The Twentieth Century. There are two great works
about symmetry in the twentieth century, and, by
content though not by exposition, the present article
is intended to be, without hyperbole, only a supple-
ment to these. One of the works is the large post-
Victorian treatise On Growth and Form by D'Arcy
Wentworth Thompson, classicist, naturalist, biologist,
and a translator of Aristotle's Historia animalium. The
other is the small mid-century volume Symmetry by
Hermann Weyl, leading mathematician and connois-
seur of physics, with an acute sense of philosophy and
poetry. There are books from this century by other
authors, some quite learned, but they in no wise com-
pare with these.
Nowadays symmetry may be conceived narrowly or
broadly, specifically or comprehensively. Our concep-
tion of it will be a fully comprehensive one, and it
is only from an approach as broad as ours that the
above-mentioned treatise of Thompson appears to be
a work on symmetry, perhaps even the leading one.
Still, the chapter on Bilateral Symmetry in Weyl's book
(pp. 3-38) is, on the whole, unsurpassable.
Also, nowadays, symmetries, if broadly conceived,
seem to occur everywhere and anywhere; in nature,
in cognition, even in perception; in moral and religious
tenets; in aesthetic expressions and aspirations; and,
generally, in mimetic experiences of any kind. The
mimesis involved may be rigorous or proximate, faith-
ful or distorted, inward or outward, sensuous or ra-
tional, realistic or idealistic.
Any meditation on symmetry must also account for
various modes or distortions of symmetry. It also be-
comes necessary to distinguish between mere distor-
tions of symmetry and direct viclations of symmetry,
and between outright contrapuntal asymmetry and
more complementary nonsymmetry.
In nature, a deviation from symmetry may be quite
small, or quite large. For instance, a honeycomb is
renowned for its hexagonal and dodecahedral sym-
metries. Its construction is a testimonial to the intelli-
gence, industry, and social instincts of the bee. In actual
physical detail, the symmetries are not quite as regular
as proverbially assumed (Weyl, p. 91), but the approxi-
mation to symmetries is a really good one.
A tree in nature is another matter. In its Platonic
idea, as it were, the tree is nature's most imposing
model for cylindrical symmetry, and there are speci-
mens that are impressively regular. However, a tree
may also be gnarled, very much so, and this need not
impair its health, and may even enhance its beauty.
Ordinarily a painter would not take out the gnarls
merely for the sake of restoring the “ideal” symmetry
that was “ideally” intended. A “modern” painter, of
whatever persuasion of modernity, is even likely to
distort the distortion over-realistically, if he is inter-
ested in the tree at all.
This is perhaps the place at which to cite a pro-
nouncement of Dagobert Frey, which, however “con-
temporary,” is only a pale replica of the shining origi-
nal of Plotinus: “Symmetry signifies rest and binding,
asymmetry motion and loosening, the one order and
law, the other arbitrariness and accident, the one
formal rigidity and constraint, the other life, play, and
freedom” (cf. Weyl, p. 16).
It is noteworthy that a very special instance of this
pronouncement had been uttered by Democritus, many
centuries before Plotinus: “according to Theophrastus,
Democritus says that plants with straight stems have
shorter lives than those with crooked stems because
it is harder for the sap to mount straight up than
sideways” (Regnéll, p. 51).
