University of Virginia Library

SYMMETRY AND ASYMMETRY

Non habet Latinum nomen symmetria (“Latin does
not have a word for 'symmetry'”). This eye-opening
remark occurs in the midst of Pliny's Natural History
(Book 34, Ch. 65; Loeb edition, Vol. 9, 174/76). Having
made this remark Pliny uses the word several times
as if it were a well-established loan-word (from the
Greek). There is corroborating evidence that indeed

it was. Vitruvius, a near contemporary of Pliny, also
uses the word several times in his De architectura, and
its connotations indicate more or less the meaning
which it may have in a textbook on architecture of
today.

Antiquity and After. “Symmetry” is a Greek term
and a Greek conception, and, as Pliny already sensed,
there is perhaps no proper verbal equivalent for it in
any European idiom. However, the term does not occur
in Homer, and it may have indeed been post-Homeric
by formation. Homeric terms have a peculiar verbal
strength, and they also have a central meaning which
they usually retain even if their later connotations are
spread over variant possibilities. But the term sym-
metry was not of this kind. Rather, in the classical
Hellenic era, the term belonged to a group of terms
and locutions that designated harmony, rhythm, bal-
ance, equipoise, stability, good proportions, and even-
ness of structure.

When translating from the Greek for the general
reader it is best to follow Pliny's (and Vitruvius') exam-
ple and let the term symmetry stand as it does, rather
than render it by a locution that, for a scholar, might
perhaps better fit the context. It is true that the dic-
tionary meaning of the term symmetry has shifted since
antiquity, but none of the original connotations has
become obsolete, certainly not entirely so. What has
seriously changed is this, that one of the connotations
that originally was barely there—in a dictionary sense,
that is—has gradually grown to prominence, and even
paramountcy. It is the connotation of “bilateral”
symmetry, or, what is the same, of mirror symmetry.

This symmetry allows a strictly geometric definition,
which can be applied to a visual tableau of any dimen-
sion. If the tableau is spatial, that is, three-dimensional,
a “mirror” is any (two-dimensional) plane in its entire
extension, and it decomposes the tableau into two
half-spaces, such that a design in one of the half-spaces
has, by reflection, a mirror image in the other. A design
and its image are geometrically congruent, except that
they differ in a sense of orientation, as a right-hand
glove differs from its left-hand mate.

In a two-dimensional tableau a “mirror” is a straight
line, any straight line, and on a one-dimensional axis
it is a point. Right and left, above and below, front
and back, when paired in a three-dimensional tableau
refer to mirror reflections with respect to three mutu-
ally perpendicular planes. “Before” and “after” corre-
spond to a bilateral symmetry on the time axis, if, as
usual, this axis is represented by a (Euclidean) straight
line.

Greek Dualities. From our retrospect, Greek philos-
ophy was little affected by symmetries and asym-
metries, but, from the first, had dualities in its thought
patterns, perhaps even to a fault. Book 13 of Euclid's
Elements is a splendid essay on the existence, con-
struction, and uniqueness of regular solids in space, and
as such it is a triumphant exercise in mathematical
symmetry in our present-day sense. It is even a hall-
mark and acme of Greek originality. And yet it is an
isolated achievement of Greek rationality, exclusive,
and compartmentalized.

However, almost every great Greek philosopher had
a thematic duality in his thinking. In Anaximander it
was injustice and retribution; in Heraclitus it was
change and constancy in the cosmos; in Parmenides
it was the ontological contrast between Being and
Appearance; in Empedocles it was, quite primitively,
and insensitively, Love and Strife; in Anaxagoras it was
mind versus the senses; in Democritus it was the work-
ing physicist's contrast between the material and the
void, or between the full and the empty, or between
the particle and the field; in Plato it was the
epistemologist's difference between opinion and
knowledge and the idealist's dualism between body and
soul, either or both of which Plato may have inherited
from Socrates; and in Aristotle it was the hardiest
duality of all, the gigantic contrariety between the
Potential and the Actual.

