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