VI. THE INFINITELY DISTANT
The standard perspective of the visual arts, which
was created in the sixteenth century, features a
“vanishing point.” This is a concrete specific point in
the total mimetic tableau, yet, in a sense, it represents
an infinitely distant point in an underlying geometry
(Christian Wiener, Introduction; E. Panofsky, Albrecht
Dürer). Mathematics since then, and especially in the
nineteenth century, has introduced various mathe-
matical constructs with infinitely distant points in
them, and we will briefly report on some of them.
There were no such tangible developments before
the Renaissance. Aristotle, in his Poetics and elsewhere,
speaks of the art of painting, but not of vanishing points
or other infinitely distant points in geometry. In antiq-
uity altogether, only later antiquity had some adum-
brations (Panofsky, “Die Perspective”...). In medieval
architecture, Gothic arches and spires would “vanish”
into the upper reaches of the aether; but they would
stay there and not converge towards concrete specific
points in the total tableau.
But the Renaissance produced perspective; and it
also began to create novel theories of vision (V.
Ronchi). Furthermore, since around A.D. 1600 mathe-
matics began to construct, concretely, infinitely distant
points, and in the first construction, an implicit one,
the Euclidean plane E2 was “closed off” in all direc-
tions by the addition of a point at infinity on each ray
emanating from a fixed point. That is, E2 was viewed
(as already in De rerum natura of Lucretius) as an
“open” disk of infinite radius; it was made, geometri-
cally, into a “closed” disk by the addition of a “hoop”
of infinite radius around it. This construction was not
performed explicitly or intentionally, but was implied
in the following assumption. By Euclid's own definition,
two straight lines are parallel if, being in the same
plane, and being produced indefinitely in both direc-
tions, they do not meet one another in either direction
(T. L. Heath, I, 190). Now, around 1600 some mathe-
maticians began to assume, as a matter of course, that
Euclid's definition is equivalent to the description that
two straight lines in the same plane are parallel, if,
after being produced indefinitely, they meet at two
infinitely distant points at both ends of the configura-
tion (and only there). To assume this is, from our pres-
ent retrospect, equivalent to assuming that there is
around
E2 the kind of hoop that we have described.
The same mathematicians soon began to sense, in
their own manner, that to close off E2 in this fashion
is neither intellectually original nor operationally
profitable. They began to “experiment” with other
procedures for closing off E2. These “experiments”
were a part, even a significant part, of the sustained
efforts to erect the doctrines of descriptive and projec-
tive geometries, and they were satisfactorily completed
in the course of the nineteenth century only.
The outcome was as follows. It is pertinent to install
our hoop around E2, but this is only a first step. The
total hoop is too wide, that is, not sufficiently restric-
tive, and it is necessary to “reduce” it in size by
“identifying” or “matching” various points of it with
each other.
First and foremost, it is very appropriate to “iden-
tify” all points of the hoop with each other, that is
to “constrict” the hoop to a single point. By the addi-
tion of this single point, the plane E2 becomes “sealed
off” as infinity, and the resulting two-dimensional
figure is topologically a spherical surface S2. Con-
versely, if one starts out with an S2, say with an ideally
smoothed-out surface of our earth, and removes one
point, say the North Pole, then the remaining surface
can be “spread out” topologically onto the E2. Such
a spreading out is done in cartography by means of
the so-called stereographic projection. This projection
of a punctured sphere S2 on the Euclidean E2 is not
only topological, that is bi-continuous, but also con-
formal, that is angle-preserving; and this was already
known to the astronomer and geographer Ptolemy in
the second century A.D. in his Geography (M. R. Cohen
and I. E. Drabkin, pp. 169-79).
The one-point completion which we have just de-
scribed can be performed for the Euclidean (or rather
Cartesian) space En of any dimension n, and the result
is the n-dimensional sphere Sn. Topologically there is
no difference between various dimensions, but alge-
braically there is. First, for n = 2, the plane E2 can
be viewed as the space of the complex numbers
z = x + iy, and the added point at infinity can be
interpreted as a complex number ∞, for which, sym-
bolically,
This interpretation is commonly attributed to C. F.
