The statistical theory of continued fractions, as will be seen below, will solve the problem of analyzing. Suppose we start with a rational number, then euclids algorithm terminates in nitely. Every rational number has a simple continued fraction expansion which is nite and every nite simple continued fraction expansion is a rational number. The most common type of continued fraction is that of continued fractions for real numbers. Continued fractions are important in many branches of mathematics. Applications of hyperbolic geometry to continued fractions. It turns out, however, that fractions of this form, called continued fractions, provide much insight into many mathematical problems, particularly into the nature of numbers. Since the simple classical numbertheoretic continued fractions are the subject of the book, this proof clearly includes all such continued fractions. More recently, continued fractions have come up in studying the dynamics of flows.
The geometry of continued fractions anton lukyanenko it turns out that any real number can be written as a continued fraction. The book of fractions presents one of the primary concepts of middle and high school mathematics. Pdf in this small paper we bring together various open problems on geometric multidimensional continued fractions. Recognize and generate simple equivalent fractions, e. Geometry of continued fractions oleg karpenkov springer. The geometry of continued fractions and the topology of surface singularities patrick popescupampu abstract. Hyperbolic geometry, continued fractions and classi. Nov 29, 2014 we introduce the concepts of continued fractions informally. This paper on the geometry, algebra and arithmetics of continued fractions is based on a lecture for students, teachers and a nonspecialist audience, beginning with the history of the golden. Well see how working with continued fractions leads to thinking about hyperbolic geometry, and then talk about generalizations to complex continued fractions and heisenberg con. There is another way to represent real numbers called continued fractions.
Continued fractions and modular symbols introduction to fractal geometry and chaos matilde marcolli mat1845hs winter 2020, university of toronto m 56 and t 1012 ba6180 matilde marcolli continued fractions and modular symbols introduction to fractal geometry and chaos. Request pdf geometry of continued fractions preface. The chapter discusses simple continued fractions, nearest integer continued fractions over the euclidean imaginary quadratic elds, and includes a summary of a. In a natural way, slopes parametrize simple riesz groups g g. The continued fraction expansions of two numbers which di. The geometry of continued fractions as analysed by. Thus, the length of the continued fraction expansion of pqis the number of division steps in the euclidean algorithm applied to p. Draw the fan to illustrate the answer the the above 5. This minor excursion from number theory and algebra is a significant advantage to this particular book as it provides a bedrock for later rate of convergence discussions. The geometry of continued fractions and the topology of surface. We explore this representation using the isometric action of the group of mobius. Geometry of continued fractions with real elements and the second keplers law 12.
The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by hirzebruch be tween the continued fraction. This idea is expanded upon and clarified in a series of papers by caroline series. The approach to gaussian integer continued fractions is this paper has been geometric, allowing the lens of hyperbolic geometry to be used to illuminate the image of a continued fraction in hyperbolic space. Continued fractions, pells equation, and other applications jeremy booher continued fractions usually get shortchanged at promys, but they are interesting in their own right and useful in other areas of number theory. Such continued fractions are called simple continued fractions and have been extensively studied by numbertheorists 1,2,3. Explain why the fractions are equivalent by using a visual fraction model. This book introduces a new geometric vision of continued fractions. The geometrical interpretation of pq being a best approximation to a is that. A corresponding graphical representation diagram is. A continued fraction is a sequence of integers arranged in a nested fraction format.
We also show that f can be interpreted as a graph, and that the simple continued fraction expansion of any real number can be can be found by tracing a unique path on this graph. A geometric representation of continued fractions alan f. Visualize, describe, and make models of geometric solids e. Although we do not limit ourselves to this case in the course, it will be used. Continued fractions have been used in mathematics since the 16th century mainly as a tool for evaluating or approximating real numbers. Hyperbolic geometry and continued fraction theory i. The fractional dimensional theory of continued fractions. The geometric properties of continued fractions are apparent from the fact that a continued fraction is a composition of a. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations of continued fractions. Continued fractions m obius maps hyperbolic geometry toplogical groups three more reasons for using m obius maps there is already a well developed theory of m obius maps.
Lecture 1 we describe the farey tessellation fand give a very quick introduction to the basic facts we need from hyperbolic geometry, using the upper half plane model. Continued fractions and modular symbols introduction to. Check whether the following are neighboring pairs of farey numbers. Geometry of continued fractions 697 say, where piqn is in its lowest terms. Continued fractions from euclid to the present day contents.
