Exploring New Frontiers in Geometry:
in the World Around Us and in Our Classrooms
A two-session mathematics symposium at the
American Association for the Advancement of Science
Annual Meeting and Science Innovation Exposition
Friday, 13th February 1998
Philadelphia Convention Center, Room 113-A
9:00am-11:30am and 3:30pm-6:00pm
Organizers:
Colm Mulcahy
Associate Professor of Mathematics
Spelman College
Atlanta, GA 30314
(404) 223-7627 (ph), (404) 223-7662 (fax)
colm@spelman.edu
http://www.spelman.edu/~colm
David Henderson
Professor of Mathematics
Cornell University
Ithaca, NY 14853
(607) 255-3523 (ph), (607) 255-7149 (fax)
dwh2@cornell.edu
http://www.math.cornell.edu/~dwh
Rationale:
Geometry may be one of the oldest areas of mathematics, but it certainly isn't ``static'' or ``dead''--on the contrary. Geometry is thriving, both as a field of mathematics, and in its ever-growing list of applications in science and technology. For instance, good geometric intuition plays a larger role than ever today as we make increased use of computers to visualize complex data sets. Applications such as these continue to drive researchers to uncover more basic information about geometry itself. All of this points to a need for people with better training in geometry, which in turn means that we must re-insert geometry as a dynamic subject into the curriculum. Furthermore, we can only hope to keep geometry alive in the classroom if we find new ways to excite teachers and students alike about it.
In the morning session (``Geometry is Alive!'') the continually-evolving nature of geometry will be highlighted, together with some cutting edge applications to, and interactions with, science and technology. These presentations will be very accessible to a general audience, and will make extensive use of software and video as aids in understanding and discovery.
In the afternoon session (``Long Live Geometry!'') we focus on vital pedagogical issues as we prepare to enter a new millennium. A diverse range of presenters, each with a unique background in different aspects of teaching geometry, will share their visions of how we should teach this subject in the next century, and why. Their viewpoints will be supported by specific examples from their own extensive experiences and from their innovative current projects.
Geometry is Alive! (morning symposium)
Speakers and abstracts:
Carolyn Gordon (Dartmouth College)
CAN YOU HEAR THE SHAPE OF A DRUM?
Benjamin Cheney Professor of Mathematics
Dartmouth College
Hanover, NH 03755
(603) 646-3047 (ph), (603) 646-1312 (fax)
carolyn.s.gordon@dartmouth.edu
``In spectroscopy, one attempts to recover information about an object such as its shape or chemical decomposition from the frequency spectrum of light or sound the object emits. Recently David Webb, Scott Wolpert and I discovered that the answer to Mark Kac's 1964 question, `Can one hear the shape of a drum?' is negative [1]. In other words, there are drums with distinct geometric shapes which vibrate at the same characteristic frequencies.''
In this presentation, Prof. Gordon will explain this result with the aid of some ``isospectral music'' prepared by Dennis DeTurck, and a film made by Jean-Pierre Bourguignon et al. For a popular account of this topic see Gordan and Webb's article in American Scientist [2], or Ivars Peterson's recent piece in MAA Online.
Some relevant web sites: Toby Driscoll has constructed animations of sound-alike drums vibrating. S. Sridhar has come up with microwave cavities in the isospectral drum shapes, which provide experimental verification of Gordon, Webb and Wolpert's mathematical result; and finally Peter Buser, John H. Conway, Peter Doyle and Klaus-Dieter Semmler have constructed a whole ``gallery'' of sound-alike drums.
Herbert Edelsbrunner (Univ. of Illinois at Urbana-Champagne)
COMPUTING ORGANIC SHAPES ON A MICRO AND A MACRO SCALE [CANCELLED]
Professor of Computer Science
University of Illinois at Urbana-Champagne
Urbana, IL 61801
(217) 333-6903 (ph), (217) 333-3501 (fax)
edels@cs.uiuc.edu
http://www.cs.uiuc.edu/~edelsbrunner
Prof. Edelsbrunner works in computational geometry, and his recent work focuses on geometric shapes and their topological properties, thus expanding computational geometry into what might be called computational topology
Topological questions of shape arise, for example, in applications of geometry to molecular biology, which will be the focus of this talk. Geometric models of proteins will be discussed and how they relate to computing volume and surface area, voids and pockets, symmetry and non-symmetry, docking between proteins and ligand, and deformations of proteins [3, 4, 5]. (See here.) Related though different questions arise in the reconstruction of macroscopic organic shapes measured by 3D scanners [6].
