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Dictionary of the History of Ideas | ||

*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 *E*2 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, *E*2 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

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

*E*2 the kind of hoop that we have described.

The same mathematicians soon began to sense, in

their own manner, that to close off *E*2 in this fashion

is neither intellectually original nor operationally

profitable. They began to “experiment” with other

procedures for closing off *E*2. 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 *E*2, 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 *E*2 becomes “sealed

off” as infinity, and the resulting two-dimensional

figure is topologically a spherical surface *S*2. Con-

versely, if one starts out with an *S*2, 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 *E*2. Such

a spreading out is done in cartography by means of

the so-called stereographic projection. This projection

of a punctured sphere *S*2 on the Euclidean *E*2 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 *E*2 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 *E*8, 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 *P*2, that is the space of

two dimensional elliptic geometry, is not orientable,

but *P*3 is. Thus, in *P*2 a fully mobile society cannot

distinguish between right- and left-handed screws, but

in *P*3 it can.

Nineteenth-century mathematics has created many

other completions of *En* which have become the sub-

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.

Dictionary of the History of Ideas | ||