What Is a Fractal?

A fractal generated mountain range using a Macintosh computer

Benoit Mandeibrot coined the term "fractal" for a book written in 1975. The term has become popular among mathematicians, among scientists, among philosophers, and (of course) among the lunatic fringes surrounding these areas. Mandeibrot argues that the mathematical tools traditionally used to study nature, such as Euclidean geometry and calculus, can only approximate the real thing. "Islands are not circles, mountains are not cones, clouds are not spheres." One way of interpreting the quantum mechanical model of a single free particle tells us that the path of the particle is continuous, but nowhere differentiable - such a path is a good candidate for a fractal set. Fractals are being used to model the surface of a protein; viscous fluid flow; geological fractures and faults; the cosmological distribution of matter; and much more.

But what is a fractal? I confess that I don't know. But it seems that I can recognize one when I see it.

Mandelbrot provides a technical mathematical definition in one of his books. But even he later expressed dissatisfaction with the definition. It should only be considered an approximation to the "correct" definition, which is not yet formulated. Here is the definition: A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension (B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman & Co., 1982, page 15.) If you didn't fully understand that, relax! It won't be on the test and I won't get still more technical in order to explain it here. The technical definitions (but not proofs) can be found in the Mandelbrot book. Instead, I will discuss an example-the Koch curve, or snowflake curve.

To generate the snowflake curve, start with an equilateral triangle, as in the illustration. On the center of each of its sides, add another equilateral triangle with side one-third the length of the original. Then, on each of the twelve sides of the resulting figure, add an equilateral triangle with side one-third of that. And so on. The snowflake curve is the "boundary" between the black and the white in the final, limiting, figure.

The snowflake curve has "topological dimension 1." In fact, that is why we call it a "curve." There is a one-to-one correspondence between an ordinary circle and the snowflake curve that is a continuous function in both directions. Since a circle has dimension 1, this means that the topological dimension of the snowflake curve is also 1. But the "Hausdorff-Besicovitch dimension" of the snowflake curve is approximately 1.26, so it is larger than 1. To see what this means, we may consider a related notion of dimension, the "similarity dimension." (Whenever the similarity dimension exists, so does the Hausdorff-Besicovitch dimension, and they are equal.)

Let's start with a line segment. I can divide it into (say) five congruent smaller line segments.

Next, let's consider a triangle, including the inside. I can divide it into (say) four congruent smaller triangles, each similar to the whole, with shrinking factor 1/2. This time the similarity dimension, d, is the solution to the equation.

What about the snowflake curve? Well, in order for the similarity dimension to exist, let's take just the top third of the curve (corresponding to just one of the three sides of the original equilateral triangle.) It can be divided into four parts, each one similar to the whole, with shrinking factor 1/3. So the similarity dimension, d, is the solution of the equation

Mandelbrot argues that sets that occur in the real world are often not the smooth objects traditionally used for mathematical modelling of nature, but are better modelled by fractal sets. This is useful for theoretical purposes for example in explanations of the formation of mountains. It has been used in computer graphics. A few computer instructions can generate in seconds a database representing a mountain chain that would have taken centuries to enter into the computer by conventional means. For example, fractals were used to produce the "Genesis Effect" sequence in one of the Star Trek films.

I have available a program in logo to draw the snowflake curve, as well as a short bibliography of non-technical articles on fractals. If you would like me to send you a copy, indicate this when you write to Dr. Leitzel (as suggested elsewhere in this issue).

(This article was written by Gerald Edgar and was based on a talk given at the Math-Science day for high school students sponsored by the College of Mathematical and Physical Sciences.)

Comments from the Chair

Let me begin my first column as Chairman with a note of thanks to all those who have assisted me in the transition to this new office. In particular, I thank Alan Woods who not only continues to provide valuable advice and encouragement but who left a smoothly functioning administrative structure which has presented me with a minimum of unpleasant surprises.

As Autumn Quarter has begun, I have watched with amazement as dedicated faculty and staff members have smoothly put into motion our massive enterprise- 16,500 students enrolled in mathematics classes, taught by 115 faculty members (including some 30 visiting faculty) and 235 teaching associates. We have been able this year to staff all courses for which there is a demand. Very few students needing mathematics classes have been closed out of the class of their choice. This is not to say that each student has been able to schedule the professor, time, and classroom of choice, but each has been offered a reasonable alternative. Some innovations this fall which take advantage of, or prepare students for, the computer technology explosion-a new graduate level sequence in numerical techniques and an experimental section of Math 578 taught in a special computer equipped classroom.

