“Nor do such theories provide a ready account for the equivalence of the slopes of the reaction-time functions for the picture-plane and depth pairs. For, in order to explain the dependence of reaction time on angular difference, we must suppose that the features that are being compared are the features of the two-dimensional drawings, which differ more and more with angular departure, and not the features of the three-dimensional objects, which are the same regardless of orientation.”

Source: Mental images and their transformations. 1982, p. 66; as cited in Niall (1997)

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Roger Shepard10

American psychologist 1929

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“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.

“Most studies employing three-dimensional objects as stimuli have used simultaneous presentation whereas most studies employing two-dimensional objects have used comparison of a single visual stimulus with a memory presentation. We suspect that it is this procedural difference rather than the difference in dimensionality that is the principal determiner of rate of mental rotation.”

Roger Shepard (1929) American psychologist

Source: Mental images and their transformations. 1982, p. 178; as cited in Niall (1997)

“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.”

Max Beckmann (1884–1950) German painter, draftsman, printmaker, sculptor and writer

Source: 1930s, On my Painting (1938), p. 16

“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 (1882–1963) French painter and sculptor

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

“If you are describing any occurrence… make two or more distinct reports at different times… We discriminate at first only a few features, and we need to reconsider our experience from many points of view and in various moods in order to perceive the whole.”

Henry David Thoreau (1817–1862) 1817-1862 American poet, essayist, naturalist, and abolitionist

March 24, 1857
Journals (1838-1859)

“Intelligent creatures [that] evolved to live deep within the atmosphere of a gas giant planet could be deluded, for eons, into thinking that the Universe is an approximately homogeneous expanse of gas, filling a three-dimensional space, but featuring anisotropic laws of motion (which we would ascribe to the planet’s gravitational field). Are we human scientists comparably blinkered?”

Frank Wilczek (1951) physicist

p, 125
"Multiversality" (2013)

“Since one cannot create '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 (1880–1966) American artist

'Search for the Real in the Visual Arts', p. 44
Search for the Real and Other Essays (1948)

“But after a while, some higher brain-center cut in, and I began being mentally able to fit the wildly changing scenery into a coherent four-dimensional whole. The process was really no more devious than the process by which one integrates the two hundred lines of a TV screen into a single two-dimensional image... which in turn is interpreted as a three-dimensional scene. It's just a matter of processing information. Impossible? I saw.”

Rudy Rucker (1946) American mathematician, computer scientist, science fiction author and philosopher

Source: The Sex Sphere (1983), p. 106

“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

Kosmos (1932)

“In classical Newtonian physics there was a clear understanding of 'what reality is'. Indeed in this classical view, reality at a certain time is the collection of all what is actual at this time, and this is contained in 'the present'. Often it is stated that three dimensional space and one dimensional time have been substituted by four dimensional space-time in relativity theory, and as a consequence the classical concept of reality, as that what is 'present', cannot be retained. Is reality then the four dimensional manifold of relativity theory? And if so, what is then the meaning of 'change in time?”

Diederik Aerts (1953) Belgian theoretical physicist

Aerts, D. (1996). " Relativity theory: what is reality? http://www.vub.ac.be/CLEA/aerts/publications/1996RelReal.pdf". Foundations of Physics, 26, pp. 1627-1644

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Roger Shepard10

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