Lesser Greek philosophers dwelt on lesser dualisms,
unimaginative ones. Aristotle reports (Metaphysica
986a 23-986b 4) that a school of Pythagoreans drew
up a list of ten opposites, viz., Limit-Unlimited, Odd-
Even, Unity-Plurality, Right-Left, Male-Female, Rest-
Motion, Straight-Crooked, Light-Darkness, Good-Evil,
Square-Oblong. Aristotle is apparently not impressed
with the particular selection of pairs in this list, because
he adds that Alcmaeon of Croton (of whom Aristotle
does not know whether he inspired the Pythagoreans
or they him) held similar views, but stated that there
is nothing fixed about pairs of contraries and that they
can be made up as the context demands it. And
Aristotle firmly adds that the real outcome of such
reflections is only this that “contraries are first princi-
ples of things.” If “contraries” are meant to be polari-
ties and dualities then this finding of Aristotle is just
as much a leitmotif in the natural philosophy of our
century as it was in classical Greece.

In addition to dualities from nature and knowledge
as listed above, there was also an all-Greek antithesis
of nomos (“human law,” or “norm”) and physis (“natu-
ral law”), and it was a dominant trait in statements
of Sophists on moral, social, and political issues
(Guthrie, III, 55-134). As a curiosity we note that an
anonymous Sophist (cf. Iamblichus, Protrepticus, Ch.
20) dwells on the contrast between lawlessness
(anomia) and order (eunomia) in such a manner as to
make him very like a champion of “law (nomos) and

order (eunomia)” in our sense today (Guthrie, III, 71).

Plotinus versus Romantics. The Greeks have some-
how created the impression on romantics of all ages—
perhaps beginning in antiquity with early Hellenophile
Stoics, and certainly in modern times with the very
early romantic Johann Joachim Winckelmann—that
dualities were the embodiment of an indissoluble union
of beauty, harmony, symmetry, etc., and that promi-
nent Greeks were likely to have statues of Pindaresque
symmetry, even if a Herodotus, Socrates, Aristotle, and
even Plato would hardly conform to this physical ideal
au naturel.

The first to dissent was, most unbelievably, Plotinus,
one of the most nonrealistic of philosophers, and he
turned his dissent into a major philosopheme about
symmetry, which he presented in his renowned essay
“On Beauty” (Enneads I, 6) and in some other passages.
Plotinus asserts and reasons that symmetry (ἡ ουμμετρπία,
τὸ ούμμετρον)
is neither a necessary nor a sufficient
prerequisite for beauty (τὸ καλόν), even if, admittedly,
“beauty is in the eye of the beholder” (this cliché is
Plotinus').

The context suggests that it is the purpose of Plotinus
to refute a widely held tenet. Being a good philosopher,
he first gives a clearly formulated version of what it
is that he is going to refute. He announces that he is
going to refute the thesis

(cf. Beardsley, p. 80;
trans. Stephen McKenna).

Plotinus' refutation of this thesis goes as follows. A
thing cannot be endowed with symmetry unless it can
be decomposed into parts which are symmetrically
paired. Therefore, if symmetry were a necessary con-
dition of beauty, a beautiful thing would have to be
decomposable, and a simple, that is, indecomposable
thing could not be beautiful. This would, according
to Plotinus, exclude from the contest for beauty such
things as monochromatic colors, single tones, the light
of the sun, gold, night lighting, and so on (ibid.), which
Plotinus finds absurd. On the other hand,

(ibid;
also, Enneads VI, 7, 22).

Plotinus of course, did not convert a single romantic.
In the nineteenth century, the ultra-romantic historian
(a very good one) Johann Gustav Droysen impressed
an Apollo-like “symmetry” on a monumentally de-
signed figure of Alexander the Great, which the twen-
tieth century is taking pains to redress (cf. G. T.
Griffith).

Still, there have been indomitable romantics even
in the twentieth century, as there always will be.
Hermann Weyl, after quoting a brief poem in adoration
of symmetry—the poem was published in 1921 by the
poetess Anna Wickham—also transcribes it into purple
prose thus:

(Weyl, p. 5).

To which a Plotinus could retort that it might also be
the beauty of a corpse, or the order and perfection
of a row of tombstones.

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).

