February 1995

Duke Math News

Duke University
Department of Mathematics
Box 90320
Durham, North Carolina 27708-0320


In accordance with a directive from the Arts and Science Council, the Department is in the process of developing a minor in mathematics, which may include cross disciplinary study. Such a minor would consist of at least five courses in mathematics and related fields coordinated with the student's Major. The mathematics courses must be above the level of Mathematics 103. A student must apply for a Minor by submitting for approval a list of proposed courses to the Director of Undergraduate Studies.

The Department is in the process of formulating examples of courses of study, such as Applied Mathematics, Numerical Mathematics, Mathematical Biology, etc. Please send your comments and suggestions to the Director of Undergraduate Studies, R. Scoville, ras@math.duke.edu, or Box 90320. [For an update on the mathematics minor, see MATHEMATICS MINOR in April's edition of the Duke Math News.]

Taking mathematics from the beginning of the world
to the time when Newton lived,
what he had done was much the better half.--Leibnitz

                  Quoted by F.R. Moulton:
                  Introduction to Astronomy
                  (New York, 1906), p. 199.  


This summer, the National Science Foundation will again sponsor a number of Research Experiences for Undergraduates (REU) programs at universities around the country. Undergraduate research fellows work for 6 to 10 weeks under the supervision of a mathematics professor with the aim of producing original research. The fellows receive a stipend of between $2000 and $3000 in addition to a travel allowance. Each of the past few summers, several Duke students have participated in such programs. For more information and application forms, see Joan McLaughlin in room 135.


Take six unit spheres (i.e., spheres of radius 1) and place them symmetrically with their centers on the positive and negative x, y and z axes, packed so that each sphere just touches its nearest four neighbors. Thus the centers will be at the vertices of a regular octahedron.

  1. What is the volume of the smallest sphere which contains these six spheres?
  2. If the six unit spheres were solid metal balls, would there be more metal or air inside this surrounding sphere?


Craig Gentry '95 presented a paper at the 101st Annual American Mathematical Society meetings in San Francisco. Approximately 5000 mathematicians attended these meetings held in early January. Craig's presentation,``Exact values and bounds for the half-half case of the problem of Zarankiewicz'' was developed while attending an REU program last summer. Other undergraduates who have presented papers recently at American Mathematical Society meetings include Jeff Vanderkam '94 and Paul Dreyer '95. The Mathematics Department partially supports travel expenses.

Newton could not admit that there was any difference between him and other men, except in the possession of such habits as . . . perseverance and vigilance. When he was asked how he made his discoveries, he answered, ``by always thinking about them;'' and at another time he declared that if he had done anything, it was due to nothing but industry and patient thought: ``I keep the subject of my inquiry constantly before me, and wait till the first dawning opens gradually, by little and little, into a full and clear light.''-- Whewell, W.

               History of the Inductive Sciences
               Bk. 7, chap. 2, sect. 5 


Several times each year, the Mathematics Department invites a distinguished mathematician to present a lecture at the undergraduate level. This popular series is run almost entirely by the students. A math major keeps in touch with the speaker, arranges the publicity, introduces the speaker, and organizes the refreshments. After the lecture, a group of undergraduates take the speaker to dinner, with no Duke faculty allowed. The student hosts this year have been Paul Dreyer and Elizabeth Ayer.

On Thursday, December 8, Dr. Donald Knuth of the Stanford University Computer Science Department continued the series with an enjoyable talk entitled ``Leaper Graphs.'' In this talk, Dr. Knuth discussed an abstraction of the movements of a knight, or leaper, on a chessboard. The movements of generalized knights on a board can be modelled as graphs that, under certain circumstances, are connected, i.e., the leaper can reach any square on the board from any other.

Dr. Knuth has authored numerous works on virtually all areas of computer science and related mathematics, including a very influential series called ``The Art of Computer Programming.'' His first publication appeared in Mad Magazine.

The series began this year with ``The Mathematics of the Mandelbrot Set'' by Robert Devaney in September. Speakers in previous years include Thomas Banchoff, John H. Conway, Joseph Gallian, Persi Diaconis, Peter Hilton, and Carl Pomerance. This series is made possible by a grant from the Cigna Foundation.

Our thanks to those who have attended -- please keep your eyes open for the next lecture in the series.


Several insurance companies actively recruit Duke students for summer internships as well as for permanent positions. Among these are Cigna Corporation of Philadelphia and both the Office of Personnel Management and the Wyatt Company in Washington, DC. Please see Joan McLaughlin in room 135 for more information.

The following actuarial exams have been scheduled for this spring. Through a grant from the Cigna Foundation, the mathematics department is able to the reimburse registration fees. Please bring your registration receipt to Carolyn Sessoms, 135A Math-Physics Building.


The Department needs a few more graders and lab TAs. Please see Cynthia Wilkerson, Room 116, for more information.

