References

1

Carolyn Gordon, David Webb, and Scott Wolpert, ``Isospectral plane domains and surfaces via Riemannian orbifolds'', Inventiones Math. 110 (1992), 1-22.

2

Carolyn Gordon and David Webb, ``You can't hear the shape of a drum'', American Scientist 84 (Jan.-Feb. 1996), 46-55.

3

N. Akkiraju, H. Edelsbrunner, P. Fu and J. Qiang, ``Viewing geometric protein structures from inside a CAVE'', IEEE Comput. Graphics Appl. 16 (1996), 58-61.

4

H. Edelsbrunner, M. A. Facello and J. Liang, ``On the definition and the construction of pockets in macromolecules'', in Proc. Pacific Symposium on Biocomputing 1996, World Scientific.

5

H. Edelsbrunner, M. A. Facello, P. Fu and J. Liang, ``Measuring proteins and voids in proteins'', in Proc. 28th Annual Hawaii International Conference on System Sciences, 1995, vol. V: Biotechnology Computing, 256-264. (Best paper award.)

6

H. Edelsbrunner, M. A. Facello, P. Fu, D. V. Nekhayev and J. Qian. ``Wrapping 3D scanning data'' in Proc. IS&N/SPIE Symposium on Electronic Imaging, January 1998, San Jose, California, to appear.

7

Walter Whiteley, ``Representing geometric objects, Learning and Geometry: Computational Approaches'', D. Kueker and C. Smith (editors), 143-178, Birkhäuser, Boston MA, 1996. KLEIN'S HIERARCHY OF GEOMETRIES AND ITS CONNECTION TO 'KNOWLEDGE REPRESENTATION' AND COMPUTATIONAL WORK WITH SIMPLE GEOMETRIC OBJECTS [WHITELEY]

8

Walter Whiteley ``How to design or describe a polyhedron'', J. of Intelligent and Robotic Systems 11, 135-160, 1994. A DESCRIPTION OF SOME QUESTIONS AND RESULTS CENTRAL TO WAYS OF DESCRIBING OF DESIGNING THE SIMPLEST 3-D OBJECTS [WHITELEY]

9

Henry Crapo and Walter Whiteley, ``Spaces of stresses, projections, and parallel drawings for spherical polyhedra'', in Beitraege zur Algebra und Geometrie (Contributions to Algebra and Geometry) 35, 259-281, 1994. THE CONNECTION BETWEEN STATIC STRESSES IS PLANE FRAMEWORKS AND CORRECT PLANE PICTURES OF SPATIAL POLYHEDRA, WITH CAREFULL ANALYSIS OF THE MANY RELATED GEOMETRIC CONCEPTS AND SPACES [WHITELEY]

10

Robert Connelly and Walter Whiteley, ``The stability of tensegrity frameworks'', J. Space Structures 7, 153-163, 1992. A SUMMARY OF RECENT RESULTS ON LEVELS OF RIGIDITY IN FRAMEWORKS: STATIC RIGIDITY, SECOND-ORDER RIGIDITY, RIGIDITY, FLEXIBILITY [WHITELEY]

11

Walter Whiteley, ``Rigidity and Scene Analysis'' in The Handbook of Discrete and Computational Geometry, edited by J. Goodman and J. O'Rourke, CRC Press, Boca Raton FL, 1998. A SUMMARY OF DEFINITIONS AND RESULTS FOR TWO AREAS OF DISCRETE APPLIED GEOMETRY, RELATED BY A SUPRISING RESULT [WHITELEY]

12

Walter Whiteley ``Analytic vs. synthetic geometry for computers'', in Learning and Geometry: Computational Approaches, edited by D. Kueker and C. Smith, 121-141, Birkhäuser, Boston MA, 1996. THE PROBLEM OF TRANSLATING BACK AND FORTH DEFINITIONS, THEOREMS, AND PROOFS, BETWEEN PROJECTIVE GEOMETRY EXPRESSED IN SYNTHETIC FORMS (POINTS LINES, CONSTRUCTIONS) AND ANALYTICALLY [WHITELEY]