Good and Evil. The contrast between Good and Evil
in Paradise Lost, or between Light and Darkness in,
say, Zoroastrianism is, from our broad approach, a
symmetry by polarity. As a religious tenet in advanced
theological stages, this symmetry is rational and ideal-
istic, in earlier creedal it is sensuous and realistic.

It is remarkable that in the Old Testament there is
very little of this polarity in the advanced theocratic
message of the leading prophets, much less than in the
fully religious or only semi-secular thinking of an
Anaximander, Xenophanes, Aeschylus, Empedocles,
and Plato, in classical Greece. However, by religious
intensity, Greece was probably less theocratic than Old
Israel.

The Human Body. The role of symmetry in animate
life is both crude and subtle, disquieting and incom-
prehensible.

The human body is outwardly endowed with a
bilateral symmetry which seems to be a near
prerequisite for most physical activities, such as walk-
ing, seeing, hearing, using one's hands, etc. In the
internal anatomy, some organs, like lungs and kidneys
conform to this symmetry, but others, very basic ones,
the heart and the alimentary canal do not. Why this
should be so is most baffling, to the general reader at
any rate. Even the outward symmetry is not very
rigorous, especially in the adult. In fact, the outward
deviation from symmetry, especially in facial contours
frequently bespeak character and personality, even
superiority.

For the comprehension of those things it is not at
all helpful to read in a very scholarly (and equally dull)
book that “all asymmetries occurring [in the human
body] are of secondary character” (cf. Weyl, p. 26).
More helpful is the suggestion, which, in depth, may
have been articulated by Weyl himself, that “the
deeper chemical constitution of our human body shows
a screw, a screw that is turning the same way in every
one of us.” But some of the explanatory details bearing
on this vitalistic “turning of the screw” are very
disquieting, inasmuch as a “wrong” turn of the screw
may be vindictively lethal (Weyl, pp. 30-38).

Circularity. A circle is rich in symmetries. It admits
mirror symmetry with respect to everyone of its in-
finitely many diameters. As a mathematical conse-
quence of this the circle can also be rotated into itself
around the center by an arbitrary angle; in fact, if two
diameters form an angle α, then mirror reflection with
respect to one diameter followed by a reflection with
respect to the second diameter will rotate the circle
by the double angle 2α.

Apparently because of this wealth of symmetry, for
2000 years, from Plato to Tycho Brahe, and including
Copernicus, scientific astronomers somehow took it for
granted that a celestial orbit of the kind that came
under their observation is, or ought to be, a circle, or
a circle rolling off on a circle (epicycle), or a figure
mathematically equivalent to such a one. They
undoubtedly had it in their thinking that what is
aesthetically (and ontologically) appealing is also
kinematically distinguished and dynamically prefera-
ble. But a mechanical preference from outward math-
ematical symmetry, while frequently profitable, can
also be misleading, and in the present case it certainly
was the latter. It was miraculously divined by Kepler,
and then mathematically rationalized by Newton that,
under gravitation, the closed orbit of one celestial body
around another—in an “ideal” two-body setting—is not
just a circle, but an ellipse, any ellipse. The ellipse can
have any eccentricity, that is any measure of deviation
from a circle. A circle can also occur; but it occurs
then only as a case of an ellipse whose eccentricity
happens to be zero. But since the eccentricity can be

any real number between 0 and 1, this is a most
unlikely value to occur. Even if, by an unlikely chance,
a pure circle does eventuate, that form undoubtedly
is very unstable; the smallest perturbation would
quickly make it into an ellipse of a small but non-zero
eccentricity. Thus, in this case, the figure with a wealth
of symmetries is exceptional within a large family of
figures each having only a few symmetries; and the
wealth of symmetries makes the exceptional figure very
unstable and most unlikely to occur.

