"Escape from 3-D, Visiting Higher Dimensions"
by K. C. Cole,   from DISCOVER magazine,  July 1993

"There is a fifth dimension beyond that which is known to man."  
In the 1960s Rod Serling's deep voice intoned that familiar 
mantra to introduce his popular TV series, The Twilight Zone.  
Serling's spooky pronouncement was  clearly an invitation to 
enter the world of the weird.  But to mathematicians, the journey 
to a higher dimension is about as mundane as a trip across town 
in a taxi.  They travel routinely not only to the fifth dimension 
but also in the seventh, the tenth and the twenty-sixth.  "It's 
nothing special," says Albert Marden, director of the Geometry 
Center in Minneapolis.  "To a mathematician, it's an everyday 
event."

	Why would mathematicians want to leave the comfort of our 
familiar three dimensional world?  Because, curiously, by poking 
their heads up into higher dimensions, they can get  clearer view 
of complex problems - They can see relationships that look 
hopelessly tangled in the squashed and compacted universe of 
lover dimensions.  Similarly, astrophysicists enter higher 
dimensions to see patterns in star clusters, particle physicists 
to look for unified theories, engineers to analyze mechanical 
linkages and communications specialists to find ways to pack 
information into tight spaces.
	There's nothing like hopping into a higher dimension to make 
a complex problem easier.  If that sounds counterintuitive, just 
think about what going to a higher dimension really means.  Say 
you're living in a one-dimensional line.  You can move forward or 
backward, like a train on its track.  But you can't move 
sideways.  It's not only out of bounds, it's out of your 
universe.  Now imagine that your universe suddenly spreads out 
into two dimensions.  You can roam freely over the entire 
surface: east, west, north, south or any direction in between.  
Or, better yet, imagine that you're a movie character, living 
your life on a two-dimensional screen.  You can simply walk away 
from that gunman about to shoot you.  Thanks to that extra 
dimension, you have new freedom to move about. 

To a mathematician, a dimension is just that: a degree of 
freedom.  For example, take a knotted piece of string.  As long 
as you stay in three dimensions, you're stuck." Says Sylvain 
Cappell, associate director of New York University's Courant 
Institute of Mathematical Sciences.  "You can't unknot it.  But 
if you could slip a bit of string through another dimension, you 
could go around the obstacle and solve it.  No matter how knotted 
it looks, you could go to a higher dimension and solve it."  
Cappell should know.  Among other things, he studies the 
properties of eight-dimensional knots in ten-dimensional space.
	The simplest way to think about a dimension is as a variable 
- that is, as a quantity that can have any of a number of 
different values.  It can represent latitude or longitude, time 
or speed, apples or oranges, particles or stars.  You can 
describe a weather pattern by plugging in values for temperature, 
humidity, wind velocity, precipitation, and so on.  If you need 
12 variables to describe a situation, you have a 12 dimensional 
problem.
	But variables alone don't add up to geometry.  As William 
Thurston, director of the Math Sciences Research Institute (MSRI) 
in Berkeley , California, points out, "Having three variables is 
not the same as having three-dimensional space."  And for 
understanding complex relationships, the shape of the space is 
often just as important as the number of dimensions it occupies.  
Take a standard two-dimensional relationship - say, a graph 
relating interest to consumer spending.  Neither has anything to 
do with geometry, but you can get a better grasp of the situation 
by looking at the shape of the line.  You can easily see where it 
peaks or bottoms out.  You can see the slope of the curve. 
	The same holds true in five - or even ten-dimensional 
models.  "Logically, it may seem like the geometry is lost, that 
it's just numbers," says Cappell.  "But the geometry can tell you 
things that the numbers alone can't: how a curve reaches a 
maximum, how you get from there to here.  "You can see hills and 
valley, sharp turns and smooth transitions; holes in  a doughnut-
shaped time-dimensional model might indicate realms where no 
solutions lie.
	Forming pictures of these intricate objects is easiest if 
you think about adding one dimension at a time, each spreading 
into space in a different direction.  Start with a point. Stretch 
the point out along one dimension and you get a line, bounded by 
two points.  Pull out the line in a perpendicular direction, and 
you sweep out a square, an area bounded by four lines.  To get a 
cube, blow up the square into the next dimension, and you get a 
solid figure bounded by six squares.  To get a four-dimensional 
cube, or hypercube, "simply" puff up the cube into yet another 
dimension you'll have an object whose borders are eight cubes.  
Similarly, you can rotate a two-dimensional disk around in a 
third dimension and get a sphere.  You can twirl a sphere around 
in a fourth dimension and get a hypersphere.
	In this sense, adding a new dimension is a kind of 
"unfolding." Explains Adam Frank, an astrophysicist at the 
University of Minnesota who works with six-dimensional spaces and 
looks at globular clusters by analyzing the geometry of 60,000-
dimensional cigars.  "You stick your head up there and the whole 
landscape changes," he says. "It's extraordinary how all these 
complicated motions can reduce to say, a simple 67-dimensional 
doughnut.  It's gorgeous!"
	Naturally, of course, you want to ask:  but where are those 
dimensions?  Or to start at the beginning, where is our nearest 
neighbor, the fourth dimension?  The answer is simple: it's 
perpendicular to all the other dimensions, just as the dimension 
we call height is perpendicular to what we call length.  
Perceiving this other dimension, unfortunately, is not  so 
simple.  But it's not impossible, either.
	Bill Thurston is known among at least some of his colleagues 
as the world's greatest living geometer.  At age 36 he won the 
Field Medal, the mathematical equivalent of a Nobel Prize.  These 
days, Thurston is trying to transform MSRI, which is 
affectionately known as "misery" in the field, into a more 
effective communicator of the pleasures of mathematics to the 
outside world.  To that end, he's begun pronouncing the acronym 
"emissary" instead.
	To a visitor, Thurston comes across as just a big kid who 
likes to play around with shapes.  Like a kindergarten classroom, 
his office has a small round table that's covered with little 
plastic triangles and pentagons in primary colors.  What Thurston 
does with them, owever, is anything but elementary.  He snaps 
four trangles together to form a tetrahedron: he then arranges 
five tetrahedrons around a continous center like a bouquet, 
pointing out how they almost - but not quite - fit snugly 
together.  That little angle left over, he says, is just enough 
in the roomy world of four dimensions to pack 600 of these 
tetrahedrons into a hypersphere.
	Perhaps