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MIRROR WORLDS 131
not possible; perhaps in a few centuries we will explore hyperspace in ways today
only dreamed about in science fiction.
Many creatures in our world, including ourselves, are bilaterally symmetric;
that is, their left and right sides are similar (Fig. 5.5). For example, on each side
of our bilaterally symmetric body is an eye, ear, nostril, nipple, leg, and arm.
Beneath the skin, our guts do not exhibit this remarkable symmetry. The heart
occupies the left side of the chest; the liver resides on the right. The right lung
has more lobes than the left. Biologists trying to explain the origins of left-right
asymmetries have recently discovered several genes that prefer to act in just one
side of a developing embryo. Without these genes, the internal organs and
blood vessels go awry in usually fatal ways. Mutations in these genes help
explain the occurrences of children born with their internal organs inverted
along the left-right axis, a birth defect that generates remarkably few medical
problems. It is imaginative to consider this as a "disease" of the fourth dimension.
If we had 4-D powers, we might be able to reverse some of these strange
asymmetries.
One way to visualize the flipping of objects in higher space is to consider the
two triangles in Figure 5.6. These are called "scalene" triangles because they
have three different side lengths. They make an "enantiomorphic pair" because
they are congruent but not superimposable without lifting one out of the
plane. Similarly, in our 3-D world, there are many examples of enantiomorphic
pairs—these consist of asymmetric solid figures such as your right and left
hands. (If you place them together, palm to palm, you will see each is a mirror
reflection of the other.) The scalene triangles, like your two hands, cannot be
superimposed, no matter how you rotate and slide them. However, by rotating
the triangles around a line in space, we can superimpose one triangle on its
reflected image. Similarly, your own body could be changed into its mirror
image by rotating it around a plane in four-space. (See Appendix B for information
on Wells's "The Plattner Story" and the adventures of a chemistry
teacher whose body is rotated in the fourth dimension.)
In four dimensions, figures are mirrored by a solid. Mirrors are always one
dimension less than the space in which they operate.
If there were a hyperperson in four-space looking at our right and left
hands, to him they would be superimposable because he could conceive of
rotating them in the fourth dimension. The same would apply to seashells with
clockwise and counterclockwise spirals as in Figure 5.7.
Can you think of other examples of enantiomorphic pairs in our universe?
For example, your ears are enantiomorphs. (I like to imagine a race of aliens