On one occasion, when he was giving a dinner to some friends at the university, he left the table to get them a bottle of wine; but, on his way to the cellar, he fell into reflection, forgot his errand and his company, went to his chamber, put on his surplice, and proceeded to the chapel. Sometimes he would go into the street half dressed, and on discovering his condition, run back in great haste, much abashed. Often, while strolling in his garden, he would suddenly stop, and then run rapidly to his room, and begin to write, standing, on the first piece of paper that presented itself. Intending to dine in the public hall, he would go out in a brown study, take the wrong turn, walk a while, and then return to his room, having totally forgotten the dinner. Once having dismounted from his horse to lead him up a hill, the horse slipped his head out of the bridle; but Newton, oblivious, never discovered it till, on reaching a tollgate at the top of the hill, he turned to remount and perceived that the bridle which he held in his hand had no horse attached to it. His secretary records that his forgetfulness of his dinner was an excellent thing for his old housekeeper, who ``sometimes found both dinner and supper scarcely tasted of, which the old woman has very pleasantly and mumpingly gone away with.'' On getting out of bed in the morning, he has been discovered to sit on his bedside for hours without dressing himself, utterly absorbed in thought.-- Parton, James [referring to Isaac Newton]


DUMU hosted its third annual high school math meet on Saturday morning, January 21. According to the director, Paul Dreyer '95, fifteen teams, each of five students, made their way to Duke from high schools in North Carolina, Virginia and Alabama for the multi-part competition. As in past years, the contest was run entirely by Duke students. Thomas Jefferson High School (Fairfax, VA) took first place while Vestavia Hills High School (Birmingham, AL) came in second and NCSSM placed third. West Springfield High School from West Springfield, VA was the top team for Division II. Individual awards went to Joon Pahk from Thomas Jefferson, Mathew Crawford from Vestavia Hills, Michael Westover from Thomas Jefferson, and Frank Thorne from NCSSM. Other individual winners were Ty McGill and Nathan Curtis from Thomas Jefferson and Jeff Mermin from Chapel Hill HS.

I don't know what I may seem to the world, but, as to myself, I seem to have been only as a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.-- Newton, I.

                        Quoted by Rev. J. Spence:
                        Anecdotes, Observations, and
                        Characters of Books and Men
                        (London, 1858), p. 40.  

Lewis Blake

The teaching of calculus is gradually changing nationwide, and Duke is one of the leaders. When you hear of our laboratory calculus you may conjure up visions of a lecture on tangent lines followed by a computer demonstration. But the reality is far different.

Our first-semester students use Euler's method to approximate solutions of differential equations. They spend an entire class day performing a physical experiment with a pendulum. They first encounter tangent lines in the context of local linearity. By the middle of the first semester they have already developed natural logarithms. And all of these mathematical concepts are developed in the context of real-world applications.

Students learn to generate models from data, including data gathered themselves from the pendulum experiment. They find an exponential function that models the first six months' growth of Professor Smith's dog, and they ``discover'' Kepler's formula that relates the period and size of planetary orbits. They model a real epidemic and are expected to discuss weaknesses and strengths of the models. In an application from economics, they encounter geometric series in the same example that exhibits marked differences between continuous and discrete models. No more do we hear the old student question, ``What's all this good for?''

We employ a technological tool in this course that not only enables us to handle certain problems that were heretofore impossible in first year calculus, but it also has a significant pedagogical effect. We require all students to come to lab and class with an HP48G calculator. All the labs that were previously done on computers can now be done with the calculators, and the teachers have the advantage of using the calculators for classroom activities that keep the students involved. For example, to begin the examination of trig functions we have students graph the difference quotient for sine t, with a fixed small value for Delta t, and when they see the cosine curve appear on the screen, they have learned more than the formal proof can teach them.

We require students to write reports on their laboratory experiences and on their other mathematical projects. This writing serves simultaneously to guide students into organizing their thoughts clearly and also to reveal to teachers the exact degree of understanding that the students have achieved. Writing in first year calculus requires training of teachers and adjustments on behalf of students, but the extra effort is rewarded with better understanding and clearer thinking.

Our current schedule has all Calculus I students in the fall and all Calculus I and II students in the spring working in the laboratory format. Although most students who place out of Calculus I are now taking a traditional version of calculus, we are developing a laboratory course into which entering (AP) Calculus II students can place. The question of how to proceed on this matter is now under consideration by our first year calculus committee.

The obvious question is ``How do these students perform in subsequent math-related courses?'' Our studies shows that 23% of the veterans of this course study two or more math courses beyond Calculus III, whereas only 13% of those from the traditional course do so. Furthermore, the performance of the students in the higher level course is at least as good as that of the students in the traditional course.

As we see students learning how to use calculus while they're participating in the learning process, and as we see the students succeeding in mathematical fields, we are motivated to examine ways to expand this type of teaching to other courses. This is an exciting and challenging time to be involved with teaching calculus, and Duke is the place to be!

Nature and Nature's laws lay hid in night:
God said, ``Let Newton be!'' and all was light.
                               Pope, A.
                               Epitaph intended for
                               Sir Isaac Newton. 

See also the previous issue of Duke Math News, dated October 1994, and the next issue, dated April 1995.