Graduate level enrollments suggest that boom times may be returning for our graduate program. Spurred by recent improvements in the job outlook for new Ph.D.'s, and by the increasing visibility of Ohio State as a center for mathematical research, a total of 90 new students have enrolled in graduate programs in mathematics for the 1986-87 academic year. Among these are some 30 secondary mathematics teachers who seek an M.A. degree, providing themselves with a mathematical basis to keep abreast of curricular changes developing in our high schools.

Once again space constraints have forced dispersal of mathematics faculty and students across campus. This year we are situated in the Mathematics building, Cockins Hall, Archer House, Lazenby Hall, and Enarson Hall (Student Services Building). Plans for the new Mathematics Annex, hopefully to be a reality in 1989, seem to be following a slow but predictable course so we remain optimistic that these space problems will be soon be solved.

As a final note, thanks to all of you who have contributed to the ongoing mission of the department through gifts to the Development Fund earmarked for the Department of Mathematics. Special thanks to Donna Mickle for establishing a scholarship fund in memory of her husband, our long-time, valued colleague, Earl Mickle.

Departmental News

Cindy Bernlohr (M.A. '85), a lecturer in mathematics at the Mansfield Regional Campus, was selected by students there to receive the distinguished teaching award.

Ernest Brickell (Ph.D. '81) has recently developed a method to execute high precision division. He is now working at Bell Communications Research in New Jersey and became interested in the problem of overcoming the clumsiness of existing division methods while examining the design process for computer chips that implement RSA cryptography schemes.

Henry Moscovici gave a series of four lectures on the Selberg trace formula and invariants of locally symmetric manifolds at the invitation of the faculty at the College de France, Paris.

Paul Nevai is the organizer of a special session, Orthogonal Polynomials and the Moment Problem, at the AMS-MAA Winter meetings in San Antonio, January, 1987.

Cary Rader, Newark Regional Campus, received the Thomas J. Evans Teaching Excellence Award on that campus Spring, 1986.

The following new faculty members have joined the department this Autumn Quarter: Guido Mislin, Professor, from ETH, Zurich (Topology); George Majda, Associate Professor, from Brown University (Numerical Analysis); David Terman, Associate Professor, from University of Michigan (Partial Differential Equations); Edward Overman, Associate Professor from University of Pittsburgh (Numerical Analysis); Vitaly Bergelson, Assistant Professor, from Tel Aviv University (Probability and combinatorics); Crichton Ogle, Assistant Professor, from Wayne State University (Topology); Matthew Foreman, Assistant Professor, from CalTech (Logic); Alvaro Vinancua, Research Instructor, from New York University (Applied Mathematica); Paul Yiu, Research Instructor, from University of British Columbia (Quadratic Forms); Patrick Dowling, Instructor, from Kent State University (Functional Analysis); Christopher Bose, Instructor, from University of Toronto (Ergodic Theory); Oussama Hijazi, Instructor, from Max Plank Institut fur Mathematik (Differential Geometry).

Those of us who teach at the college level frequently lament that students seem to remember less and less about their previous coursework. In elementary calculus, the most standard algebraic formulas or trigonometric identities are not in the students' memory banks. In higher level courses, the derivative and integral of many frequently used elementary functions are not readily recalled. Mathematics builds new material by relying on previously established results.

To those of you who are current users of mathematics in a variety of settings, I issue the following challenge: Submit a list of twenty formulas that everyone working in mathematics should know by heart. We may not all agree on the content of such a list, but in a future issue I will publish the twenty most frequently submitted.

Undergraduate and graduate students will benefit from new scholarship funds that have been recently established through gifts to the Development Fund.

The Earl J. Mickle Memorial Fund has been created with gifts from the family and friends. Earl Mickle retired as professor emeritus of mathematics in 1980 having been on the staff at Ohio State since he joined the department as a graduate teaching assistant in 1937. He received his Ph.D. in 1941. He authored more than twenty-five research papers in the fields of real variables, measure theory, and topological groups. Twelve students completed doctoral work under his direction. During his tenure on the staff, he served the department and University in many influential ways and had strong loyalty and dedication to the department's development.

