clifford_a-_pickover_surfing_through_hyperspacebookfi-org

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A good student exercise is to draw a graph of y = a"/n!for a fixed a. You'll see the
same kind of increase in y followed by a decrease as you do for hyperspheres.
Draw a 3-D plot showing the relationship between sphere hypervolume, dimension,
and radius.
Plot the ratio of a ^-dimensional hypersphere's volume to the ^-dimensional
cube's volume that encloses the hypersphere. Plot this as a function of k. (Note
that a box with an two-inch-long edge will contain a ball of radius one inch.
Therefore, for this case, the box's hypervolume is simply 2*.) Here's a hint: It
turns out that an w-dimensional ball fits better in an ra-dimensional cube than an
w-cube fits in an w-ball, if and only if n is eight or less. In nine-space (or higher)
the volume ratio of an w-ball to an w-cube is smaller than the ratio of an w-cube
to an »-ball.
Plot the ratio of the volumes of the (k + l)-th dimensional sphere to the £thdimensional
sphere for a given radius r.
For more technical readers, compute the hypervolume of a fractal hypersphere of
dimension 4.5. To compute factorials for non-integers, you'll have to use a
mathematical function called the "gamma function." The even and odd formulas
given in this chapter yield the same results by interpreting k! = T(k +1).
Can you derive a formula for the surface area of a ^-dimensional sphere? How
does surface area change as you increase the dimension?
Chapter 5
I. Astronomers actively search for evidence of the universe's shape by looking at
detailed maps of temperature fluctuations throughout space. These studies are aided by
the sun-orbiting Microwave Anisotropy Probe spacecraft and the European Space
Agency's Planck satellite. In a closed, "hyperbolic" universe, what astronomers might
think is a distant galaxy could actually be our own Milky Way—seen at a much
younger age because the light has taken billions of years to travel around the universe.
Cambridge University's Neil Cornish and other astronomers suggest that "if we are fortunate
enough to live in a compact hyperbolic universe, we can look out and see our
own beginnings."
According to Einstein's theory of general relativity, the overall density of our universe
determines both its fate and its geometry. If our universe has sufficient mass,
gravity would eventually collapse the universe back in a Big Crunch. In effect, such a
universe would curve back on itself to form a closed space of finite volume. The space
is said to have "positive curvature" and resembles the surface of a sphere. A rocket traveling
in a straight line would return to its point of origin. If our universe had less mass,
the universe would expand forever while its rate of expansion gets closer and closer to
zero. The geometry of this universe is "flat" or "Euclidean." If the universe had even