clifford_a-_pickover_surfing_through_hyperspacebookfi-org

THE DIVINITY OF HIGHER DIMENSIONS 49
something impossible to achieve on a plane without self-intersection—our
brains would not be possible. However, as Martin Gardner notes in The Unexpected
Hanging, one can imagine self-intersecting networks along which electrical
impulses travel across intersections without turning corners. 7
As Thomas Banchoff points out in Beyond the Third Dimension, the first edition
of Flatland appeared as only one thousand copies in November 1884, but
since then interest and sales have dramatically increased. Edwin Abbott Abbott
was not the first person to consider a 2-D universe inhabited by flat creatures, but
he was the first to explore what it would mean for 2-D creatures to interact with a
higher-dimensional world. Today computer graphic projections of 4-D objects
bring us a step closer to higher-dimensional phenomena, but even the most brilliant
mathematicians are often unable to grasp the fourth dimension just as the
square protagonist of Flatland had trouble understanding the third dimension.
Prior to Abbott's work, several individuals considered analogies between 2-
D and 3-D worlds. For example, psychologist and physiologist Gustave Fechner
wrote Space Has Four Dimensions in which a 2-D creature is a shadow man
projected to a vertical screen by an opaque projector. The creature could interact
with other shadows, but, based on its limited experience, could not conceive
of a direction perpendicular to its screen. The idea of 2-D creatures dates
back to Plato's Allegory of the Cave, in the seventh book of The Republic, where
shadows are representations of objects viewed by 3-D observers constrained to
watch the lower-dimensional views. Unlike Fechner, Plato does not suggest
that the shadows have the capability of interacting with one another.
How to Hide from a Four-Dimensional Creature
Virtually all books about the fourth dimension suggest that it is impossible for us to
hide from 4-D beings who could see inside our homes even with the doors closed.
However, a lower-dimensional being could learn to hide from higher-dimensional
beings by taking advantage of objects in the higher-dimensional world.
Let's start with a Flatland analogy. Consider a 3-D piece of Swiss cheese. If
you were to drop this on Flatland so that it intersected the 2-D universe, a
Flatlander could hide within the 2-D cross section of a hole in the cheese. The
lucky Flatlander in this hole cannot be seen by a 3-D being. Of course, it might
be difficult for the Flatlander to distinguish this hole from an ordinary 2-D circle;
however, if there were some way for Flatlanders to sense and crawl into these
holes, natural selection might evolve Flatlanders with such abilities. By analogy,