Mathematical graphics
The MSRI Summer School  Reed College, July 13  July 26, 2003
The problem: how to produce mathematical images?
The solutions:
Other themes
Mathematics and art are strongly related, most recently in the art
of M. C. Escher
(De Smit).
Mathematics is heavily involved in the production of good
graphics
(Fix).
Technology is indispensable, but style is
also crucial to efficient utilization of graphics.
Furthermore, good use of graphics in the exposition of mathematics
makes mathematics available to a larger audience.
(SaintAubin)
Schedule
First week:

Monday, July 14

Tuesday, July 15

9:10  10:30: Bill Casselman  PostScript II

10:40  12:00: David Austin  Java II

1:30  5:30: Workshop

7:00  9:00: Workshop

Wednesday, July 16

9:10  10:30: Bill Casselman  PostScript III

10:40  12:00: David Austin  Java III

1:00  3:00: Workshop

Rest of the day free

Thursday, July 17

9:10  10:30: Bill Casselman  PostScript IV

10:40  12:00: David Austin  Java IV


12:10  Picnic lunch  in the breezeway in between the two Steele dorms

1:00  Discussion with Bart de Smit about how he produced the Escher graphics  in
the usual lecture room

1:30  5:30: Workshop

7:30: Bart de Smit on Escher and the Droste effect (open to the public)
Speaker's outline:
One of M.C. Escher's most intriguing works depicts a man standing in a
gallery who looks at a print of a city that contains the building that
he is standing in himself. This picture, with the title Print Gallery,
contains a mysterious white hole in the middle.
In a paper of Hendrik Lenstra and the speaker in the April 2003 issue of
the Notices of the AMS it is shown that well known mathematical results
about elliptic curves imply that what Escher was trying to achieve in
this work has a unique mathematical solution. This discovery opened
up the way to filling the void in the print. With help from artists
and computer scientists a completion of the picture was constructed
at the Universiteit Leiden. The white hole turns out to contain the
entire image on a smaller scale, which in the Dutch language is known
as the Droste effect, after the Dutch chocolate maker Droste.
In the talk the mathematics behind Escher's print and the process of
making the completion will be explained and visualized with computer
animations.
Many more aspects of this topic can be found at
http://escherdroste.math.leidenuniv.nl/.


Friday, July 18

9:10  10:30: Bill Casselman  PostScript V

10:40  12:00: David Austin  Java V

1:30  5:30: Workshop

7:30: John Sullivan on Optimal Geometry as Art
(open to the public)
Speaker's outline:
Topology studies those properties of curves or surfaces that are
unchanged under deformation, while geometry studies properties of
particular shapes. For any topological object, we can ask for
its optimal geometric shape, minimizing some geometric energy.
A classical example is a soap bubble which is round because it
minimizes surface area while enclosing a fixed volume.
Other examples, at the frontier of current mathematical research,
include knots tied tight in thick rope, which minimize their length,
and surfaces which minimize elastic bending energy. The resulting
shapes are not only mathematically elegant, but often exhibit striking
visual beauty.
I will show two computergenerated videos, illustrating optimal
shapes for knots and a mathematical way to turn a sphere inside out
(controlled by surface bending energy). I will discuss the artistic
choices that went into the making these films, and will show other
examples of mathematical art arising from optimal geometry, including
computergenerated sculpture.


Saturday  free for your weekend pleasure.

Sunday  boat trip (speaking of weekend pleasure)  the bus to Cascade
Locks will be at the upper
parking lot next to the Steele dormitories at 9:00 AM. We'll board the boat at the Locks
and, I hope, be back about 5:00 Sunday evening. Food provided on the boat
at M.S.R.I. expense.
Second week:

Monday, July 21

About 9:00  Jim Fix  Geometric data structures & algorithms  I
Speaker's outline:
Some mathematical graphics projects face the challenge of
storing, manipulating, and rendering 1000s of geometric
objects. When this is the case, naive algorithms and
data structures can be prohibitively expensive, especially
for an interactive demonstration or illustration.
In these lectures I will give an introduction to basic
(2D) geometric algorithms and the data structures
that support them. Building from simple nongeometric
methods (day 1), I will describe algorithms for doing fast
intersection tests and finding decompositions of the
plane (day 2). I will also discuss practical search
structures for querying and rendering large collections
of geometric objects (day 3).
All these methods will be motivated by computational
problems that arose from trying to generate a specific
figure (related to the book Indra's Pearls by
Mumford, Series, and Wright) at
the beginning of this summer.