Still, circular motion does play a role, in all parts
of physics, as a constituent of any wave-like event; of
an ordinary wave on the water or in the air; of an
electromagnetic wave in the propagation of light, as
a dual to the photon; and of a de Broglie wave, as
a dual to the corpuscular aspect of any elementary
particle of matter. A wave, wherever and however
occurring, is a composite bundle of “simple” waves,
so called “monochromatic” ones, and the mathematical
structure of a simple wave is always the same. The
pulse of a monochromatic ray of energy is rigorously
invariant in time and thus constitutes a most “depend-
able” clock (atomic clock). The Greeks were already
groping for such a clock. Aristotle reports that some
philosopher(s) before him not only measured time by
the daily rotation of the celestrial sphere but even
defined it quantitatively in this way. Surprisingly,
Aristotle frowns on this definition (Physica Book 4, Ch.
X, 218b 1-5).

Returning to gravitation we note that physicists
nowadays, out of their fertile imaginations, have
imputed a nuclear structure to gravitation too, com-
plete with corpuscles, hopefully named gravitons, and
with dual de Broglie waves, hopefully spoken of as
gravitational waves. Whenever these will be con-
clusively verified to “exist,” circularity will have finally
come to gravitation too; and how a Eudoxus and a
Ptolemy would welcome such a newcomer would be
worth knowing.

Time's Arrow. For whatever reasons, perhaps for
“magical” ones, an asymmetry, when pronounced, can
be perturbing, both to our reason and to our psyche;
and nothing can be more, or more universally perturb-
ing than the asymmetry of time, which cosmology and
evolution like to call the arrow of time, and physics
its irreversibility.

When Saint Augustine asked, as others had already
asked before him, what God had been doing before
creating the world, his query was a challenge to creed
and theology. But when Schoolmen, believers in Crea-
tion, asked the parallel question how and why God
had chosen to create the world at the instant of time
at which he had done so, then this question was not
addressed to theology only, but, in a sense, to scientific
reasoning too. In the twentieth century updated ver-
sions of this question arise in any evolutionary theory
of cosmology as well, especially when there is seem-
ingly conflicting evidence that the evolutionary process
might have started ten, or thirteen, or fifteen billion
years ago. Most importantly, however, while in our
stream of consciousness and variety of experience,
especially experience of and in the mind, there is a
qualitative difference between “past” and “future,”
“before” and “after,” yet there is also great need—in
any kind of organized knowledge, scientific or his-
torical, legal or medical, ethical or religious—for the
representation of time on a geometrically interpreted
time axis, on which the present is a point, and past
and future are half lines that are bilaterally symmetric
with regard to this point. And bilateral symmetry, in
its mathematical conception, does not provide for a
distinction of the two halves that are symmetrically
opposed to each other or paired with each other.

The philosopher Henri Bergson was very insistent
that, because of this trait and of related ones, geo-
metrically controlled time be banished from the pre-
cincts of his élan vital and évolution créatrice and
the related vitalistic manifestations. Bergson never
seriously proposed to show how to keep this geometri-
cally controlled time from intruding into the precincts
of his vital processes. But even if he had done so
successfully, the problem of the symmetry and
asymmetry of time within the general intellectual cli-
mate of our time would not have been thereby resolved
by half, because the problem is fully encountered in
exact science, even in the case of physics itself. In fact,
nineteenth-century physics arrived at conclusions
which as a package were ill-assorted, and physicists
around 1900 were discomfited by them. The following
were several of the items of the odd assortment.

(1) Some physical processes are reversible, meaning
that it is theoretically possible—though perhaps none
too probable—that they be run totally in reverse, as
when a movie is shown backwards. Of such kind are
all purely mechanical processes that, schematically, do
not involve the macroscopic production and propaga-
tion of heat.

(2) However, the creation and propagation of heat
is irreversible, meaning that a physical process in a
closed system involving them cannot be totally re-
versed so as to restore the initial situation in its
entirety. Rudolf J. E. Clausius also introduced (1850)
a quantitative measure of irreversibility which he
termed entropy, and he posited the so-called second
law of thermodynamics by which for a closed physical
system the total entropy of the system cannot decrease
in time but only increase or at most remain constant.

(3) And yet, the nineteenth century also erected the

“kinetic theory of matter” which, qualitatively and
quantitatively, interpreted thermal energy in terms of
mechanical motions and collision of molecules. Now,
mechanical processes of this kind are totally reversible,
and

(Ehrenfest,
p. 13).

