There is in nature another representation of hyper-dimensionality
which, though difficult to demonstrate, is too interesting and
significant to be omitted here.
Imagine a helix, intersected, in its vertical dimension, by a moving
plane. If necessary to assist the mind, suspend a spiral spring
above a pail of water, then raise the pail until the coils, one
after another, become immersed. The spring would represent the helix,
and the surface of the water the moving plane. Concentrating
attention upon this surface, you would see a point--the elliptical
cross-section of the wire where it intersected the plane--moving
round and round in a circle. Next conceive of the wire itself as a
lesser helix of many convolutions, and repeat the experiment. The
point of intersection would then continually return upon its own
track in a series of minute loops forming those lesser loops, which,
moving circle-wise, registered the involvement of the helix in the
plane.
It is easy to go on imagining complicated structures of the nature
of the spiral, and to suppose also that these structures are
distinguishable from each other at every section. If we think of the
intersection of these with the rising surface, as the atoms, or
physical units, of a plane universe, we shall have a world of
apparent motion, with bodies moving harmoniously amongst one another,
each a cross-section of some part of an unchanging and unmoving
three-dimensional entity.
Now augment the whole by an additional dimension--raise everything
one space. The helix of many helices would become four-dimensional,
and superficial space would change to solid space: each tiny circle
of intersection would become a sphere of the same diameter,
describing, instead of loops, helices. Here we would be among
familiar forms, describing familiar motions: the forms, for example,
of the earth and the moon and of their motion about the sun; of the
atom, as we imagine it, the molecule and the cell. For is not the
sphere, or ovoid, the unit form of nature; and is not the spiral
vortex its characteristic motion, from that of the nebula in the sky
to the electron in the atom? Thus, on the hypothesis that our space
is traversing four-dimensional space, and that the forms of our
space are cross-sections of four-dimensional forms, the unity and
harmony of nature would be accounted for in a remarkably simple
manner.
The above exercise of the imagination is a good preparation for the
next demand upon it. Conceive a dichotomous tree--one that always
divides into two branches--to pass through a plane. We should have,
as a plane section, a circle of changing size, which would elongate
and divide into two circles, each of which would do the same. This
reminds us of the segmentation of cell life observed under the
microscope, as though a four-dimensional figure were registering its
passage through our space.