13

Edwin Abbot, Flatland: a romance of many dimensions, Seeley, London, 1814. Reprinted by Dover, New York, 1992. THIS IS THE BEST PLACE TO BE INTRODUCED INTO THE CONCEPT OF WHAT INSIDERS AND OUTSIDERS SEE, STARTING WITH THE EXAMPLE OF A. SQUARE IN THE PLANE [MAXWELL]

14

Judith Wechsler (editor), On Aesthetics in Science, MIT Press, Cambridge, 1981. INTERESTING COLLECTION OF ESSAYS BY SIX AUTHORS, MOSTLY SCIENTISTS. [MAXWELL]

15

George K. Francis, A Topological Picturebook, Springer, New York, 1987. MANY DRAWINGS OF DIFFERENT SPHERE EVERSIONS. SILVIO LEVY, WHO WROTE THE BRIEF HISTORY OF SPHERE EVERSIONS IN THIS PAPER, GOT A LOT OF HIS MATERIAL FROM GEORGE FRANCIS [MAXWELL]

16

Anthony Phillips, ``Turning a surface inside out'', Scientific American 214 (May 1966), 112-120.

17

Dana Scott, ``Hyperbolic Space in Not Knot'', in Der Prix Ars Electronica (edited by Christina Schopf and Margit Knipp), VERITAS-Verlag, Linz, Austria, 1992. THIS ESSAY GIVES A GOOD NON-TECHNICAL OVERVIEW OF THE IDEAS IN NOT KNOT, WHICH WAS AWARDED A PRIZE AT THIS ARS ELECTRONICA VIDEO FESTIVAL. [MAXWELL]

18

William P. Thurston and Silvio Levy, Three-Dimensional Geometry and Topology, Princeton University Press, Princeton, NJ, 1997. ALL OF THE DETAILS! [MAXWELL]

19

Nelson Max, Turning a Sphere Inside Out (video), International Film Bureau, Chicago, 1977. THE ORIGINAL VIDEO DONE OF A SPHERE EVERSION. THE RENDERING OF THE EVOLVING SPHERE WAS DONE BY JIM BLINN [MAXWELL]

20

Charlie Gunn and Delle Maxwell, Not Knot (video), produced by the Geometry Center, University of Minnesota (http://www.geom.umn.edu/video/NotKnot) (1991), available from A.K. Peters, Wellesley, MA.

21

David Epstein and Charlie Gunn, Supplement to Not Knot, Geometry Center, University of Minnesota. First published by Jones and Bartlett, Boston, MA, 1991, now published by AK Peters, Wellesley, MA. THIS IS THE BOOKLET THAT ACCOMPANIES THE VIDEO NOT KNOT. THERE A NUMEROUS HISTORICAL REFERENCES AND EXERCISES IN THE BOOKLET. IT IS ESSENTIAL IN PROVIDING SOME MORE BACKGROUND ON THE MATERIAL [MAXWELL]

22

Silvio Levy, Delle Maxwell, and Tamara Munzner, Outside In (video), produced by the Geometry Center, University of Minnesota (http://www.geom.umn.edu/locate/oi/) (1994), available from A.K. Peters, Wellesley, MA.

23

Silvio Levy, Making Waves, A Guide to the Ideas behind Outside In, Geometry Center, University of Minnesota. Published by A.K. Peters, Wellesley MA, 1995. THIS IS THE MATERIAL THAT SUPPLEMENTS THE VIDEO OUTSIDE IN [MAXWELL]

24

Al Marden, Afterword in the above Outside In supplement.

25

William P. Thurston, ``Making Waves; The Theory of Corrugations'', also in the Outside In supplement.

26

Tamara Munzner, Silvio Levy, Lori Thomson, Delle Maxwell, Charlie Gunn and Jeffrey Weeks, Shape of Space (video), produced by the Geometry Center, University of Minnesota (http://www.geom.umn.edu/locate/sos/) (1995), available from the Minnesota Council of Teachers of Mathematics at http://www.mctm.org/products/ssvideo.html.

27

William P. Thurston and Jeffrey Weeks, ``The mathematics of three-dimensional manifolds'', Scientific American, July 1984.