This paradox was somehow overcome (Ehrenfest, p.
3) by an argument from probability. There is in the
argument a step that is intuitive, but, in full, the argu-
ment is “technical,” and a faint residue of discomfort
seems to linger on. Also, a parallel difficulty, if not
a paradox, from the organic world picture remains, and
it is the following.

(4) If the second law of thermodynamics is presumed
to apply to the entire universe as one physical system,
as physics around 1900 was bound to presume, then
the total entropy of the system must be constantly on
the increase. However, “entropy also measures the
randomness or lack of orderliness of the system, the
greater the randomness the greater the entropy” (Blum,
p. 15). On the other hand,

(Blum, p. 99).

Thus, whenever a form or unit of life comes into being
on any spot of the universe then any such occurrence,
taken by itself, runs counter to the general trend to-
wards an increase of the entropy, and, in fact,

(ibid.).

In sum, since any organization of parts and phenomena
of the universe represents, in some appropriate sense,
the creation of design and symmetry in the order of
the universe, and randomness represents the opposite
of design and symmetry, therefore, by the second law
of thermodynamics, that is, by the arrow of time, the
order of the universe cannot but gradually disintegrate
and dissolve. It is true that in the immediate vicinity
of a rise of life there is a preservation and even a small
strengthening of the symmetry. But these features are
confined by location, and fleeting by duration, and they
are compensated for by an acceleration of the process
of decline in the remainder of the universe.

In the twentieth century it has been difficult to
reconcile this bleak vista with the outlook of most
cosmological theories emerging, even when no tech-
nical inconsistencies could be argued. An even greater
difficulty, from entropy, is posed by any theory of a
cosmogonic creation of the universe, assuming as most
theories do, that there really was a Creation. In theories
of today—as in fact, much more naively, in the theories
of the pre-Socratics—almost any cosmogonic theory
of creation starts out with a physical state in which
there is some kind of turbulence, or less systematically,
turbulent disorder, or at least formlessness. Out of this
initial state develops some kind of gaseous or galactic
organization on a comprehensive universal scale, and
together with this various standard physical processes
begin to take place, some reversible and some
irreversible. How all this could accord with the second
law of thermodynamics nobody dares to suggest, but
there seems to be an understanding among cosmologists
in the second half of the twentieth century not to allow
this difficulty to prevent speculation.

Present-day Dualities. In bilateral symmetry, how-
ever much it might deviate towards asymmetry, the
parts that are “symmetrically” opposed are expected
to be, at least in a recognizable approximation, equal
and conformable, by congruence or other modes of
equality. In a duality however—or in a polarity, which
is an intensified duality—the entities that are opposed
are expected to be different, and even contrary, by
contrast, or otherwise. Usually, a duality contraposes
two contrasting aspects of the same whole, and an
asymmetry contraposes two complementary parts of
a whole larger than either part. But this criterion is
sometimes not easy to apply, and, altogether, sym-
metries, asymmetries, and dualities overlap in impor-
tant ways.

Thus, Plato's dichotomy, which is meant to be a
logical procedure for arriving at a definition of any-
thing definable (Sophist 218D-231B) proceeds by
exhibiting a succession of dyads; and it cannot be
readily made out whether in Plato's construction the
two elements in a dyad are symmetric, asymmetric,
or dual. Leaping ahead from Plato's procedure by an
ingenuous dichotomy to a present-day procedure by
an intricate computer, in which so-called “informa-
tion” is coded, produced, and transmitted by a succes-
sion of dyadic yes-or-no signals, it is again not easy
to decide whether the two possibilities “yes” and “no”
are symmetric, asymmetric, or dual.

But there are significant contemporary cases in
which there are no such doubts. For instance, the
opposition between corpuscles and de Broglie waves
in particle physics is a pronounced duality, because
the selfsame particle is sometimes a corpuscle and
sometimes a wave, depending on the context. That is,
an elementary particle exhibits properties that can be

best explained by endowing the particle with features
of both a corpuscle and a wave. In an ontological
interpretation of the entire theory, corpuscle and wave
may perhaps be viewed as complementary parts of a
unit larger than either, but in the prevalent inter-
pretation in working physics they are different but
coextensive aspects of the same whole. Furthermore,
in its purely mathematical apparatus, this duality is
but an instance of an extremely comprehensive duality,
the only one of its kind, which is spread through all
parts of mathematical analysis, and in which the
two magnitudes which are contraposed are distinctly
heterogeneous.

