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Topic: 4D Geometry
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Gene Klotz

Posts: 336
Registered: 12/3/04
4D Geometry
Posted: May 13, 1993 1:53 PM
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[This is from the Hampshire College high school summer program mailing
list. I've promised to forward all postings. G.K.]

This is to anyone who has some knowledge of computer
graphics, and would like to help me out with a maxi I'm teaching at
Hampshire this summer. The course is called "Images From the Fourth
Dimension", and I'll be teaching it for the third time (I offered it
at both HCSSiM '91 and '92). There's a description at the end of
this note. I'll be covering much the same material as I have in the
past, but I'm planning to use computer graphics in a more fundamental
way than I have in previous summers. I'm a relative new-comer to
computers, although I've been doing a lot over this past year.

I'm interested in hearing about what's out there in terms of
software, computer-generated videos, etc. that relates to
4-dimensional graphics and animation (I'm already familiar with
Banchoff's older stuff). I'll have a Macintosh in the classroom and
will expect all students to make use of the animation and graphics
features in Mathematica. Those students who are so inclined will
have the opportunity to use other computer facilities on campus, but
I'm not sure at the moment exactly what those will be. Any
information, suggestions, leads, etc., would be appreciated.

Thanks,

Allen Shepard
Allegheny College
Jr. Staff, HCSSiM '77--'80
Sr. Staff, HCSSiM '91 & '92


IMAGES FROM THE FOURTH DIMENSION (maxi)
HCSSiM '91 and '92
Allen Shepard



This course will cover a variety of topics dealing with the
4th dimension. Just as in 2- and 3-dimensional space, 4-dimensional
space can be studied either geometrically or algebraically, and we'll
make use of both approaches. The algebra is useful in providing
careful definitions of what the fourth dimension is and for rigorous
proofs; but our most important goal will be to gain some sense of
what 4-dimensional space and objects living in it "look like." Two
major tools for doing this will be computer graphics and our own
imaginations.

Topics to be explored include:

4-dimensional elementary school geometry: What are the
analogues of 3-dimensional shapes such as cones, cylinders, spheres
and pyramids? We'll find formulas for (hyper) volumes, etc.

Regular polyhedra in 4 dimensions: You've seen the
hypercube, but are there also hypertetrahedra, hyperoctahedra...?
Are there regular polyhedra in 4-dimensional space that have no
counterparts in lower dimensions? Can we classify all regular
4-dimensional polyhedra? (Answer to the last question: Yes!)

Topology: We will study topological shapes such as the
Klein bottle and the projective plane that can only be visualized in
4-space.

Graphs of complex functions: Since complex numbers can be
represented by points of the 2-dimensional plane, we need
4-dimensional graph paper for functions from the complex numbers to
complex numbers. We'll get the computer to help us see what these
graphs look like for various functions.

Quaternions: Quaternions are like complex numbers, except
that they use three imaginary numbers, i, j, and k; therefore, a
quaternion is of the form
a + bi + cj + dk, a,b,c,d real numbers. Just as complex numbers can
be viewed as points in the plane, quaternions can be viewed as points
in 4-space. The multiplication structure of quaternions turns out to
have some very interesting geometrical properties that generalize the
rules for multiplying complex numbers geometrically. We will use
quaternionic multiplication to help us understand how things rotate
in 4-dimensional space.

Physics: Here the 4th dimension is frequently interpreted as
time. However, according to relativity theory, time is not the same
for all observers--clocks will actually speed up and slow down as
they move in different ways in space. There are formulas which
describe how this happens, and these formulas are based on a notion
of distance that is different from the usual one. We will learn a
bit about Einstein's theory of special relativity and how the
geometry of this model differs from "ordinary" 4-dimensional space.
"Paradoxes" of relativity (such as the one in which one twin takes a
journey in a spaceship and returns to earth much younger than the
twin who stayed behind) will be discussed along with the breakdown of
simultaneity. Finally, we will answer the question that's on
everyone's mind, "what happens if you fall into a black hole?"
If money is no object (ha!) use mathematica on a silicon graphics
machine. Maybe even better is matlab on any decent workstation.
It is possible to make things spin in real time in 1993, but the
machinery is still expensive.

This is one amateur's opinion on the subject. You might want
to look at the SIGGRAPH (correct speling of acronym?) conferences
to see what is possible in 4d. Most of my plots are 3d.
If I were giving this course, I would be tempted to try to throw in some
topology. It's easy to see why two circles won't link in 4-space,
but can you "see" how a circle links with a sphere? Can you visualize
the process whereby a strip of paper with a 360 degree twist can be
continuously untwisted? Can you visualize the Hopf fibration of S^3
or a flatly embedded 2-torus? It seems to me feasible to attack
problems of this kind with wire-frame graphics.

-Michael Larsen
JS '81, '83


If I were giving this course, I would be tempted to try to throw in some
topology. It's easy to see why two circles won't link in 4-space,
but can you "see" how a circle links with a sphere? Can you visualize
the process whereby a strip of paper with a 360 degree twist can be
continuously untwisted? Can you visualize the Hopf fibration of S^3
or a flatly embedded 2-torus? It seems to me feasible to attack
problems of this kind with wire-frame graphics.

-Michael Larsen
JS '81, '83





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