within those horizons. Now because, on the Higher
Space Hypothesis, each space is the container of all phenomena of
its own order, the futility, for practical purposes, of going
outside is at once apparent. The highly intelligent threadworm
neither knows nor cares that the point of intersection of two lines
in his diagram _represents_ a point in a space to which he is a
stranger. The point is there, on his page: it is what he calls a
_fact_. "Why raise" (he says) "these puzzling and merely academic
questions? Why attempt to turn the universe completely upside down?"

But though no _proofs_ of hyper-dimensionality have been found in
nature, there are equally no contradictions of it, and by using a
method not inductive, but deductive, the Higher Space Hypothesis
is plausibly confirmed. Nature affords a sufficient number of
_representations_ of four-dimensional forms and movements to justify
their consideration.

SYMMETRY

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 mus