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Science  21 Apr 2006:
Vol. 312, Issue 5772, pp. 368b-369b
DOI: 10.1126/science.1126513

Shadows of Reality

The Fourth Dimension in Relativity, Cubism, and Modern Thought

by Tony Robbin

Yale University Press, New Haven, CT, 2006. 151 pp. $40, £30. ISBN 0-300-11039-1.

In addition to untying a knot in a cord whose ends were sealed together without touching the cord itself, [the 19th-century American spiritualist Henry] Slade claimed to have joined solid wooden rings together, transported objects out of closed containers, and written on pages tightly pressed between two slates—all supposedly under scientific conditions.” In an entertaining early chapter of Shadows of Reality, Tony Robbin discusses Slade's 1876 trial for fraud in London. Physicists argued that such feats could be explained by access to an extraspatial dimension, and the trial “did little to dampen enthusiasm for the spiritualism of the fourth dimension.” Yet 19th-century explorers of the then-arcane fourth dimension bequeathed to us a toolkit for imagining and imaging, inter alia, the hypercube, the cross-polytope, and the pentatope.

Robbin used multiple perspectives in this painting (acrylic on canvas, 1978) to place the viewer in several positions at once as a depiction of hyperspace.

CREDIT: TONY ROBBIN

The principal tools are slices and shadows (1). We can, like A. Square in Flatland (2), train our mind's eye to “see” four-dimensional objects as stacks of lower dimensional cross sections. Or we can study their projected images (in our three-dimensional space) and imagine lifting them back whence they came. Thanks to these tools and to 20th-century revolutions in thought (the physics of spacetime) and technology (modern computer graphics), the fourth dimension is no longer a spirit abode, accessible only to magicians and mediums. Physicists can't do without it. Mathematicians have tamed its geometry. Artists, including Robbin, delight in exploring it. Hyperspace has become respectable, even conventional.

Shadows of Reality is a fascinating flythrough of the diverse, intellectually rigorous climes in which Robbin finds tracks and traces of hyperspace. Impressed by the power of x-rays to reveal heretofore unglimpsed reality, he tells us, Picasso studied four-dimensional geometry and used it in his 1910 painting Seated Woman with a Book “to show his audience the reality they knew existed but could not otherwise see.” Hermann Minkowski's formulation of space-time transformed four-dimensional geometry from an idea into truth. Quasicrystals—which Robbin identifies with Penrose tilings—can be derived by projection from four-dimensional space. The intrepid author even tackles Roger Penrose's twistor theory, quantum entanglement, and category theory. The book is a persuasive case for restoring geometry to its once-hallowed place among the liberal arts.

Unfortunately, that is not the case Robbin is trying to make. Shadows of Reality is a polemic on behalf of one visualization method and against the other: shadows over slices. According to Robbin, the slicing model, promulgated in Flatland, gets all the credit, while the projection model does all the work. “Consider this book a modest proposal to rid our thinking of the slicing model of four-dimensional figures and spacetime in favor of the projection model,” he writes, adding that “the proposal means developing a distaste for the slicing model.…” It's hard to tell whether he really believes this—if his doctor orders a CT scan, would he demand an x-ray instead?—or whether he's trying to spice up a book he fears might otherwise be bland. If he does believe it, I fail to understand why. The projection model has received more than equal time for the past 60 years, thanks to the unstinting efforts and ever-widening influence of the great, late geometer H. S. M. Coxeter and his classic Regular Polytopes (3). In any case, why privilege either of these two invaluable and complementary methods over the other? Most of us poor three-dimensional creatures need all the help we can get.

Robbin's zeal sometimes leads him astray. For example, in the chapter “Patterns, Crystals, and Projections” he briefly considers three methods for generating Penrose tilings: matching the tiles together in accordance with specified rules; cutting the tiles into smaller tiles and then inflating them, ad infinitum, to construct a tiling that repeats on all scales; and projection from a periodic tiling of a higher dimension (four or five) onto a plane. Each of these methods tells us things about Penrose tilings (and many other nonperiodic tilings) that the others do not. All three are powerful and fruitful tools in the study of aperiodic order. Yet Robbin dismisses the first on specious grounds and the second as “of limited use.” “It is now understood,” he asserts, that N. G. de Bruijn's projection method “is the most general, informative, and foolproof of the three.” (By the way, the projection method is a two-step operation: first slice, then project that—so it is also known as “cut and project.”)

Robbin is, of course, entitled to his opinions. And let's not quibble over details. Any author of such a bold interdisciplinary adventure is likely to get some details wrong, and he is no exception. (Still, one wishes some of the more egregious had been caught at the proof stage.) The panorama he sketches in Shadows of Reality is rich and fascinating. Enjoy the forest, just don't look too closely at the trees.

References and Notes

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