“I started to introduce letters into my pictures [his first collage art]. These were forms which could not be deformed, because, being two-dimensional, they existed outside three-dimensional space; their inclusion in a picture allowed a distinction to be made between objects which were situated in space and those which belonged outside space”

— Georges Braque

the letters
Source: posthumous quotes, Braque', (1968), p. 68

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Georges Braque 43

French painter and sculptor 1882–1963

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“The triangles that we can measure are not large enough… to detect the curvature. Fortunately, however, we are, in a way, able to communicate with the fourth dimension. The theory of relativity has given us an insight into the structure of the real universe: …a four-dimensional structure. The study of the way in which the three space-dimensions are interwoven with the time-dimension affords a kind of outside point of view of the three-dimensional space… from this outside point of view we might be able to perceive the curvature of the three-dimensional world.”

Willem de Sitter (1872–1934) Dutch cosmologist

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“A glimpse heavenward at a constellation or even at a single star only suggests infinity; actually our vision is limited. We cannot perceive unlimited space; it is immeasurable. The universe, as we know it through our visual experience, is limited. It first came into existence with the formation of matter, and will end with the complete dissolution of matter.<!-- I TRIMMED this portion to simply cut down a little on quotes from this one essay. ~ 2009·03·21 02:36 Kalki} Where there is matter and action, there is space. Pictorial space exists two dimensionally. When the two dimensionality of a picture is destroyed, it falls into parts — it creates the effect of a naturalistic space. When a picture conveys only naturalistic space, it represents a special case, a portion of what is felt about three-dimensional experience. This expression of the artist’s experience is thus incomplete.”

Hans Hofmann (1880–1966) American artist

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'Search for the Real in the Visual Arts', p. 43
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“To help us to understand three-dimensional spaces, two-dimensional analogies may be very useful… A two-dimensional space of zero curvature is a plane, say a sheet of paper. The two-dimensional space of positive curvature is a convex surface, such as the shell of an egg. It is bent away from the plane towards the same side in all directions. The curvature of the egg, however, is not constant: it is strongest at the small end. The surface of constant positive curvature is the sphere… The two-dimensional space of negative curvature is a surface that is convex in some directions and concave in others, such as the surface of a saddle or the middle part of an hour glass. Of these two-dimensional surfaces we can form a mental picture because we can view them from outside… But… a being… unable to leave the surface… could only decide of which kind his surface was by studying the properties of geometrical figures drawn on it. …On the sheet of paper the sum of the three angles of a triangle is equal to two right angles, on the egg, or the sphere, it is larger, on the saddle it is smaller. …The spaces of zero and negative curvature are infinite, that of positive curvature is finite. …the inhabitant of the two-dimensional surface could determine its curvature if he were able to study very large triangles or very long straight lines. If the curvature were so minute that the sum of the angles of the largest triangle that he could measure would… differ… by an amount too small to be appreciable… then he would be unable to determine the curvature, unless he had some means of communicating with somebody living in the third dimension…. our case with reference to three-dimensional space is exactly similar. …we must study very large triangles and rays of light coming from very great distances. Thus the decision must necessarily depend on astronomical observations.”

Willem de Sitter (1872–1934) Dutch cosmologist

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“The concepts which now prove to be fundamental to our understanding of nature—a space which is finite; a space which is empty, so that one point [of our 'material' world] differs from another solely in the properties of space itself; four-dimensional, seven- and more dimensional spaces; a space which for ever expands; a sequence of events which follows the laws of probability instead of the law of causation—or alternatively, a sequence of events which can only be fully and consistently described by going outside of space and time—all these concepts seem to my mind to be structures of pure thought, incapable of realisation in any sense which would properly be described as material.”

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“Sound and talking undoubtedly increase the entertainment value of a picture. There is a distinct conflict, however, between a pictorial and sound elements, which cannot be entirely avoided until third dimensional pictures are made.”

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Jill Lepore, The Secret History of Wonder Woman (2014), p. 139.

“The white room is an interior to be made devoid of any specific sensualism emanated by objects. Ultimately it is classic white canvas expanded into three-dimensional space.”

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“One thing is sure – we have to transform the three-dimensional world of objects into the two-dimensional world of the canvas... To transform three into two dimensions is for me an experience full of magic in which I glimpse for a moment that fourth dimension which my whole being is seeking.”

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“Minkowski… supposed that this fourth dimension of time was not detached from and independent of the three dimensions of space. He introduced a new four-dimensional space to which ordinary space contributed three dimensions, and time one; we may call it 'space-time'…. The succession of positions which a particle occupied in ordinary space at a succession of instants of time would be represented by a line in space-time; this he called the 'world-line' of the particle…. Newton's absolute space and absolute time fell out of science, and they carried much with them in their fall. First to go was the concept of simultaneity…. It now became necessary to find a way of treating gravitation which should not involve simultaneity. Einstein found through the medium of his 'Principle of Equivalence'.”

James Jeans (1877–1946) British mathematician and astronomer

The Growth of Physical Science (1947)

“The universality, invariance, and elegance of principles governing the universe may be reflected in principles of the minds that have evolved in that universe - provided that the mental principles are formulated with respect to the abstract spaces appropriate for the representation of biologically significant objects and their properties. (1) Positions and motions of objects conserve their shapes in the geometrically fullest and simplest way when represented as points and connecting geodesic paths in the six-dimensional manifold jointly determined by the Euclidean group of three· dimensional space and the symmetry group of each object. (2) Colors of objects attain constancy when represented as points in a three-dimensional vector space in which each variation in natural illumination is cancelled by application of its inverse from the three-dimensional linear group of terrestrial transformations of the invariant solar source. (3) Kinds of objects support optimal generalization and categorization when represented, in an evolutionarily shaped space of possible objects, as connected regions with associated weights determined by Bayesian revision of maximum entropy priors”

Roger Shepard (1929) American psychologist

R. N. Shepard, (1994). "Perceptual-cognitive universals as reflections of the world." Psychonomic Bulletin and Review, 1, 2–28.

“The subject detects the presence and interrelationships of the basic components of one of the two-dimensional drawings - particularly, the variously oriented straight lines, the several types of vertices by which they are connected and, presumably, something of the structural relationships among these components within the two-dimensional pattern. Then, on the basis of some higher-level processing of these extracted features and their interrelationships, an internal representation, code, or verbal description is generated for each picture separately that captures the intrinsic structure of the three-dimensional object in a form that is independent of the particular orientation in which that object happens to be displayed.”

Roger Shepard (1929) American psychologist

Source: Mental images and their transformations. 1982, p. 64; as cited in: Keith K. Niall, "‘Mental rotation’, pictured rotation, and tandem rotation in depth." Acta psychologica 95.1 (1997): 31-83.

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Georges Braque 43

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