CHEMISTRY
Joel Miller
Thanh N. Truong
Peter J. Stang
MATHEMATICS
Graeme W. Milton
Jim Carlson
PHYSICS
Charles Jui
Charles Jui
Craig Taylor
Valy Vardeny
Valy Vardeny
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The 4th Dimension
By Lee Siegel
The Salt Lake Tribune
We live in three dimensions: up-down, backward-forward and side-to-side. But we are immersed in a multidimensional mathematical world that makes possible things like
supercomputers, opinion polls and static-free music.
"What's exciting to me is we can imagine things we cannot see," said Jim Carlson, mathematics chairman at the University of Utah. "We can imagine shapes in four, five and more dimensions.
"When you deal with data or measurements that involve more than three variables at a time, then you are dealing with higher-dimensional spaces. . . . It's hidden in our technology
-- for example, the error-correcting [static-reducing] chip in your cellular phone or CD player."
Carlson discussed how the 3-D world benefits from multidimensional math in a recent interview and public lecture.
Human thought was pretty much limited to three dimensions until 1637, when RenZ Descartes invented analytic geometry. It was a crucial step for math, physics and other
sciences because it made it possible to solve geometry problems using algebra.
Descartes' "idea was to describe points in a plane by pairs of numbers or coordinates," Carlson said.
A point on a sheet of paper or other flat surface is described by two coordinates: the point's distance from a horizontal axis and from a vertical axis. A point in three dimensions
is described by three coordinates -- for example, latitude, longitude and altitude.
"Take the next step and think of points in four dimensions given by four numbers," Carlson said. "Geometrically, that means we can construct three lines that meet each other at
right angles, but we can't make a fourth line that meets all the others at right angles."
But formulas can be devised to solve problems represented by four or more dimensions. Carlson gave examples of everyday multidimensional math.
Political Polling: Imagine 100 people are asked 12 questions, each of which involves rating their feelings on a scale of zero to four. Each person's overall political philosophy can
be thought of as a single point represented by 12 numbers or coordinates in 12-dimensional space.
If 12 dimensions could be plotted on a piece of paper, the 100 points would fall into clusters. (Such calculations actually are done by computers.) Each cluster of points
represents people with similar philosophies on issues covered by the poll, for example, those likely to vote for a certain candidate, Carlson said.
Pattern Recognition: Multiple dimensions can help recognize patterns from other kinds of data, Carlson said.
"Let's suppose you collect a lot of fossil dinosaur teeth and would like to determine which ones belong to which species. You can take measurements of different features of each
tooth: length, width, a little knob here or there. If you end up with 10 measurements for each tooth, it's very difficult to draw any conclusion from that long list of numbers."
But if one could draw a graph in 10-dimensional space -- with 10 measurements serving as coordinates for a point representing each tooth -- then dinosaurs of distinct species
should appear as distinct clusters of points on an imaginary 10-dimensional graph, he said.
Preventing Noise: Multidimensional math helps prevent static or interference in digital signals on cellular phones and compact discs.
"To figure out how, think about how you send a digital signal. It's just a stream of zeros and ones," Carlson said. Noise occurs "when you send a zero that might get flipped to a
one, or vice versa."
To reduce noise, send the signal in the form of 000s and 111s instead of single zeros and ones. Any combination of zeros and ones other than 000 and 111 represents an incorrect
signal or "noise."
Static is much more likely to change one digit rather than two. So it is likely that 001, 010 and 100 really should have been 000, while 011, 101 and 110 really should have been 111.
Carlson said that is a three-dimensional error-correcting code because each signal is represented by three digits. He said the first error-correcting codes for computers and cell
phones were seven-dimensional, while today most are 64-dimensional.
Supercomputing: Imagine a supercomputer built with 4,096 smaller computers, Carlson said.
"If you take any three-dimensional cube and draw a picture of it on a piece of paper, then you are representing something three-dimensional on a two-dimensional surface," he
said. "If we could project a 12-dimensional cube onto a piece of paper, we would see 4,096 little dots corresponding to the corners of the 12-dimensional cube. You would find
these dots connected by a bunch of little lines which would correspond to the edges of this 12-dimensional cube. That's the wiring diagram for this machine."
Economics: "Suppose you are a refinery. In producing gasoline you may use 10 different feedstocks -- crude oil from different manufacturers, naphtha, various additives,"
Carlson said.
"The final product has to satisfy certain conditions -- a certain viscosity, a certain boiling point -- and be as cheap to produce as possible," he continued. "What you've got is 10
different ingredients and a bunch of possible ratios in which to combine those ingredients. If you impose all the requirements, what you've got is a geometric figure in
10-dimensional space, a shape with many flat faces but in 10 dimensions."
The solution -- the best combination of ingredients to make gas at the cheapest cost -- is represented by one of that 10-D shape's numerous corners, Carlson said.
This multidimensional method, called linear programming, is widely used "to minimize costs for any industrial product with many ingredients," he said.
Originally published May 4, 2000, in The Salt Lake Tribune.
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