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