28

Jeffrey Weeks, The Shape of Space, Marcel Dekker Inc., New York NY, 1985. A GOOD INTRODUCTION TO TOPOLOGY; LOTS OF EXERCISES TO HELP BUILD UP VISUALIZATION SKILLS. THERE IS A FUN QUIZ ON FINDING TOPOLOGICALLY EQUIVALENT OBJECTS IN THE BEGINNING OF THE BOOK. THE DOUGHNUT AND THE COFFEE CUP ARE EQUIVALENT! [MAXWELL]

29

Jeff Weeks, ``Reconstructing the global topology of the universe from the microwave background radiation'', in preparation for Classical and Quantum Gravity.

30

David W. Henderson, Experiencing Geometry on Plane and Sphere, Prentice Hall, Upper Saddle River NJ, 1996. A PROBLEM-BASED TEXT FOR A JUNIOR LEVEL COURSE THAT INVITES READERS TO EXPLORE THE BASIC IDEAS OF GEOMETRY ON THE PLANE AND SPHERE USING LIVING (NON-AXIOMATIC) PROOFS [HENDERSON]

31

David W. Henderson, Differential Geometry: a Geometric Introduction, Prentice Hall, Upper Saddle River NJ, 1998. A PROBLEM-BASED TEXT FOR A SENIOR LEVEL COURSE THAT USES LIVING PROOFS TO LEAD THE READER FROM THE BASIC INTUITIONS OF SURFACES TO THE UNDERLYING FORMALISMS [HENDERSON]

32

David W. Henderson with (with J.J. Lo and K. Gaddis), ``Learning mathematics through personal experiences: A geometry course in action'', For The Learning of Mathematics, v16, 46-52, 1996. A DESCRIPTION AND DISCUSSION OF A COURSE USING EXPERIENCING GEOMETRY ON PLANE AND SPHERE AND THE STUDENTS' REACTIONS TO IT [HENDERSON]

33

David W. Henderson, ``I Learn Mathematics From My Students--Multiculturalism In Action'', For The Learning of Mathematics, v16, 34-40, 1996. A DESCRIPTION OF MATHEMATICS LEARNED FROM STUDENTS AND A DISCUSSION OF WHY IT MAY BE TRUE THAT EACH SEMESTER 30-40% OF THE STUDENTS (ESPECIALLY WOMEN AND PERSONS OF COLOR) HAVE SHOWN THE AUTHOR MATHEMATICS NEW TO HIM [HENDERSON]

34

H. Abelson and A. di Sessa, Turtle Geometry, MIT Press, Cambridge MA, 1984.

35

James Casey, ``Using a Surface Triangle to Explore Curvature'', Mathematics Teacher, Vol. 87, 69-77, 1994.

36

John E. Gilbert, Tricia Goodman, Janet Cornett Gummerman, Gary Hamrick, Cynthia J. Hays, Susan K. May, Kim Meyer, Laura Moss, Diane Radin, Elizabeth Traister, Geometry from a high school perspective, Texteam Geometry Institute, 1997. AVAILABLE FROM: THE CHARLES A. DANA CENTER, THE UNIVERSITY OF TEXAS AT AUSTIN, 2901 NORTH IH-35, SUITE 2.200, AUSTIN, TX 78722-2348 [KING]

37

James King and Doris Schattschneider (edtors), Geometry Turned On: Dynamic Software in Learning, Teaching, and Research, Math Notes 41, Mathematical Association of America, 1997.

38

James King, Geometry Through the Circle with The Geometer's Sketchpad, Key Curriculum Press, Berkeley CA, 1996.

39

Istvan Lenart, Non-Euclidean Adventures on the Lenart Sphere, Key Curriculum Press, Berkeley CA, 1996.

40

Daniel Scher, Exploring Conic Sections with The Geometer's Sketchpad, Key Curriculum Press, Berkeley CA, 1995.

41

Michael Serra, Patty Paper Geometry, Key Curriculum Press, Berkeley CA, 1994.

42

The Platonic Solids Video, Activity Book and Manipulative Kit, Visual Math Project, Key Curriculum Press, Berkeley CA, 1991.

43

David Wells, Hidden Connections, Double Meanings, Cambridge University Press, New York, 1988.