The annual income from the fund will be used to provide merit based scholarship or fellowship awards to undergraduate students with majors in the department.

The Robert C. Tumbleson Memorial Scholarship Fund was created through a bequest of Frances Ruth Tumbleson. Both Robert and Frances were graduates of Ohio State. She received a bachelor's degree in business administration in 1931 and he received a bachelor of arts degree in mathematics in 1934. Prior to setting up residence in Oberlin, Ohio in 1955 as vice-president of the Oberlin School of Commerce, Robert Tumbleson was head of the Office of Scientific Information at the National Science Foundation. He also served in public relations positions for the department of Agriculture and Commerce, the Atomic Energy Commission, the Public Health Service, and the wartime Board of Economics Warfare.

The earnings on this fund will be allocated to scholarships for undergraduate students.

As part of the University's Capital Campaign, members of the department donated contributions to create the Mathematics Scholarship Fund and the Mathematics Graduate Student Support Fund. The income from the first of these mentioned funds will be used to award scholarships to undergraduate students. The earnings from the second will be used to give small grants in support of graduate students' research or for travel to present papers at national or regional meetings

If you wish to make a contribution to any of these funds, or to other activities of the department, you are cordially invited to do so. Just send your check, made payable to The Ohio State University Development Fund, with a note stating how it should be designated, to the Editor of Math Matrix, Department of Mathematics, 231 West 18th Avenue, Columbus, OH 43210-1174.

Penny Higgins Dunham, M.Sc. '71, recently received the Baynham Award for Outstanding Teaching at Hanover College, Hanover, Indiana. In addition to a medal, the recognition also carries a cash award. Her husband, Bill, Ph.D. 75, is an associate professor of mathematics at Hanover and received the Baynham Award in 1981.

Julie Ferrar, B.Sc. '84, currently an actuarial associate with Nationale Nederlanden in Washington, D.C. The firm is based in The Hague, Netherlands and Julie has the unique talent of being fluent in Dutch.

Hank Heiberg, M.Sc. '68, Ph.D. '71, currently a Fellow in the Society of Actuaries and employed at Safeco Life Insurance Company in Seattle, Washington.

Bernard Ploeger, SM, Ph.D. '75, currently Vice President for Administration at the University of Dayton, Dayton, Ohio. In that capacity he is responsible for activities in personnel, finance, computing, and general university facilities.

Neal Raber, Ph.D.'72, an associate professor of mathematics at the University of Akron, Akron, Ohio, has received the 1986 Outstanding Teacher Award from that institution. He has been active in course development, written several texts for use in classes there, served as faculty advisor to the student mathematics club, and has been involved in all levels of University committee work. He is currently president of the Greater Akron Mathematics Educators Society.

For eight weeks this past Summer talented young high school students were challenged by the problems distributed on a daily basis in Arnold Ross's special program. Sixty-five students, representing 16 states and Canada, lived and worked together in a stimulating environment. They were nurtured in their endeavors by outstanding faculty. In addition to the core course in number theory for first year participants, courses in combinatorics, error correcting codes, probability, and representation theory were offered to provide a rich experience for students beyond the first year.

As part of the general program activities, special speakers were invited to visit the department. Their talks, while open to the entire department, were specially directed to the young students in the summer program. Professor Tom Banchoff of Brown University spoke on "The Fourth Dimension and Computer Animated Geometry." Professor Victor Klee of the University of Washington presented "Some Unsolved Problems in Discrete Mathematics." The speakers communicated enthusiasm for their subject and the students responded with good questions and much appreciation.

Ross's program is highly regarded in all parts of the mathematics community. It serves a broad audience of talented young students. Partial payment of the expenses of the program comes from gifts to the Development Fund.

(Material from this column is prepared by Jerry Edgar)

Another chance to try your problem solving skills! Send your solutions to the Editor. The best (or most interesting) solutions will be included in a future issue of Math Matrix. You are encouraged to submit problems for inclusion in this corner. Problems with an applied flavor are especially welcome.