Walter Whiteley (York University)
OLD QUESTIONS - NEW ANSWERS: GEOMETRY FOR COMPUTER AIDED DESIGN
Professor of Mathematics and Statistics
York University
4700 Keele Street, North York
Ontario M3J 1P3, Canada
(416) 736-2100 X 33971 (ph), (416) 736-5757 (fax)
whiteley@mathstat.yorku.ca
http://www.math.yorku.ca/Who/Faculty/Whiteley
``At the core of many applications of mathematics lies the need to describe and predict the shape, form, and static and kinematic behaviour of discrete structures. Underlying these developments is an exciting blend of combinatorial, symbolic, and numerical methods, organized around a deep understanding of the classical and modern geometries. Applications in our (approximately) Euclidean world require methods from across the range of geometries: topological, combinatorial, projective, affine, Euclidean. Statics, local kinematics, computer vision, and multivariate splines all present a common core of combinatorial and projective geometric patterns, while the methods and unsolved problems of plane parametric CAD (Computer Aided Design) run the full gamut of geometries.'' [7, 8, 9, 10, 11, 12]
Delle Maxwell (Silicon Graphics)
SHARING THE MIND'S EYE: COMPUTER ANIMATION IN VISUALIZING GEOMETRY
Computer Graphic Designer
Silicon Graphics, Inc., MS 10U 982
2011 North Shoreline Blvd.
Mountain View, CA 94039
(415) 933-4545 (ph), (415) 933 0255 (fax)
delle@sgi.com
http://reality.sgi.com/employees/delle
In recent years, the Geometry Center at the University of Minnesota has broken new ground in the art of communicating mathematics by making several high quality videos to help communicate sophisticated ideas and discoveries in geometry to young audiences. Delle Maxwell, who served as graphic designer/animator and co-director for these landmark videos, will show highlights from the award winning Outside In (1994) [22] concerning Thurston's beautiful sphere eversion [15, 23, 25, 16] and Not Knot (1991) [20] which explains the recently discovered connections between knots and the hyperbolic geometry associated with their complements [21, 17]. She will also talk about the mathematical/design/programming collaborations and challenges involved in communicating geometry through the medium of video.
Jeffrey Weeks (Canton, NY)
THE SHAPE OF SPACE
88 State Street
Canton, NY 13617
(315) 379-0237 (ph)
weeks@geom.umn.edu
http://www.northnet.org/weeks
Dr. Weeks will wrap up the morning session by considering ``The Shape of Space'', bringing together ideas explored in his book of the same name [28], and the new Geometry Center video The Shape of Space [26] created by Weeks, Maxwell and their colleagues. In particular, he will address a vital question which has puzzled philosophers and astronomers alike down through the ages: Is the universe finite or infinite?
Dr. Weeks: ``That ancient question may finally be answered using observational data to be available in the year 2001 [29]. This presentation will show how a universe can be finite, yet have no boundary. Freely available Topology Games java applets, used in grades 5-12, will introduce the basic concept. The Shape of Space video will then take the viewer on a computer-animated tour of several possible shapes for the universe. The presentation will conclude with an explanation of how the upcoming measurements of the cosmic microwave background radiation may be used to determine the exact shape of the real universe, assuming it's small enough that we can see at least `half way around'.'' (See here for the paper itself).
Dr. Weeks is a pioneer in the design and development of interactive geometry games as a tool for getting students to experience the geometry of manifolds without having to learn the theory first. In addition to the java games mentioned above, see here.
Long Live Geometry! (afternoon symposium)
Speakers and abstracts:
David Henderson (Cornell University)
OPENING STUDENTS' MINDS: EXPERIENCING NON-AXIOMATIC GEOMETRY IN THE CLASSROOM
Professor of Mathematics
Cornell University
Ithaca, NY 14853
(607) 255-3523 (ph), (607) 255-7149 (fax)
dwh2@cornell.edu
http://www.math.cornell.edu/~dwh
Prof. David Henderson will open the proceedings with a presentation which echoes the philosophy and methods of his recent books Experiencing Geometry on Plane and Sphere [30], and Differential Geometry: A Geometric Introduction [31]. These provide the framework for hands-on, cooperative-learning based explorations of geometry, setting aside the restrictions of formalism. This distinctive approach stimulates and challenges students to develop a deeper understanding of mathematics through participation and through starting from intuitive understanding.