Yet, there is another case from particle physics, a
baffling one, which again belongs to the doubtful cate-
gory, namely the opposition between matter and anti-
matter. The transformation which creates the opposi-
tion is “charge conjugation” that is the replacement
of electrons by positrons (=anti-electrons) and of
positrons by electrons throughout a physical system in
its entirety. It would be a case of symmetry rather than
of asymmetry, except for the fact that in our part of
the universe, at any rate, the anti-particles are much
less stable than the particles. Furthermore, the unstable
“free” positron becomes very stable when bound with
a neutron to form a proton. And the resulting pair
composed of electron and proton is in a sense only
a duality, because by its mass the proton is quite
unequal to the electron, being about 1835 times larger.

General Symmetries. Let us suppose that a three-
dimensional tableau consists of a mirror, a design in
front of the mirror, and its reflection in the back of
the mirror. If we “erase” the physical trace of the
mirror, making it thus “invisible” and “two-sided,” then
there is no front and back of it, and the design and
its reflection have become indistinguishable and inter-
changeable. That is, mathematically, a mirror reflection
does not actually break up the space into two halves,
one in front and the other in back, and transform the
front half into the back half, but it transforms the entire
space into itself. And this transformation happens to
be such that any of the two halves goes into the other
and that the points on the mirror itself remain each
where it is. Now, a tableau in the space is (bilaterally)
symmetric with respect to a mirror, if the trans-
formation we have just described leaves the tableau
unchanged, that is, leaves it looking after the trans-
formation as it looked before the transformation.

Inasmuch as a mirror reflection of the space is a
transformation of the entire space into itself it is called
an automorphism. In general, an automorphism of a
space, any space, is an invertible transformation of the
entire space into itself, and such a transformation is
nothing other than a rearrangement of the totality of
the points of the space from their given ordering into
any other. In order to arrive at a general mathe-
matically oriented notion of symmetry, it is also neces-
sary to consider not only single automorphisms but
certain assemblages of automorphisms called groups
(for the definition of a “group” see Weyl, pp. 47, 144,
and passages on other pages listed in the index to the
book under “group”). Now, the mathematical notion
of symmetry demands that there shall be given some
group of automorphisms. If such a group is given and
held fast, then a figure in space is called symmetric
if each automorphism of the group leaves it unchanged.
Thus, symmetry is a relative concept. A figure is not
just symmetric tout court, but it is symmetric relative
to a given group of automorphisms, which, in a logical
sense, has to be given first.

The symmetry of a figure is “interesting,” or “mean-
ingful,” or “relevant,” mathematically or aesthetically,
if the automorphisms of the underlying group are
“interesting,” or “meaningful,” or “relevant”; in short,
if the underlying automorphisms are “good” auto-
morphisms.

In our Euclidean space, by “normal” standards of
taste, mathematical or aesthetic, the “best” auto-
morphisms are those that transform figures into
“congruent” ones, thus leaving (Euclidean) distances
and angles unchanged. These are the so-called
orthogonal transformations. They consist of transla-
tions, rotations, mirror reflections, and combinations
of such. Different from these, yet still very “normal”
are so-called dilations (Weyl, pp. 65, 68). A dilation
is merely a change of scale; all distances are changed
in the same ratio, and angles remain the same. Weyl
lists large numbers of finite groups of orthogonal trans-
formations for plane and space, and notable physical
and ornamental designs which are symmetric relative
to these; he also asserts (pp. 66, 99), that in the case
of the plane these groups were already determined in
substance by Leonardo da Vinci, but he does not give
a reference showing where to find them in Leonardo's
works.

