clifford_a-_pickover_surfing_through_hyperspacebookfi-org

12 - surfing through hyperspace
theoretical physics, providing the foundation for the concepts and methods
later used in relativity theory. Riemann replaced the 2-D world of Zarf with
our 3-D world crumpled in the fourth dimension. It would not be obvious to
us that our universe was warped, except that we might feel its effects. Riemann
believed that electricity, magnetism, and gravity are all caused by crumpling of
our 3-D universe in an unseen fourth dimension. If our space were sufficiently
curved like the surface of a sphere, we might be able to determine that parallel
lines can meet (just as longitude lines do on a globe), and the sum of angles of
a triangle can exceed 180 degrees (as exhibited by triangles drawn on a globe).
Around 300 B.C. Euclid told us that the sum of the three angles in any triangle
drawn on a piece of paper is 180 degrees. However, this is true only on a
flat piece of paper. On the spherical surface, you can draw a triangle in which
each of the angles is 90 degrees! (To verify this, look at a globe and lightly trace
a line along the equator, then go down a longitude line to the South Pole, and
then make a 90-degree turn and go back up another longitude line to the equator.
You have formed a triangle in which each angle is 90 degrees.)
Let's return to our 2-D aliens on Zarf. If they measured the sum of the
angles in a small triangle, that sum could be quite close to 180 degrees even in
a curved universe, but for large triangles the results could be quite different
because the curvature of their world would be more apparent. The geometry
discovered by the Zarfians would be the intrinsic geometry of the surface. This
geometry depends only on their measurements made along the surface. In the
mid-nineteenth century in our own world, there was considerable interest in
non-Euclidean geometries, that is, geometries where parallel lines can intersect.
When physicist Hermann von Helmholtz (1821-1894) wrote about this subject,
he had readers imagine the difficulty of a 2-D creature moving along a
surface as it tried to understand its world's intrinsic geometry without the benefit
of a 3-D perspective revealing the world's curvature properties all at once.
Bernhard Riemann also introduced intrinsic measurements on abstract spaces
and did not require reference to a containing space of higher dimension in
which material objects were "curved."
The extrinsic geometry of Zarf depends on the way the surface sits in a highdimensional
space. As difficult as it may seem, it is possible for Zarfians to
understand their extrinsic geometry just by making measurements along the
surface of their universe. In other words, a Zarfian could study the curvature of
its universe without ever leaving the universe—just as we can learn about the
curvature of our universe, even if we are confined to it. To show that our space
is curved, perhaps all we have to do is measure the sums of angles of large tri-