Gauss (1777-1855). Next, for n = 4, En can be inter-
preted as the space of quaternions a + ib + jc + kd,
which were created by William Rowan Hamilton
(1805-65), and the point at infinity can be interpreted
as a quaternion ∞ for which (°) holds. This can still
be done for E8, if it be viewed as a space of so-called
Cayley numbers (= pairs of quaternions), but no other
such cases of so-called “real division algebras” are
known (N. Steenrod, pp. 105-15; M. T. Greenberg, p.
87). As regards quaternions it is worth recording, as
a phenomenon in the history of ideas, that around 1900
there was an international organization of partisans
who believed that quaternions were one of the most
potent operational tools which the twentieth century
was about to inherit from the preceding one; the orga-
nization has been long extinct.
After the spheres, the next important spaces which
arise from En by a suitable addition of points at infinity
are so-called projective spaces; we will speak only of
“real” projective spaces, and denote them by Pn. (Other
projective spaces are those over complex numbers,
quaternions, or Cayley numbers; see Steenrod, Green-
berg, loc. cit.) For each dimension n, Pn arises from
En if one identifies each infinitely distant point of the
“hoop” around En with its antipodal point, that is, if
for each straight line through the origin of En the two
infinite points at the opposite ends of it are identified
(that is “glued together”). The resulting space is a
closed manifold (without any boundary), and it is the
carrier of the so-called elliptic non-Euclidean geometry
of F. Klein (S. M. Coxeter, p. 13). Klein's purpose in
devising his geometry was to remove a “blemish” from
the spherical (non-Euclidean) geometry of B. Riemann.
In Riemann's geometry any two “straight” (i.e.,
geodesic) lines intersect in precisely two points,
whereas in Klein's variant on it they intersect in pre-
cisely one point only.
The Pn, that is the real projective spaces, have a
remarkable property: for even dimensions n they are
nonorientable, but for odd dimensions orientable. A
space is orientable, if a tornado (or any other spinning
top), when moving along any closed path, returns to
its starting point with the same sense of gyration with
which it started, and it is nonorientable if along some
closed path the sense of gyration is reversed. In the
case of a Pn with an even-dimensional n the sense of
gyration is reversed each time the path “crosses” in-
finity. In particular, the space P2, that is the space of
two dimensional elliptic geometry, is not orientable,
but P3 is. Thus, in P2 a fully mobile society cannot
distinguish between right- and left-handed screws, but
in P3 it can.
Nineteenth-century mathematics has created many
other completions of En which have become the sub-
stance of the theory of Riemann surfaces and of alge-
braic geometry. Twentieth-century mathematics has
produced a one-point “compactification” (P. Alexan-
droff, “Über die Metrisation...”), which has spread
into all of general topology, and a theory of prime-
ends (C. Carathéodory, “Über die Begrenzung...”),
which in one form or another is of consequence in con-
formal mapping, potential theory, probability theory,
and even group theory.
In the nineteenth century, while mathematics was
tightening the looseness-at-infinity of Euclidean struc-
ture, French painting was loosening the tightness-at-
inifinity of perspective structure. The French movement
is already discernible in Dominique Ingres, but the
acknowledged leader of it was Paul Cézanne. Cézanne
was not an “anarchist,” wanting only to “overthrow”
classical perspective without caring what to put in its
place, but analysts find it difficult to say what it was
that he was aspiring to replace perspective by. We
once suggested, for the comprehension of Cézanne, an
analogy to developments in mechanics (Bochner, The
Role of Mathematics..., pp. 191-201), and in the
present context we wish to point out, in another vein,
that Cézanne was trying to loosen up the traditional
perspective by permitting several vanishing points
instead of one (E. Loran, Cézanne's Composition), and
by giving to lines of composition considerable freedom
in their mode of convergence towards their vanishing
points (M. Schapiro, Paul Cézanne). This particular
suggestion may be off the mark, but the problem of
a parallelism between nineteenth-century develop-
ments in geometry and in the arts does exist.
