The Fourth Dimension

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The Fourth Dimension

A presentation by Dr. Thomas Banchoff
at Fairfield University 10 February, 1998

Tom Banchoff is a scholar who has ennobled the teaching profession, has inspired his students to live a value-oriented life and has earned a world-wide reputation for his remarkable scholarship concerning the Fourth Dimension. Tom has worked on our campus of Fairfield University five summers, twice participating in Clavius Group mathematical seminars, last summer assisting in a workshop regarding Catholic faculty on America's campuses and on two other occasions has made presentations to our faculty and students. Concerning these latter presentations one of the Fairfield University seniors later wrote in the Senior report that Tom's presentation was "the most interesting and significant thing he had experienced in four years at Fairfield."

A world conference on the Fourth Dimension was held a decade ago at Brown University attracting over 700 philosophers, artists and scientists. Brown is home for Tom Banchoff, a pioneer in applying computer graphics to the Fourth Dimension thereby permitting illustration of this illusive and abstract concept. Tom is known world-wide as a research mathematician, specializing in the Fourth Dimension. In fact he has been called the Fourth Dimension's chief apostle. With the help of the computers at Brown he has developed sensational films of surfaces in the third as well as the Fourth Dimension. There is rarely a meeting of the American Mathematical Society and the Mathematical Association of America in which he does not participate explaining one of his many films. Some 6,000 mathematicians attend the world-wide meetings of the International Congress of Mathematicians held every four years (somewhat like the Olympics). Tom always makes a presentation and, without exaggeration, he attracts some of the largest audiences. In fact in 1978 at Helsinki so many had to be turned away from his presentation that the mathematicians demanded (and got) a second presentation. It almost never happens that the I C M changes their sacrosanct schedule to accommodate the demand for a speaker. Tom is very widely known and respected.

During Tom's past visits to Fairfield, both faculty and students had a chance to visit the Fourth Dimension under the direction of one of the world's most prominent scholars in this area. One of Tom's favorite books is Flatland, the satire by Edwin Abbott Abbott concerning two dimensional creatures trying to understand a three dimensional world. Tom's presentation invites his audience to cope with understanding something as transcendental as the Fourth Dimension. This has been a valuable experience for both our students and faculty. We who inhabit three dimensional space feel a little uneasy when physicists nonchalantly announce that these three familiar dimensions are a remnant of a universe which had been contained originally in eleven dimensions. Nevertheless a fourth dimension has intrigued philosophers, theologians, artists, scientists and mathematicians since the 16th century. Interest in a fourth dimension is not the sole domain of scientific fiction writers and of space age fans blasting enemies into hyperspace. Scientists are already finding practical applications for a fourth dimension in trying to present a coherent explanation of the universe being finite, unbounded and curving back on itself.

Tom's presentation is non-trivial mathematics, but a non-mathematician is not at a disadvantage since Tom is quite clear. Everyone seems to come away from his films and explanations with some new insight. Magazine articles about Tom frequently show in the background the surrealist depiction of the crucifixion by Salvador Dali (who by the way was another admirer and personal friend of Tom).

Tom has been a member of the Clavius Mathematical Research Group (consisting of mathematicians both lay and Jesuit) for many years. He was inspired by insights of the Jesuit mathematician Roger Boscovich concerning moving to higher dimensions through geometrical visualizations.

Tom has shown an amazing grasp of Jesuit goals as well as the Jesuit tradition of scholarship. He has many friends who are engaged in the Jesuit Apostolates - not only of scholarship but also of concern for Faith and Justice.

Dr. Thomas Banchoff

Tom has been chosen for many honors and received many prizes as a teacher who is concerned about the well being of his students. He has filled many posts and most recently has been chosen as president of the Mathematical Association of America.

Here follows some comments by his students and colleagues concerning Tom Banchoff the teacher and geometer.

Somehow, Tom is a private tutor for 100 students at a time. He is always aware of who is doing well (and who is not) on his assignments, helping those in need. His system of student-teacher response papers is a very labor-intensive endeavor on Professor Banchoff's part, but his fire sparks everybody else in the class to work even harder.

Even though he is probably the busiest professor at Brown, Professor Banchoff is also one of the most easily accessible and helpful.

"My memories of Brown have since faded, but Tom's class is a striking exception. It was fun, exciting while I learned more about the learning and exploration than any other course at Brown."

Although many teachers are enthusiastic, few combine that enthusiasm with Banchoff's intellectual power, discipline, and faith in students.

