But how could readymades be third-dimension shadows of his fourth-dimension Large Glass machine? For an answer we can look to the great mathematician of the late-nineteenth and early-twentieth century, Henri Poincaré, who continues to be regarded as one of history's great mathematicians, was also a famous popularizer of scientific ideas. Many artists, at the beginning of modern art in the early-twentieth century, knew and discussed Poincaré's works (Henderson, 1983). Poincaré had developed a specific geometric technique (see Illustration 5), where two-dimensional shadows could be used to express the existence of a three-dimensional sphere without the observer ever actually seeing the three-dimensional object (Davis, 1993, pp. 138-139).13 From a two-dimensional creature's perspective, by mentally putting together (in a series) the relations of two-dimensional shadows projected from the sphere, we can, through logic, extrapolate and therefore "know" or see in our minds the higher dimensional object.14

Duchamp had also said that he wanted the titles of his readymades "to carry the mind of the spectator towards other regions more verbal" (Sanouillet & Peterson, 1973, pp. 141-142). For Duchamp, one cannot physically see the fourth dimension (Sanouillet & Peterson, p. 98). For two-dimensional creatures, Poincaré's 2-D shadows would lead to the 3-D sphere only if they were to use the inductive powers of their minds to "see" the existence of a sphere they could never physically perceive. According to Poincaré's definition of shadow projections, and by dimensional analogy, we should be able to use Duchamp's 3-D readymade shadows to lead ourselves to the higher fourth-dimensional perspective of the Large Glass. Duchamp defined the fourth-dimension as as beyond direct sensory experience, whereas the second and third dimensions can be experienced by the senses (Sanouillet & Peterson, 1973, p. 98).15 In other words, two-dimensional creatures would have to use their minds to evaluate the relations among the sizes of the shadow circles in order to get to the sphere; and by analogy, we have to use our minds to evaluate the relations among the readymades to mentally "see" or understand what the Large Glass is in the fourth dimension.

Let us return to what Duchamp called his "rectified readymade," Apolinère Enameled. Think of it as a shadow of the Large Glass, as defined by Poincaré in his projection technique (see Davis, 1993). Will it bridge the different dimensions and enable us to see beyond our three-dimensional limited perspectives to the next higher dimension?

Gervais (1984, p. 115) has made the general observation that the bed is an "impossible object." One assumes, since he does not cite "impossible figure or object" as a psychological category of optical illusions, that he uses this term as a vernacular description of the perspective problems in the bed without linking them to the Penroses' formal idea and term. Gervais cites three problems: (1) the right foot of the headboard is attached to the front mattress rail; (2) the back mattress rail cuts diagonally from the mattress rail; and (3) the painted rungs, four in the footboard and five in the headboard, should be equal in number, but are not (p. 115). I had to make a three-dimensional model of the Apolinère Enameled room and objects and do a computer analysis of the entire picture in order to realize that these three problems (that Gervais and I had noticed independently) are not the only examples of false perspective with respect to the bed. Moreover, the entire room-the rug, the dresser, the walls, the girl-is all "out of whack." (I discovered that even the reflection in the mirror of the back of the girl's head that Duchamp said was "missing" cannot be right when you consider the necessary angles for reflections and the girl's closeness to the wall.) Although we accept the whole picture as a Gestalt, each individual object, in relation to the others, exists in an independent world that we have to force ourselves to see.

Part I of Rhonda Roland Shearer's essay was originally published in
Art & Academe (ISSN: 1020-7812), Vol. 10, No. 1 (Fall 1997): 26-62.
Copyright © 1997 Visual Arts Press Ltd.
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