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DEGREES OF FREEDOM 13<br />
angles and look for sums that are not 180 degrees. Mathematical physicist Carl<br />
Friedrich Gauss (1777-1855)—one of the greatest mathematicians of all<br />
time—actually attempted this experiment by shining lights along the tops of<br />
mountains to form one big triangle. Unfortunately, his experiments were inconclusive<br />
because the angle sums were 180 degrees up to the accuracy of the surveying<br />
instruments. We still don't know for sure whether parallel lines intersect<br />
in our universe, but we do know that light rays should not be used to test ideas<br />
on the overall curvature of space because light rays are deflected as they pass<br />
nearby massive objects. This means that light bends as it passes a star, thus altering<br />
the angle sums for large triangles. However, this bending of starlight also<br />
suggests that pockets of our space are curved in an unseen dimension beyond<br />
our spatial comprehension. Spatial curvature is also suggested by the planet<br />
Mercury's elliptical orbit around the sun that shifts in orientation, or precesses,<br />
by a very small amount each year due to the small curvature of space around the<br />
sun. Albert Einstein argued that the force of gravity between massive objects is a<br />
consequence of the curved space nearby the mass, and that traveling objects<br />
merely follow straight lines in this curved space like longitude lines on a globe. 1<br />
In the 1980s and 1990s various astrophysicists have tried to experimentally<br />
determine if our entire universe is curved. For example, some have wondered if<br />
our 3-D universe might be curved back on itself in the same way a 2-D surface<br />
on a sphere is curved back on itself. We can restate this in the language of the<br />
fourth dimension. In the same way that the 2-D surface of the Earth is finite<br />
but unbounded (because it is bent in three dimensions into a sphere), many<br />
have imagined the 3-D space of our universe as being bent (in some 4-D space)<br />
into a 4-D sphere called a hypersphere. Unfortunately, astrophysicists' experimental<br />
results contain uncertainties that make it impossible to draw definitive<br />
conclusions. The effort continues.<br />
A Loom with Tiny Strings<br />
In heterotic string theory . . . the right-handed bosons (carrier particles) go<br />
counterclockwise around the loop, their vibrations penetrating 22 compacted<br />
dimensions. The bosons live in a space of 26 dimensions (including<br />
time) of which 6 are the compacted "real" dimensions, 4 are the dimensions<br />
of ordinary space-time, and the other 16 are deemed "interior<br />
spaces"—mathematical artifacts to make everything work out right.<br />
—Martin Gardner, The Ambidextrous Universe