Problem 8. Suppose x is given. Let a(1) = x and a(n) = x^a(n-l) for n = 2,3,4.... Put A = lim_n->ooa(n). Express x in terms of A. [This perennial favorite was suggested by Ronald D. Selby (class of '73)]

Problem 9. Provide an elementary proof of this fact: Suppose a sequence (p_n) has the property that whenever (X_n) is a bounded sequence, the sequence (y_n) given by

Solutions to previous problems

Problem 7. For notational convenience, write a^b for exponentiation a^b Consider five 2's:

Solutions were received from Joe Damico (B.A. '72) and Kenneth E. Williams (An MA student in Ohio State's Department of Communication and Linguistics). This is an abridged version of Damico's solution.

There are 14 unique ways (using parentheses) to evaluate an expression containing 5 constants and 4 operators. If the 5 constants (a,b,c,d,e) are all equal to 2, and the 4 operators ($) are all exponentiation, then these 14 expressions yield only 4 different values, as shown below:

The number of ways to evaluate an expression with one, two, or three operators are easily listed:

The results shown above for 1, 2, 3, or 4 operators can be summarized in terms of the number of leading left parentheses. This table can also be extended to calculate the total number of expressions for n = 6, 7, 8, and 9 operators.

In this table each new row is formed by summing entries from the previous row. Thus, row 6 is calculated from row 5 as follows:

This picture was found in the archives and carries the notation "Mathematics Society - 1917." The Editor would appreciate your help in identifying individuals in the photo and the building in front of which the picture was taken.

The Ohio State University is soon to become a state Supercomputer Center. A Cray YMP series machine is to be installed in 1988, as soon as the Federal Government agencies have their orders filled. In the interim, a Cray XMP 24 machine will be installed to help the university get started into this business. For approximately two years, a university wide supercomputer committee has studied various possibilities for machine purchase. During that time, the capabilities of several machines and university needs were assessed. Bill Davis represented the Department of Mathematics on that committee. The Cray machines are in a class of machines called, in the trade, vector architecture machines. Roughly, but only roughly, this means that vectors with real or complex entries are loaded into special registers, and multiplied, added, or compared componentwise with single instructions. This, of course, makes many of the standard operations in linear algebra much faster than they can ever be executed on a scalar, i.e. component by component, machine.

The administration of the supercomputer project has been assigned to C. William McCurdy and Russell Pitzer of the Department of Chemistry, with strong cooperation from Charles Csuri, the director of the Computer Graphics Research Group. Committees have been formed to get the installation and implementation underway. It is expected that the actual installation of the XMP will take place in Spring, 1987. Among these committees is a Research Committee which will likely be the one most closely aligned with the Department of Mathematics' Center for Scientific Computation. In fact, several members of the department, including Ed Overman, George Majda, Dave Terman, and, possibly, Ted Scheick and Bill Davis will be participants in the activities of the new Center.

Anyone interested in using the new system from our department will have relatively easy access through communication links established from the Math-Stat Pyramid computer. In fact, in the longer run, anyone with access to nearly any machine on campuus can connect with the Cray. The Supercomputer Center will be funded by a special allocation from the legislature to the Board of Regents and will not be a major added expense to the University budget. It includes networking and general access to the Cray from all of the other State Universities, and the possibility of access by interested private institutions, both academic and industrial, in the state. This will require serious networking of remote Systems, as well as education and consultation services which will need to be provided by the Center.

C. William Kern, formerly with the Office of Science Technology Policy in Washington, D.C., assumed the duties of Dean of the College of Mathematical and Physical Sciences on November 1, 1986. Dr. Kern received his B.Sc. from Carnegie Institute of Technology and his Ph.D. in physical chemistry from the University of Minnesota. After serving as an assistant professor of chemistry at the State University of New York at Stony Brook, he joined the staff of Battelle Memorial Institute as a research scientist in 1966. While at Battelle, he served as manager of the Chemical Physics Section and later as Director of the Battelle Institute Program. While employed at Battelle, Dr. Kern was an adjunct professor of chemistry. He was a full-time professor of chemistry from 1976-1980. In 1980 he joined the National Science Foundation as senior staff associate and director of the Computer Science Research Network Project. Subsequent to that he served at NSF as program director for structural chemistry and thermodynamics, acting director of the Division of Chemistry, and head of the Physical Chemistry and Chemical Dynamics Section.

We welcome Dr. Kern back to Ohio State in his new role as Dean of the College and look forward to the leadership and direction he will provide in these important times.