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(Stuart Davis).. .. is one of but few, who realized his canvas as a.. .. two-dimensional surface plane. (1931)
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Stuart Davis, Arshile Gorky, Creative Art 9, September 1931, p. 213

 

Arshile Gorky

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The surfaces of three-dimensional space are distinguished from each other not only by their curvature but also by certain more general properties. A spherical surface, for instance, differs from a plane not only by its roundness but also by its finiteness. Finiteness is a holistic property. The sphere as a whole has a character different from that of a plane. A spherical surface made from rubber, such as a balloon, can be twisted so that its geometry changes. ...but it cannot be distorted in such a way as that it will cover a plane. All surfaces obtained by distortion of the rubber sphere possess the same holistic properties; they are closed and finite. The plane as a whole has the property of being open; its straight lines are not closed. This feature is mathematically expressed as follows. Every surface can be mapped upon another one by the coordination of each point of one surface to a point of the other surface, as illustrated by the projection of a shadow picture by light rays. For surfaces with the same holistic properties it is possible to carry through this transformation uniquely and continuously in all points. Uniquely means: one and only one point of one surface corresponds to a given point of the other surface, and vice versa. Continuously means: neighborhood relations in infinitesimal domains are preserved; no tearing of the surface or shifting of relative positions of points occur at any place. For surfaces with different holistic properties, such a transformation can be carried through locally, but there is no single transformation for the whole surface.

 

Hans Reichenbach

 

...the stereographic projection of the spherical surface. From the north pole P we draw radial lines to project every point of the surface of the sphere upon the horizontal plane [below, perpendicular to a line joining it to P and the sphere's center]. In general this transformation is unique and continuous , although the metrical relations are distorted; for the point P, however, it shows a singularity. Point P is mapped upon the infinite; i.e., no finitely located point of the plane corresponds to it. It can be shown that every transformation possesses a singularity in at least one point. The surface of the sphere is therefore called topologically different from the plane. Only a "sphere without a north pole" [point] would be topologically equivalent to a plane. ...such a sphere has a point-shaped hole without a boundary and is no longer a closed surface.

 

Hans Reichenbach

 

Since one cannot create "real depth" by carving a hole in the picture, and since one should not attempt to create the illusion of depth by tonal gradation, depth as a plastic reality must be two dimensions in a formal sense as well in the sense of color. "Depth" is not created on a flat surface as an illusion, but as a plastic reality. The nature of the picture plane makes it possible to achieve depth without destroying the two-dimensional essence of the picture plane. ... A plane is a fragment in the architecture of space. When a number of planes are opposed one to another, a spatial effect results. A plane functions in the same manner as the walls of a building. ... Planes organized within a picture create the pictorial space of its composition. ... The old masters were plane-consciousness. This makes their pictures restful as well as vital...

 

Hans Hofmann

 

It is remarkable that this generalization of plane geometry to surface geometry is identical with that generalization of geometry which originated from the analysis of the axiom of parallels. ...the construction of non-Euclidean geometries could have been equally well based upon the elimination of other axioms. It was perhaps due to an intuitive feeling for theoretical fruitfulness that the criticism always centered around the axiom of parallels. For in this way the axiomatic basis was created for that extension of geometry in which the metric appears as an independent variable. Once the significance of the metric as the characteristic feature of the plane has been recognized from the viewpoint of Gauss' plane theory, it is easy to point out, conversely, its connection with the axiom of parallels. The property of the straight line as being the shortest connection between two points can be transferred to curved surfaces, and leads to the concept of straightest line; on the surface of the sphere the great circles play the role of the shortest line of connection... analogous to that of the straight line on the plane. Yet while the great circles as "straight lines" share the most important property with those of the plane, they are distinct from the latter with respect to the axiom of the parallels: all great circles of the sphere intersect and therefore there are no parallels among these "straight lines". ...If this idea is carried through, and all axioms are formulated on the understanding that by "straight lines" are meant the great circles of the sphere and by "plane" is meant the surface of the sphere, it turns out that this system of elements satisfies the system of axioms within two dimensions which is nearly identical in all of it statements with the axiomatic system of Euclidean geometry; the only exception is the formulation of the axiom of the parallels. The geometry of the spherical surface can be viewed as the realization of a two-dimensional non-Euclidean geometry: the denial of the axiom of the parallels singles out that generalization of geometry which occurs in the transition from the plane to the curve surface.

 

Hans Reichenbach

 

..there comes a point when something catches on the canvas, something grips on the canvas. I don’t know what it is, you can put your paint on the surface? Most of the time it looks like paint, and who the hell wants paint on a surface? But there does come a time – you take it off, put it on, goes over here, moves over a foot, as you go closer you start moving in inches not feet, half-inches – there comes a point when the paint doesn’t feel like paint. I don’t know why. Some mysterious thing happens. I think you have all experienced it.. ..What counts is that the paint should really disappear, otherwise it’s craft. That’s what I mean by something grips in a canvas. The moment that happens you are then sucked into the whole thing. Like some kind of rhythm.’

 

Phillip Guston