# The Beauty of Fractals: Six Different Views

Denny Gulick
Jon Scott
Series: MAA Notes
Volume: 76
Edition: 1
Pages: 107
https://www.jstor.org/stable/10.4169/j.ctt5hh8gw

1. Front Matter
(pp. i-vi)
2. Preface
(pp. vii-viii)
Jon Scott and Denny Gulick
(pp. ix-x)
4. 1 Mathscapes—Fractal Scenery
(pp. 1-22)
Anne M. Burns

As a mathematician who started out as an art major, I became very excited when the first home computers appeared on the scene in the mid eighties. In 1987 I attended a conference at NYU (I believe it was called “Computer Graphics for the Arts and Sciences”), and marveled at the beautiful fractal pictures created using mathematical methods. On my way home I stopped at Barnes and Noble and boughtThe Beauty of Fractalsby Peitgen and Richter. I was hooked! I bought an IBM PC that had a screen resolution of 320 x 200 and 3 colors. I learned...

5. 2 Chaos, Fractals, and Tom Stoppard’s Arcadia
(pp. 23-34)
Robert L. Devaney

Tom Stoppard’s wonderful play,Arcadia, offers teachers of both mathematics and the humanities the opportunity to join forces in a unique and rewarding way. The play features not one but two mathematicians, and the mathematical ideas they are involved with form one of the main subthemes of the play. Such contemporary topics as chaos and fractals form an integral part of the plot, and even Fermat’s Last Theorem and the Second Law of Thermodynamics play important roles.

The play is set in two time periods, the early nineteenth century and the present, in the same room in an English estate,...

6. 3 Excursions Through a Forest of Golden Fractal Trees
(pp. 35-50)
T. D. Taylor

This paper presents an exploration of various features of the four self-contacting symmetric binary trees that scale according to the golden ratio. We begin with an introduction to background material. This includes relevant definitions, notations and results regarding symmetric binary fractal trees; various aspects of the golden ratio; and connections between fractals and the golden ratio. The main part of the paper consists of four subsections, with each subsection discussing a particular ‘golden tree’. Each tree possesses remarkable symmetries. Classical geometrical objects such as the pentagon and decagon make appearances, as do fractal objects such as a golden Cantor set...

7. 4 Exploring Fractal Dimension, Area, and Volume
(pp. 51-62)
Mary Ann Connors

Fractals were conceived and introduced in the context of advanced mathematical research. However, the basic simplicity of their construction and numerous elementary representations lends itself to an introduction at the secondary school or beginning college curriculum. These fascinating geometric figures are, therefore, capable of enriching the contemporary mathematics curriculum in an important and significant manner, and, simultaneously, providing a bridge to the world of mathematics for the general public. In addition, there are numerous practical applications modeled by fractals.

In this article a fractal is considered to be a geometric shape that has the following properties:

1. The shape is...

8. 5 Points in Sierpiński-like Fractals
(pp. 63-74)
Sandra Fillebrown, Joseph Pizzica, Vincent Russo and Scott Fillebrown

The Sierpiński Triangle (see Figure 5.1) is one of the most recognized fractals. It has many well-known properties and there are many different ways to define it. For example, one can define the Sierpiński Triangle as what is left after removing certain sets of points; see, for example, [2, p. 180]. Another definition of the Sierpiński Triangle, and the one we will use, is that the Sierpiński Triangle is the fixed point of an Iterated Function System (IFS). See [1] for a complete introduction to Iterated Function Systems; a brief description covering what is needed for our purposes is given...

9. 6 Fractals in the 3-Body Problem Via Symplectic Integration
(pp. 75-94)
Daniel Hemberger and James A. Walsh

The statement of then-body problem is tantalizingly simple: Given the present positions and velocities ofncelestial bodies, predict their motions under Newton’s inverse square law of gravitation for all future time and deduce them for all past time.

This simplicity belies the fact that efforts to solve this problem, beginning particularly with the work of Henri Poincaré in the late 19th century, essentially led to the creation of the field of dynamical systems. The study of then-body problem remains an active area of research (see, for example, [17]).

It is intriguing to consider how Poincaré’s research might...