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.
