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4d Visualization Part 1
Posted:
May 20, 1993 5:00 PM
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In the course of talking to people at the Geometry Center
and observing their work, I have noticed many are trying
to visualize four dimensional space. I noticed this
common element when talking to Eduardo Tabacman,
Richard McGehee, Bruce Peckham (who visited in the
fall), and most recently Brad Barber. In addition to
research, there is talk of higher dimensional graphics
software. The Center's graphics program Geomview has
four dimensional capabilities; both arbitrary
projections and slices are possible. Last week there was
a meeting to discuss the possibilities for software to
accommodate k dimensional manifolds in n dimensional
spaces. (There were no concrete conclusions as to how to
actually do this, but the meeting resulted in a lengthy
wish list.)
When I asked people how well their work made them
understand 4D, I got the following interesting
responses: Tabacman and McGehee said they still didn't
have the intuitive sense that one gets for 3D, and
therefore did not feel they understood 4D. Barber felt he
could visualize 4D, as it corresponded to theorems he
knew. These answers made me curious for two reasons;
first, they show a difference in opinion on what it means
to visualize and understand. Second, it makes me wonder
how much the understanding depends on the mathematical
context.
Inspired by these questions, I decided to approach the
subject directly; namely, I have interviewed anyone
here who looked at four dimensions in their research who
was willing to discuss it. I am making these interviews
into a series of articles, trying to specifically focus
on 4D visualization. The content of the articles varies
quite a bit since the answer to the second question above
is that the understanding and even what it means to
understand varies significantly with mathematical
context.
As a general guideline for the interviews, I asked the
following list of questions. I would be interested to
hear any comments that non Geometry Center people have on
the subject. I hope that perhaps by linking these
articles by a common theme I can inspire some
discussion?
Questions (meant as guidelines only):
In what context did you use 4D space?
Do you feel that you can understand or visualize 4D?
If so, what did it take to make you feel that you
understood? What are the mathematical ideas that help
you understand it?
If not, what would it take to make you feel that you
understood?
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