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In Japan, we have a traditional tool to produce interesting surfaces one
after another, like magic. It is called Nankin-Tama-Sudare, and is a special
bamboo blind used for party entertainment. The shapes of the surfaces made
with the bamboo blind one after another interest us very much, and the
narration of the manipulator is also very interesting.
Before reading this page, you may have thought that surfaces made with
modern weapons, computers, are more interesting than ones made with Nankin-Tama-Sudare
are. High-tech is wonderful!
These are computer graphics for an end-of-term assignment submitted by
second-year students attending the subject "Introduction to Geometry"
at the Department of Mathematical Sciences, Ibaraki University. I gave
the lectures in the latter half of the 2002 academic year. When I began
the subject, more than 100 students wanted to attend it, and the number
of students surprised me. There were an insufficient number of computers
for students. However, the high-tech classroom at Ibaraki University has
a device to make the classroom wider and we could access enough computers.
Once this crisis was over, I could start the lectures. The end-of-term
assignment was to freely invent a surface with high impression, and to
draw it with the symbolic manipulation program Mathematica. To tell the
truth, I expected that most students would try to search the Internet and
they would submit a copy of the information they found. Among submitted
graphics, we have many that are obviously copies. However, we also have
some that are more interesting than I expected. Viewing the entire submitted
graphics, I note that they are very interesting, and they excite me as
a teacher.
As the aim of the subject, I wrote in the syllabus:"As an attempt at
mathematical education with computers, we will draw various curves and surfaces
with the symbolic manipulation program Mathematica. As the basis of the advanced
learning, we understand that there are many curves and surfaces and they are very
interesting. We consider why abstract ideas like curvature are necessary."
I believe that this aim has been achieved.
We are in the age of personal computers where everyone can skillfully use
electronic devices very easily. By considering the possibilities of computers,
many people have tried to improve mathematical education. However, many
may know that some mathematicians express negative opinions. They claim
"Computers are black boxes with unknown principles. We would make
human beings who cannot think by themselves, if we use computers in mathematical
education." I understand that this is the heart of their negative
opinion. Besides, I know that they also claim "Computer graphics have
no reality. Only superficial understanding can be achieved with them."
In our assignment, I found that some students made the mistakes described
below, since we can still produce beautiful graphics automatically, even
if we input incorrect commands. We can notice mistakes easily, if we take
the step of considering the principles. Some students don't try to consider
them.
I have corrected mistakes like those above for writing web pages.
As for reality, I believe the generation that has grown up
with displays of family computers and cellular phones find computer graphics
sufficiently realistic, even though older people may not. Obviously, the word
"reality" refers to the psychological aspect of human beings, and it
depends very much on the individual.
I have often encountered my computer behaving very strangely in a way that
I cannot understand when drawing the submitted graphics to put them on
websites. This may be unavoidable, because the computer is doing computations
that were impossible for mathematicians before the twentieth century and
because the computer is performing very large computations. However, the
reason for the strange behavior may be essentially mistakes in programming.
I hope that people in developing software continue their efforts to improve
the software.
With advances in fixing bugs and further improvements in the ability of
computers, the day will come when people the world over will have software
for mathematics at home and will enjoy mathematics with it. It will be
soon.
References: Tohsuke Urabe, Let's Draw Curves and Surfaces with Mathematica!,
Computer & Education, Vol. 13, pp.29-32, 2002, Council for Improvement
of Education through Computers
04 March, 2003 Tohsuke Urabe
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