44

N.A. Court, College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools (2nd edition), Barnes and Noble, New York, 1952.

45

H.S.M. Coxeter, Introduction to Geometry (2nd edition), Wiley, New York, 1989.

46

Roger A. Johnson, Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle, Houghton Mifflin 1929, reprinted by Dover, Mineola NY.

47

Joseph Malkevitch (editor), Geometry's Future, COMAP (2nd edition), W.H. Freeman, New York, 1992. A VARIETY OF VIEWS ABOUT WHAT GEOMETRY SHOULD BE TAUGHT AS PART OF AN UNDERGRADUATE SURVEY COURSE IN GEOMETRY [MALKEVITCH]

48

Joseph Malkevitch et al, For All Practical Purposes, COMAP (4th edition), W.H. Freeman, New York, 1997. THIS IS A TEXTBOOK FOR THE LIBERAL ARTS UNDERGRADUATE SURVEY COURSE IN MATHEMATICS WHICH ILLUSTRATES THE CONNECTIONS BETWEEN GEOMETRY AND A VARIETY OF EVERYDAY APPLICATIONS OF MATHEMATICS [MALKEVITCH]

49

Joel Brinkley, Defining Vision: The Battle for the Future of Television, Harcourt Brace, New York, 1997. A RESPECTED JOURNALIST LOOKS AT THE POLITICS OF THE HIGH FINANCIAL STAKES BEHIND ALTERNATIVE WAYS TO DEVELOP HIGH DEFINITION TECHNOLOGY [MALKEVITCH]

50

Solomon Golomb, Robert Peile, and Robert Scholtz, Basic Concepts in Information Theory and Coding, Plenum Press, New York, 1994. A READABLE INTRODUCTION TO ERROR CORRECTING CODES AND INFORMATION THEORY [MALKEVITCH]

51

Arun Netravali and Barry Haskell, Digital Pictures: Representation and Compression, Plenum Press, New York, 1988.

52

Steven Roman, Introduction to Coding and Information Theory, Springer-Verlag, New York, 1997. A WELL WRITTEN INTRODUCTION TO THE THEORY OF ERROR CORRECTING CODES AND SOME MATERIAL ABOUT DATA COMPRESSION [MALKEVITCH]

53

Lawrence Shepp (editor), Computed Tomography, American Mathematical Society, Providence RI, 1983. THE PAPERS PRESENTED AT A SHORT COURSE ON TOMOGRAPHY WHICH SURVEY THE MATHEMATICS OF TOMOGRAPHY [MALKEVITCH]

54

John Truxal, The Age of Electronic Messages, McGraw Hill, New York, 1990.

55

Joel Hass, Michael Hutchings and Roger Schlafly, ``The Double Bubble Conjecture'', Electronic Research Announcements of the American Mathematical Society 01, 1995, 98-102.

56

Joel Hass and Roger Schlafly, ``Bubbles and Double Bubbles'', American Scientist (Sep.-Oct. 1996).

57

Mathematics Teacher 90 (Jan 1997).

58

Dan Litchfield, Dave Goldenheim and Charles H. Dietrich ``Euclid, Fibonacci, and Sketchpad'', Math Horizons, Feb 1997, 9-10.

59

Mary Henderson and Brenda LaFrance, Holistic Math and Science Curriculum Project, Akwesasne Curriculum Center (co-sponsored by the Aboriginal Teacher Education Program), presented at Queen's University, Kingston, Ontario, Canada, April 23, 1996.

60

Paulus Gerdes, ``On ethnomathematical research and symmetry,'' in Symmetry: Culture and Science, Vol. 1, No. 2, 154-170, Budapest, 1990.

61

Paulus Gerdes, ``Extension(s) of a reconstructed Tamil ring-pattern``, in The Pattern Book: Fractals, Art, and Nature (edited by C. Pickover), 377-379, World Scientific Press, Singapore, 1995.

62

Paulus Gerdes, ``On mirror curves and Lunda-designs'', in Computers and Graphics (an international journal of systems & applications in computer graphics), Vol. 21, No. 3, 371-378, Oxford, 1997.