Boats & Deckchairs

Natural History, Dec, 1999 by Stephen Jay Gould, Rhonda Roland Shearer

Several other White Box notes reinforce this interpretation of the boat-deckchair hybrid as a representation in four dimensions, with the alternative states as different 3-D views that cannot be seen simultaneously in our 3-D world. Duchamp begins by posing the classic conundrum: "What is the meaning of this word 4th dimension since it does not have either tactile or sensorial correspondence as do the 1st, the 2nd, the 3rd dimension?"

Duchamp then gives a remarkably concise and generalized description of the boat-deckchair: "From the 2-dimensional perspective giving the appearance of the 3-dimensional continuum, construct a 3-dimensional (or perhaps a 2-dimensional perspective) of this 4-dimensional continuum." This note sounds cryptic, but concrete translation into the boat-deckchair example resolves both meaning and intent: consider the boat and deckchair views as 2-D paintings that, at least for the boats (given the included cues for perspective), depict a world in 3-D. But both views really represent two aspects of a "hybrid" 3-D object seen simultaneously in 4-D space. We can now finally grasp what Duchamp meant when he wrote, on the back of the boat "pseudopostcard" (as quoted earlier in this essay), "to establish a 4-dimensional perspective--not by placing them in relative positions with respect to each other in space--but simply by considering the optical illusions produced by the difference in their dimensions."

We also know that Duchamp invoked the example of a 3-D cube to express the simultaneous view of an entire 3-D object in 4-D space, thus representing the boat-deckchair duality as two views on two adjacent faces of a cube, both visible at the same time in four dimensions. Duchamp describes this simultaneous sight of the entire cube, again making an analogy to our simultaneous touching of all parts of the penknife in three dimensions (I love his phrase "circum-hyperhypo-embraced," that is, "grasped all around at once, both above and below"):

   3-D perspective starts in an initial frontal plane without deformation. 4-D
   perspective will have a cube or 3-D medium as a starting point which will
   not cause deformation, i.e. in which the 3-D object is seen
   circum-hyperhypo-embraced (as if grasped with the hand and not seen with
   the eyes).

Finally, Duchamp explicitly notes that in 4-D space, two intersecting planes (the boats and the deckchairs on the two adjacent faces of a cube) can be seen at once along an axis in the higher dimension: "2 intersecting planes do not determine a space--they merge along a plane perpendicular to their common intersecting line."

Popular books on the fourth dimension often try to depict this additional factor as time, while treating the three dimensions of our everyday world as space. This common formulation expresses Duchamp's observation that in ordinary 3-D space, one can "see" the entirety of an object only through time, because one must move one's eye sequentially around a 3-D object to grasp the full form that cannot be perceived all at once. But we can express both the paradox and the reality of the fourth dimension in a more interesting (and also mathematically accurate) manner when we represent the added dimension spatially, as a fourth axis (albeit undrawable in our surrounding 3-D world) at right angles to each of our three everyday spatial axes and therefore imbued with the remarkable property of offering a simultaneous view of entire 3-D objects-if only we could leave our 3-D world and, like A Square above the plane of Flatland, gaze upon our known universe from outside.