Because he is a person who has tirelessly, selflessly and passionately dedicated his life to teaching and to his students, Professor Banchoff has been countless students' "teacher of the year."

"His courses were the most intellectually stimulating that I have taken. They taught me how to do mathematics and his verve made me want to become a mathematician. He breathes life into the subject by giving the historical perspective."

"Tom Banchoff is one of a handful of research mathematicians who has received widespread recognition for his ability to make mathematics come alive for students and teachers at all levels."

Professor Thomas Banchoff is a deeply ethical man of civility, with a devotion to his religion and a touch of elegance. He is a faculty member in love with his subject.

Tom's appreciation for music and art is as important as mathematical equations, and he incorporates beauty into his research and teaching.

Tom Banchoff stands out as one of the most creative and interesting mathematicians I have ever known who happens to be blessed with fantastic communication skills.

Curriculum Vitae of Tom Banchoff

University of Notre Dame, BA 1960;
University of California, Berkeley, M.A. 1962, Ph.D. 1964;
Teaching Experience: Benjamin Peirce Instructor, Harvard, 1964-6;
Research Associate, Universiteit van Amsterdam, 1966-7;
Brown University: Asst. Prof. '67, Assoc. Prof. '70, Professor 1973.

Teachings awards:
Danforth (1960) and Woodrow Wilson (1959)
Senior Teaching Citation, 1976 (chosen by the Brown senior class);
Lester Ford Award 1978 (for outstanding expository writing);
Joseph Priestley Medallion, Dickinson College 1987 (science award)
Bray Award for Teaching Excellence in Sciences, Brown 1993 (faculty);
Mathematical Association of America NE Section Award 1995;
MAA National Award for Distinguished College or University; Teaching of Mathematics 1996.
Publications Directly Related to Teaching:

1) Real Time Computer Graphics Techniques in Geometry, 1974 (with Charles Strauss), AMS Proceedings: Computers in Education;
2) International Congress of Mathematicians, Helsinki, 1978, invited address in the section on Pedagogy, Computer Animation and the Geometry of Surfaces in 3- and 4-Space;
3) Linear Algebra Through Geometry (with John Wermer), Springer- Verlag 1983, revised and expanded second edition 1991;
4) EDGE-The Educational Differential Geometry Environment, (with Richard Schwartz) 1987;
5) Student-Generated Interactive Software for Calculus of Surfaces in a Workstation Laboratory, UME Trends 1990;
6) Beyond the Third Dimension, Scientific American Library 1990, 1996 (200 page computer illustrated volume);
7 other articles on teaching (among 65 bibliography items).

Associate Editorships:
Mathematics Magazine 1978-81,
Amer. Math. Monthly 1982-85,
Geometriae Dedicata 1986-9,
Communications in Visual Mathematics, 1996-;

Present Offices
1997 Professor of the year award Carnegie Foundation
President of the Mathematical Association of America

Frontispiece of Flatland


(Taken from an introduction to the 1884 Flatland Centenary Exhibition at Brown)

Flatland was written one hundred years ago. Why have six new editions come out in the last year and a half? Why is Flatland more alive today than ever before?

One answer is clear: Flatland is still the best introduction to the challenge of visualizing higher dimensions. And it's still a good story. Flatland also ranks as classic satire, giving us a view of Victorian English society with all of its self-imposed limitations of perspective, which strangely enough mirrors many aspects of our modern-day world.

Flatland raises the fundamental question, "How do you react when you come face to face with the transcendental, with something which you recognize almost from the very start that you will not be able to comprehend fully? How do you organize your insights and how do you communicate them to other people?" It is that question that brings together the evangelist, the artist, and the student of higher dimensional mathematics. The Flatland analogy provides one of the best ways to approach the challenge.

Abbott presents a first person narrative of an individual in a narrow society who experiences a conversion through an encounter with the transcendental but is persecuted when he tries to bring his new insights back to the old society.

Back to: The Clavius Group

Pertinent information about members of the Clavius Group

Clavius Activities at the Institute for Advanced Studies IAS in 1989
Clavius Activities at Notre Dame University in 1990
Clavius Activities at Fairfield University in 1991
Clavius Activities at the Institut des Hautes Etudes Scientifique in 1992
Clavius Activities at Boston College in 1993
Clavius Activities at the Institute for Advanced Studies IAS in 1994
Clavius Activities at Notre Dame University in 1995
Clavius Activities at Boston College in 1996
Clavius Activities at the Institut des Hautes Etudes Scientifique in 1997

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