Prof. Henderson has been teaching geometry this way for over twenty years [32]. He will also share why he believes that most semesters he learns new geometry from about 40% of his students, especially, students who are different from him in gender, race, and cultural background (see his paper ``I Learn Mathematics From My Students - Multiculturalism In Action'' [33]).
James King (Univ. of Washington)
RESHAPING SCHOOL GEOMETRY WITH MODELS AND SOFTWARE
Associate Professor of Mathematics
University of Washington
Seattle, WA 98195
(206) 543-1915 (ph), (206) 543-0397 (fax)
king@math.washington.edu
http://www.math.washington.edu/~king/personal.html
``Mathematics teachers are living in the midst of currents of change and reform. What are appropriate methods for teaching geometry in the schools? What should geometry in the schools be? For seven years, the teachers at the IAS/Park City Mathematics Institute High School Teachers Program have wrestled with these questions while in the midst of an intense mathematics research environment. This report will describe examples of what they do in their schools, using models, projects, software, paper-folding, and balls, but also using ways of seeing mathematics and geometry everywhere.'' [34, 35, 36, 37, 38, 39, 40, 41, 42, 43]
John H. Conway (Princeton University)
THE WONDERS OF TRIANGLE GEOMETRY
John von Neumann Professor of Mathematics
Princeton University
Princeton, NJ 08544
(609) 258-6468 (ph)
conway@math.princeton.edu
``A few decades ago there was a great revolt against the teaching of projective geometry in the Universities and Colleges, and unfortunately this was later followed by a revolt against the teaching of geometry in the high schools. In my view this has had a disastrous effect on mathematical education, because even people who are unhappy with formula can appreciate geometrical results visually. In Thurston's words, `Geometry is the user interface of Mathematics'. Fortunately the situation is improving. High school students who a few years ago learned only a minuscule amount of geometry are now keenly using the excellent geometrical computer systems, and want to know more. We are about to experience a great rival of the `old' geometry, and in particular, of the classical geometry of the triangle. I have long been interested in triangle geometry, and being a professional mathematician, am keen to regard it as a coherent theory rather than a collection of individually interesting theorems. In my talk, I shall describe some famous theorems about triangles and some less famous ones, and will attempt to describe the many links between them.'' [44, 45, 46]
Joseph Malkevitch (York College (CUNY))
NEW GEOMETRY HELPS EMERGING TECHNOLOGIES FROM ROBOTICS TO HDTV
Professor of Mathematics
York College (CUNY)
Jamaica, NY 11451
(718) 262-2550 (ph), (718) 262-2027 (fax)
joeyc@cunyvm.cuny.edu
``Geometry was born in an attempt to understand physical space. However, recently geometry has grown to encompass visual phenomena in a broad sense. A surprising consequence of this expansion in geometry's domain has been widespread application of geometry to a variety of areas associated with rapidly emerging new technologies such as medical imaging robotics, compact disks, fax, wireless telephony and high definition television Since many of these new geometric discoveries can be taught in a way which builds on intuitively appealing ideas, the connection between these new geometric results and new technologies can be taught in grades K-12.''
Prof. Malkevitch has been involved in a variety of projects of the Consortium for Mathematics and its Applications (COMAP). These included GEOMAP (Geometry and its Applications Project) which developed modules about recent geometry for high school students and Geometry's Future which involved running a conference about what geometry undergraduate mathematics majors should be taught. Prof. Malkevitch edited the proceedings of this conference which was published by COMAP under the title Geometry's Future [47]. He has also been deeply involved with the International Commission on Mathematics Instruction (ICMI) in advocating reform in the teaching of geometry around the world. See [48, 49, 53, 51, 50, 52, 54] also.
Colm Mulcahy (Spelman College)
EXPLORING FRONTIERS IN GEOMETRY--LOOKING BACK AND LOOKING FORWARD
Associate Professor of Mathematics
Spelman College
Atlanta, GA 30314
(404) 223-7627 (ph), (404) 223-7662 (fax)
colm@spelman.edu
http://www.spelman.edu/~colm
``Geometry, with its intuitive pictorial appeal, is arguably the branch of mathematics most suited to communication to a general audience. Yet for many, including some with advanced mathematical training, the ancient roots of geometry can lead to a mistaken impression that while the subject is certainly elegant, it is essentially static. In this brief final presentation, I will attempt to dust off any remaining cobwebs perceived to be clinging to this field [55, 56], and try to mould the diverse viewpoints expressed in the earlier talks into a unified picture of geometry as a thriving, dynamic endeavour which is also in need of fresh talent and ideas [57, 58] to keep it alive and kicking well into the next century.''