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.
