbolic counterparts of some basic geometric constructions such as the construction of the middle point of a hyperbolic geodesic segment. Euclidean geometry and indras pearls by caroline series and david wright many people will have seen and been amazed by the beauty and intricacy of fractals like the one shown on the right.
An introduction to the apollonian fractal paul bourke email. An introduction to the apollonian fractal paul bourke. The first problem is to compare the hyperbolic and euclidean distances. Here the straight lines distance minimising lines are great circles i. Here is a short discussion of inversion with many references. On the other hand hyperbolic geometry describes spacetime more conveniently. They were discovered by apollonius of perga, a renowned greek geometer.
An apollonian packing is one of the most beautiful circle packings based on an old theorem of apollonius of perga. We prove now that every inversion is a hyperbolic re. On ther other hand, any inversion about circles are hyperbolic. Theorem 4 if a circle \d\ is inverted in another circle \c\, then \d\ is inverted to itself if and only if \c\ and \d\ are perpendicular.
Everything from geodesics to gaussbonnet, starting with a. This fractal is not particularly well known, perhaps because. On the one hand, infinitely many such generalized objects exist, but on the other, they may, in principle, be completely classified, as they fall into, only finitely, many families, all in bounded dimensions. Armed with the methods of construction given here, the. An apollonian circle packing is one of the most of beautiful circle packings whose construction can be described in a very simple manner based on an old theorem of apollonius of perga. Pdf apollonian circles and hyperbolic geometry researchgate. Jan 08, 2019 this paper studies generalizations of the classical apollonian circle packing, a beautiful geometric fractal that has a surprising underlying integral structure. The curvature of a circle is the inverse of its radius. The apollonian octets and an inversive form of krauses theorem j. The circle of apollonius in hyperbolic geometry eugen j. Definition if the hline h meets the h circle k in a single point p, then h is a hyperbolic tangent htangent to k at p. This provides a completely algebraic framework for hyperbolic geometry. In this sense, even though these two weak metrics are not conformally invariant, they both capture a certain aspect of the hyperbolic geometry realized in the two well.
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