From this approach, an “obvious” asymmetry is
frequently a regular symmetry with respect to a group
of automorphisms that are not strict orthogonal trans-
formations but obvious distortions of such. The leading
case is a reflection in a curved, or rather corrugated
mirror in an amusement park. A reflection in a cor-
rugated mirror is still a reflection, and it still produces
a “bilateral symmetry,” even if distorted. In our “neu-
tral” physical perception there is no difference be-
tween the distortion in the corrugated mirror of an
amusement park for vulgar purposes, and the distortion
in the nonrealistic painting of an artist from an exalted
inspiration.

As an aside we note that in the physical theory of
elementary particles certain approximate symmetries
have been called broken symmetries.

Homogeneity. The orthogonal transformations in
Euclidean space are not only arbiters of symmetry for
designs that are outwardly imposed on the space as
their background and framework, but their presence
also creates a certain internal evenness of the structure
of the space as such, taken by itself. For space as
substratum of the universe, and within a theologico-
metaphysical imagery, internal evenness of structure
of a space was already adumbrated by Nicholas of Cusa
in the first half of the fifteenth century, but professional
mathematics began to be properly aware of it only
in the course of the nineteenth century. In mathe-
matics, the feature of “evenness of structure” that we
have in mind, is nowadays termed homogeneity, and
is defined below. It is again a relative concept and,
again, a group of automorphisms of the space is
involved. The demand is that the group be transitive.
By this is meant that any point P of the space can
be carried into any other point Q by at least one of
the automorphisms. If this is so, then the space is
homogeneous (with respect to the given group).

Obviously the homogeneity of a space is the more
interesting if the underlying group of automorphism
is more important. If the space is a so-called metric
space, it is usually expected that the underlying
automorphism shall at least be isometric, meaning that
the transformations preserve the distance between any
pair of points.

Our common Euclidean space is certainly homo-
geneous with regard to orthogonal transforma-
tions; in fact, the mere translations suffice for homo-
geneity, so that rotations and reflections ought to se-
cure some further properties of Euclidean space, which
might perhaps be even characteristic of it. Such at
any rate was the expectation of the physicist and
physiologist Hermann von Helmholtz; he took a physi-
ologist's interest in the nature of Euclidean space,
which, to him, was the space of physiological percep-
tion. Helmholtz emphasized the fact that any two
planes which go through a point can be transformed
into each other by a rotation around the point (“local
mobility”), and he apparently was under the impression
that this, together with homogeneity, holds for
Euclidean space only. But Friedrich Heinrich Schur
(1852-1932) demonstrated that all this also holds for
any non-Euclidean space of constant curvature,
whether the curvature is negative à la Bolyai-
Lobatchevsky, or positive à la Riemann (and Beltrami).
Schur even demonstrated this for spaces of any number
of dimensions, no matter how large.

It is also a curious fact of present-day mathematics
that it is not at all easy to draw up criteria of “internal
symmetry” which fit Euclidean space and no other.

In cosmology there is a quest for homogeneity of
another, though related kind. It does not refer to the
structure of the spatial substratum of the universe but
to the mode of distribution of matter in it. This
homogeneity, when assumed present, falls under the
so-called Cosmological Principle, and a report on it
is given in the article on Space.

Symmetry in Physics. The terms “symmetry” and
“symmetry law” occur frequently in present-day
physics, and we now comment on this occurrence.

We have stated above that, in a spatial setting, an
impression of symmetry arises if a design is not changed
by a group of automorphisms of the space, and we
add to this that, as far as design is concerned, the
underlying space is a setting for it, and any auto-
morphism of the space is a certain change of this
setting. Quite generally, in mathematics and mathe-
matically controlled science, the imperviousness to
changes within a setting is technically called
“invariance” (relative to these changes). In this sense
a law of symmetry is a particular case of a law of
invariance, and, to begin with, the two are not
coextensive because symmetry involves a connotation
of space, whereas invariance is more comprehensive.
However, in physics many invariances involve space
variables, or at least space data, and in this way the
terms symmetry and invariance have drawn ever closer
together and have become almost synonymous.

Thus, a present-day physicist may view even the
nineteenth-century law of conservation of energy as
a symmetry law. The nineteenth century envisaged
various forms of energy, mechanical, thermal, electri-
cal, etc., and admitted that they may be transformed
into each other, that is, change over into each other.
But the law maintained (and continues to maintain)
that throughout such changes of form, the total amount
of energy in a closed physical system remains constant,
that is invariant. To this and the following see the first
half of Eugene P. Wigner, Symmetry and Reflection.

