Let us first flash the light of our hypothesis upon an all but
universal characteristic of living forms, yet one of the most
inexplicable--_symmetry_.
Animal life exhibits the phenomenon of the right-and left-handed
symmetry of solids. This is exemplified in the human body, wherein
the parts are symmetrical with relation to the axial _plane_.
Another more elementary type of symmetry is characteristic of the
vegetable kingdom. A leaf in its general contour is symmetrical:
here the symmetry is about a _line_--the midrib. This type of
symmetry is readily comprehensible, for it involves simply a
revolution through 180 degrees. Write a word on a piece of paper and
quickly fold it along the line of writing so that the wet ink
repeats the pattern, and you have achieved the kind of symmetry
represented in a leaf.
With the symmetry of solids, or symmetry with relation to an axial
_plane_, no such simple movement as the foregoing suffices to
produce or explain it, because symmetry about a plane implies
_four-dimensional_ movement. It is easy to see why this must be so.
In order to achieve symmetry in any space--that is, in any given
number of dimensions--there must be revolution in the next higher
space: one more dimension is necessary. To make the (two-dimensional)
ink figure symmetrical, it had to be folded over _in the third
dimension_. The revolution took place about the figure's _line_ of
symmetry, and in a _higher_ dimension. In _three_-dimensional
symmetry (the symmetry of solids) revolution must occur about the
figure's _plane_ of symmetry, and in a higher--i.e., the _fourth_
dimension. Such a movement we can reason about with mathematical
definiteness: we see the result in the right- and left-handed
symmetry of solids, but we cannot picture the movement ourselves
because it involves a space of which our senses fail to give any
account.
Now could it be shown that the two-dimensional symmetry observed in
nature is the result of a three-dimensional movement, the right-and
left-handed symmetry of solids would by analogy be the result of a
_four_-dimensional movement. Such revolution (about a plane) would
be easily achieved, natural and characteristic, in four space, just
as the analogous movement (about a line) is easy, natural, and
characteristic, in our space of three dimensions.