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.
(Saint-Aubin)
Schedule
First week:
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Monday, July 14
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Tuesday, July 15
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9:10 - 10:30: Bill Casselman - PostScript II
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10:40 - 12:00: David Austin - Java II
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1:30 - 5:30: Workshop
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7:00 - 9:00: Workshop
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Wednesday, July 16
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9:10 - 10:30: Bill Casselman - PostScript III
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10:40 - 12:00: David Austin - Java III
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1:00 - 3:00: Workshop
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Rest of the day free
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Thursday, July 17
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Friday, July 18
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9:10 - 10:30: Bill Casselman - PostScript V
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10:40 - 12:00: David Austin - Java V
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1:30 - 5:30: Workshop
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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 computer-generated 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
computer-generated sculpture.
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Saturday - free for your weekend pleasure.
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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:
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Monday, July 21
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Tuesday, July 22
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Wednesday, July 23
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9:10 - 10:30: Jim Fix - Introduction to graphics algorithms - III
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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.)
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7:00 - 9:00: Workshop
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Thursday, July 24
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Remarks about presentations (from an older discussion on my web site)
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The biggest single problem with talks is usually timing. It is very difficult for an inexperienced
person to judge how long an exposition will take. When asked what to do about this
problem, responses from the listeners tended to be extremely harsh (summary execution,
limb mutilation, expulsion, etc). Towards the end of the talks this term I worked out a
system of signaling to let speakers know how much time remained. But if the speaker
doesn't look at the instructor? An alarm clock?
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For the instructor, the most important point is not to dominate the audience. Rude
behaviour on the instructor's part will lead to the class uniting against him! Of course,
uniting the students in a class is perhaps always a good thing, but perhaps also there are
better things to rally them around.
- One minor problem was that students had a slight tendency to talk
too much and display too little. Oral mathematics is an oxymoron.
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Students also tend to try to cram too much material into one lecture, and to speed up
speaking as the talk proceeds. Like almost all the other faults pointed out here, this
problem is by no means restricted to students.
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One common error in presentations is that speakers frequently launch into
derivations without saying where there they are going. There is lots of advice freely
given on the general organization of talks; my own ideal is (1) What is the problem? (2)
History of the problem. (3) How this talk is going to solve it?
(4) What remains to be done? This format makes sense even for short expositions.
Everybody
likes a puzzle.
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10:45 - ?: Project presentations
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10:45 - Julie Tzu Chung
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11:00 - Robert Bradshaw
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11:15 - Peter Dolan
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11:30 - Adam Weyhaupt
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11:45 - Maria Voloshina
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1:00 - Stephanie Upshaw and Stefanie Krzak
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1:15 - Stephen Canon
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4:30 - Christine Devena
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4:45 - Kevin Kesseler
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5:00 - Matthew Salomone
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2:00 -3:30 - Jim Blinn -
Graphic design decisions for the telecourses The Mechanical Universe and
Project Mathematics
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Evening: Barbecue &
videos & other mathematical
hoo-haw
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Friday, July 25
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More project presentations
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10:00 - Myung-Sin Song
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10:15 - Weidong Chen
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10:30 - Bree Ettinger & G. Michael Guy
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10:45 - William (Pat) Hooper
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11:00 - Gerhard Trippen
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11:15 - Basant Pangeni
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11:30 - Brad Safnuk
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11:45 - Cliff Haithcock
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1:00 - Ned Hummel
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1:15 - Liam Watson
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1:30 - Fawntia Fowler
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1:45 - Robert (Bob) Sentinella
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2:00 - Atichart Kettapun
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2:15 - Wenjun Ying
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2:30 - Suho Park
References
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Bill Casselman,
PostScript manual for mathematicians.
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Jim Blinn, Jim Blinn's Corner: a Trip Down the Graphics Pipeline,
Morgan-Kaufmann, 1996.
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de Berg et al., Computational Geometry, Springer, 1996.
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David Flanagan, Java in a Nutshell, 4th edition, O'Reilly, 2002.
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Jonathan Knudsen, Java 2D graphics, O'Reilly, 1999.
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Sun Microsystems' documentation and tutorial
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Jim Blinn's notes on Project
Mathematics!
Lecturers
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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.
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David Austin
received his degree
from the University of Utah, working in topology.
He is currently at Grand Valley State University in Allendale, Michigan.
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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.
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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.
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Jim Fix
is currently
a computer scientist in the mathematics department of
at Reed College.
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Richard Froese
is currently at the University of British Columbia.
His field of research is mathematical physics.
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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.
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Yvan Saint-Aubin
is currently the Head of the Department at
the Université de Montréal. His field of research
is statistical mechanics.
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