It is not always easy to decide how two physical
laws of symmetry relate to each other. It can be stated,
for instance, that Newton's force of gravitational
attraction (inverse square of the distance) obeys two
laws of symmetry, a spatial and a temporal. According
to the spatial law, the force of attraction is invariant
for all orthogonal transformations of space, not distin-
guishing between points of origin or directions in
space. According to the temporal law, it is invariant
in time.

As just stated, these two laws of symmetry are sepa-
rate, and have equal standing. However, the theory
of relativity erases the separateness, and geology casts

doubt upon their equal standing. In fact, the general
theory of relativity, by the very formulation of the
phenomenon of gravitation, fuses the two laws of
symmetry into one, so that they cannot be separated.
On the other hand, in geology, the temporal invariance
is part of the general law of uniformitarianism which
seems to assert that, say, in the rise of the solar system,
or at least in the evolution of the earth, the familiar
laws of “classical” physics are “eternal,” that is
temporally invariant, and thus have always been before
what they are presently. This seems to make the
temporal symmetry of the gravitational force hier-
archically prior, and thus superior to the spatial one.

In basic physics of the twentieth century, as the
century progressed, the number of invariance proper-
ties has shown a tendency to increase (Wigner, p. 60).
There has been many a period of anguish when two
leading symmetry laws were seemingly in an irrecon-
cilable clash with each other. Occasionally such a
period of anguish was followed by a period of relief
when, to the soothing accompaniment of a Nobel prize,
the clash was somehow composed.

On the other hand, the twentieth century also undid
at least one law of the preceding century. It was the
law of similitude. Although not very important, it was
treated with respect. It asserted

(Wigner, p. 5).

We might add that the elementary charge, that is the
magnitude of the electron, seems to be the most im-
placable of the instruments of doom.

This doom reached beyond the law of similitude,
which is a not-too-important law of physics. It also
enveloped the serene vision of Leibniz that space is
nothing but order and relations, and perhaps also
with predestined harmony ensuing; it somehow also
enveloped the creeds of the eighteenth and nineteenth
centuries—“naive” creeds in the eighteenth century
and less naive ones in the nineteenth—that, in spite
of all vehemence and violence in man, everything will
in the end turn out to be continuous, controllable, and
adjustable to scale.

But these creeds overlooked the electron. James
Clerk Maxwell, at the height of Victorianism, tried to
forget about the electron, by creating a magnificent
field theory in which everything can be adjusted to
scale, but it was not enough for physics; the electron
simply had to be considered and taken very seriously
indeed. Magnetism is a close kin to electricity. When
Thales of Miletus, almost twenty-six centuries ago, saw
a magnet in action, he exclaimed that the world is full
of Gods. And so it is even today.

BIBLIOGRAPHY

Monroe C. Beardsley, Aesthetics from Classical Greece
to the Present (New York and London, 1966). Harold F.
Blum, Time's Arrow and Evolution (Princeton, 1951). Paul
and Tatiana Ehrenfest, The Conceptual Foundations of the
Statistical Approach in Mechanics (Ithaca, 1959). G. T.
Griffith, Alexander the Great, The Main Problems (Cam-
bridge and New York, 1966). W. K. C. Guthrie, A History
of Greek Philosophy, 3 vols. (Cambridge and New York,
1962; 1965; 1969), Vol. III. Iamblichus, Protrepticus, ed. H.
Pistelli (Stuttgart, 1888). Pliny, Natural History, The Loeb
Classical Library, 10 vols. (Cambridge, Mass. and London,
1938-63). H. Regnéll, Ancient Views on the Nature of
Life (Lund, 1962). D'Arcy Wentworth Thompson, On
Growth and Form (Cambridge, 1917; 1942). Hermann Weyl,
Symmetry (Princeton, 1952). Eugene P. Wigner, Symmetries
and Reflections (Bloomington and London, 1967).

SALOMON BOCHNER

[See also Beauty; Cosmology; Nature; Optics and Vision;
Space.]