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4D Geometry
Posted:
May 13, 1993 1:53 PM


[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 newcomer 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, computergenerated videos, etc. that relates to 4dimensional 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 3dimensional space, 4dimensional 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 4dimensional 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:
4dimensional elementary school geometry: What are the analogues of 3dimensional 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 4dimensional space that have no counterparts in lower dimensions? Can we classify all regular 4dimensional 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 4space. Graphs of complex functions: Since complex numbers can be represented by points of the 2dimensional plane, we need 4dimensional 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 4space. 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 4dimensional space.
Physics: Here the 4th dimension is frequently interpreted as time. However, according to relativity theory, time is not the same for all observersclocks 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" 4dimensional 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 4space, 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 2torus? It seems to me feasible to attack problems of this kind with wireframe 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 4space, 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 2torus? It seems to me feasible to attack problems of this kind with wireframe graphics.
Michael Larsen JS '81, '83