10:40  12:00: Richard Froese  A survey of Java 3D  I

1:30  5:30: Workshop

7:00  9:00: Workshop

Tuesday, July 22

Wednesday, July 23

9:10  10:30: Jim Fix  Introduction to graphics algorithms  III

10:40  12:00: Yvan St. Aubin  When is a picture worth a thousand words?
The maxim glorifying the use of graphical elements can also be used to set a
standard: if a graphic is not going to be worth a thousand words, do not
bother to draw it! I shall describe three visual elements that I have done
in my scientific carreer which I think pass the test. They were drawn
respectively for a scientific paper, for lecture notes aimed at future high
school teachers and for a general public lecture. (I hope this lecture might
be useful even though I do not plan to answer the question raised in the
title.)

7:00  9:00: Workshop

Thursday, July 24

Friday, July 25

More project presentations

10:00  MyungSin Song

10:15  Weidong Chen

10:30  Bree Ettinger & G. Michael Guy

10:45  William (Pat) Hooper

11:00  Gerhard Trippen

11:15  Basant Pangeni

11:30  Brad Safnuk

11:45  Cliff Haithcock

1:00  Ned Hummel

1:15  Liam Watson

1:30  Fawntia Fowler

1:45  Robert (Bob) Sentinella

2:00  Atichart Kettapun

2:15  Wenjun Ying

2:30  Suho Park
References

Bill Casselman,
PostScript manual for mathematicians.

Jim Blinn, Jim Blinn's Corner: a Trip Down the Graphics Pipeline,
MorganKaufmann, 1996.

de Berg et al., Computational Geometry, Springer, 1996.

David Flanagan, Java in a Nutshell, 4th edition, O'Reilly, 2002.

Jonathan Knudsen, Java 2D graphics, O'Reilly, 1999.

Sun Microsystems' documentation and tutorial

Jim Blinn's notes on Project
Mathematics!
Lecturers

Bill Casselman
received his degree at Princeton University in 1967.
His current fields of research are automorphic forms and Coxeter groups.
He is also the editor of the NOTICES of the American Mathematical Society
responsible for covers and, occasionally, other graphics.

David Austin
received his degree
from the University of Utah, working in topology.
He is currently at Grand Valley State University in Allendale, Michigan.

Bart de Smit
received his PhD in mathematics in Berkeley in 1993.
Since 1997 he has been a mathematician at the Universiteit Leiden
in the Netherlands, primarily working in number theory.

John M. Sullivan
is a professor of mathematics at the Technical
University of Berlin and at the University of Illinois. He received
his PhD from Princeton in 1990, after earlier degrees from Harvard and
Cambridge. Sullivan's research in geometry deals with finding optimal
shapes for curves and surfaces in space. Examples include clusters of
soap bubbles, which minimize their surface area, or knots tied tight
in rope, which minimze their length. Sullivan has made extensive use
of computer graphics to illustrate this work.

Jim Fix
is currently
a computer scientist in the mathematics department of
at Reed College.

Richard Froese
is currently at the University of British Columbia.
His field of research is mathematical physics.

Jim Blinn
is the well known author of a column in the IEEE journal
on computer graphics, one with a definitely mathematical flavor.
Many of his columns have been collected in books of interest to
mathematicians. He is also largely reponsible for the technical aspects of the
film project Mathematics!
He is currently at Microsoft Research.

Yvan SaintAubin
is currently the Head of the Department at
the Université de Montréal. His field of research
is statistical mechanics.