Hyperbolic geometry and continued fraction theory ii. An introduction to the modern geometry of the triangle and. In this talk we introduce continued fractions, prove their basic properties and apply these properties to solve a practical problem. Anton lukyanenko faculty, 1 graduate student, 3 undergraduate students apply by january 15, 2018 one way to write a number like pi is to specify a sequence of continued fractions that represent it. For a rational, it is exactly the euclidean algorithm. For example, they given a way to write a prime congruent to 1 modulo 4 as a sum of two squares. Continued fractions were studied by the great mathematicians of the seventeenth and eighteenth centuries and are a subject of active investigation today. Numerous classical theorems have been extended to the multidimensional case, casting light on phenomena in diverse areas of mathematics.
The geometry of gaussian integer continued fractions. See, for instance, noneuclidean geometry, continued fractions, and ergodic theory. The rise of computational geometry has resulted in renewed interest in. Interestingly, it is related to many other topics in mathematics. On the one hand, a continued fraction can be represented as the union of two broken lines with the ends of their segments at integer points that approach alternately from the bottom and from the top to the straight line y x. Different continued fractions in a block somewhat describe different ways of splitting the integer value associated to that row. The algebra and geometry of continued fractions with. The geometry uncovers more than the classical algebraic analyses and we prove the. The reasons for including this topic in the course on classical algebra are. Proceedings of the roman number theory association volume 2, number 1, march 2017, pages 6181 michel waldschmidt continued fractions. The rise of computational geometry has resulted in renewed interest in multidimensional. We also state without proof some further properties of continued fractions and provide a brief overview of some facts in this connection.
Noneuclidean geometry, continued fractions, and ergodic theory caroline series 1 the mathematical intelligencer volume 4, pages 24 31 1982 cite this article. For this reason, from now on we restrict our attention to real numbers. Patterns, fractions, and geometry digital chalkboard. Continued fractions and the geometry of numbers burton rosenberg november 26, 2003 introduction a continued fraction is an expansion of a number with certain relationship to the euclidean algorithm. Noneuclidean geometry, continued fractions, and ergodic. Lecture 2 we introduce continued fractions and explain the relationship. Geometry of continued fractions mathematical association. The book of fractions the graphical representation of a fraction 6 f02.
In section 2, we present some new results for general conformal iterated function systems. Draw the farey diagrams to check your answers to the above 3. Geometrical generalisations of continued fractions to many dimensions were introduced recently as well 8, 36 one may expect that such an observation has been made a while ago, e. One traditional way to write a continued fraction is. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems. Oleg karpenkov traditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics.
The hyperbolic geometry of continued fractions k1jbn 317 says that the converse does hold when the bn are allpositive numbers. The notion of fractional dimensions is one which is now well known. The hyperbolic geometry of continued fractions emis annales academi. Traditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics. Inspired by work of ford, we describe a geometric representation of real and complex continued fractions by chains of horocycles and horospheres in hyperbolic space. The geometry of continued fractions and the topology of surface singularities. Basic definitions and facts a continued fraction is given by two sequences of numbers bnn. In every block, there is one rational number whose continued fraction is the most dispersed. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations. Continued fractions cornell department of mathematics. The geometry of continued fractions is a specialized and rich subject in its own right 30. Geometry of continued fractions ebook by oleg karpenkov.
Geometry of continued fractions oleg karpenkov auth. Let us return to 6 c d and in particular ponder the meaning of its continued fraction. Later in this section we recall from mu some major features of general conformal iterated function systems. Lecture 1 continued fractions the theory of continued fractions is closely related to the gauss reduction theory for inde nite integral quadratic forms translated into the matrix. Continued fractions are positive natural numbers arranged is a way to eventually approximate real numbers and other numerical forms. Geometry of continued fractions mathematical association of. On the other hand, a continued fraction can be represented as a chain of the socalled ford circles tangent to the xaxis. The algebra and geometry of continued fractions with integer. There are several books devoted entirely to the subject of continued fractions e. Pdf open problems in geometry of continued fractions. Background parabola theorem parabolic region sternstolz series hyperbolic geometry hyperbolic geometry and continued fraction theory ii ian short 16 february 2010. The hyperbolic geometry of continued fractions k1 jb. We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. The object of the present paper is the investigation of the dimensional numbers of sets of points which, when expressed as continued fractions, obey some simple restriction as to their partial quotients.
Thus it is natural to look at the continued fraction expansion. Reciprocally, continued fractions have applications to the euclidean algorithm. In this paper we obtain some results about general conformal iterated function systems. Worksheet number thirteen continued fractions we are all familiar with decimal numbers, and know that a repeating decimal gives a rational number, while one that never repeats and goes on forever represents an irrational number. In particular, continued fractions have a geometric interpretation in terms of integer geometry, which we. The geometry of continued fractions and the topology of.