Shadows of Reality

Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought

Tony Robbin
Copyright Date: 2006
Published by: Yale University Press
Pages: 160
https://www.jstor.org/stable/j.ctt1npzrx
  • Cite this Item
  • Book Info
    Shadows of Reality
    Book Description:

    In this insightful book, which is a revisionist math history as well as a revisionist art history, Tony Robbin, well known for his innovative computer visualizations of hyperspace, investigates different models of the fourth dimension and how these are applied in art and physics. Robbin explores the distinction between the slicing, or Flatland, model and the projection, or shadow, model. He compares the history of these two models and their uses and misuses in popular discussions. Robbin breaks new ground with his original argument that Picasso used the projection model to invent cubism, and that Minkowski had four-dimensional projective geometry in mind when he structured special relativity. The discussion is brought to the present with an exposition of the projection model in the most creative ideas about space in contemporary mathematics such as twisters, quasicrystals, and quantum topology. Robbin clarifies these esoteric concepts with understandable drawings and diagrams.Robbin proposes that the powerful role of projective geometry in the development of current mathematical ideas has been long overlooked and that our attachment to the slicing model is essentially a conceptual block that hinders progress in understanding contemporary models of spacetime. He offers a fascinating review of how projective ideas are the source of some of today's most exciting developments in art, math, physics, and computer visualization.

    eISBN: 978-0-300-12962-5
    Subjects: Art & Art History, General Science

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Preface
    (pp. ix-xii)
  4. Acknowledgments
    (pp. xiii-xiv)
  5. Part One. Past Uses of the Projective Model

    • 1 The Origins of Four-Dimensional Geometry
      (pp. 3-18)

      In mathematician Felix Klein’s posthumously published memoirDevelopments of Mathematics in the Nineteenth Century(1926), Klein says of Hermann Grassmann that unlike ‘‘we academics [who] grow in strong competition with each other, like a tree in the midst of a forest which must stay slender and rise above the others simply to exist and to conquer its portion of light and air, he who stands alone can grow on all sides’’ (161). Grassmann never had a university position, taught only in German gymnasiums, and was consequently allowed to be a generalist: a philosopher, physicist, naturalist, and philologist who specialized in...

    • 2 Fantasies of Four-Dimensional Space
      (pp. 19-27)

      By the end of the nineteenth century, many authors were touting the superiority of thought that was based on an understanding of four-dimensional geometry, and collectively they established in popular culture the once-esoteric mathematical idea of the fourth dimension. Some propagandists and spiritualists even envisioned a kind of Superhero 4-D Man, who could pass through walls and do similar amazing feats. Other texts by serious mathematicians and hyperspace philosophers helped shift the focus of four-dimensional research from investigating the fourdimensional polytopes to include explorations of the properties of four-dimensional space and the observations of viewers situated in that space. Nevertheless,...

    • 3 The Fourth Dimension in Painting
      (pp. 28-40)

      At the turn of the twentieth century, European artists looked to the fourth dimension to structure new perceptions. These artists used the new four-dimensional geometry to make their emotional experience of the world whole. There have been valiant attempts to reconstruct, month by month, the intellectual state of play during the early years of cubism. Art historians Josep Palau i Fabre, Linda Henderson, and Pierre Daix have recaptured lost time to a remarkable degree. Their findings can be summarized with two statements about what happened in the first decade of the twentieth century: first, that artists developed a view of...

    • 4 The Truth
      (pp. 41-50)

      Before Hermann Minkowski gave his famous paper ‘‘Space and Time’’ to the 80th Assembly of German Natural Scientists and Physicians, in Cologne on 21 September 1908, the idea that space could be described by four-dimensional geometry was just that, an idea. After his speech, it was the truth. To see how it became the truth, we must trace how the contemporaneous quandary of the ether was redefined to be a problem in four-dimensional geometry. At the turn of the twentieth century, the nature of the ether (the hypothesized all-pervasive medium that propagated light waves) was the paramount puzzle in science....

    • Entr’acte

      • 5 A Very Short Course in Projective Geometry
        (pp. 53-58)

        In artist’s perspective, parallel lines converge to a vanishing point, and systems of one, two, or three vanishing-point perspectives were well developed by Renaissance painters. It was German astronomer Johannes Kepler (1571–1630) who first suggested that these vanishing points be considered points at infinity and that such points be included in the geometry handed down by the Greeks. Perhaps only an astronomer could make such a suggestion, as Greek geometry was rooted in land-based problems such as architecture, agricultural fields, and roads; there were no provisions for locations or spaces that one could not imagine measuring oneself. But as...

  6. Part Two. Present Uses of the Projective Model

    • 6 Patterns, Crystals, and Projections
      (pp. 61-71)

      For 30,000 years humans have made patterns. We put patterns on practically everything we make. InThe Sense of Order,art historian Ernst Gombrich argued that this is because patterns are fundamental to the way humans think: the propensity for making patterns is hardwired in the right brain. For all these years,patternmeant a motif that repeats at a regular interval. The mechanics of the loom enforce a periodic repeat, as does the roller that prints patterns on wallpaper, although human patterns predate these technologies. But in 1964 Robert Berger discovered that one could make a pattern using 20,000...

  7. 7 Twistors and Projections
    (pp. 72-82)

    Is space really composed of dimensionless points? High school math and common sense say that it is, but there is other math, and common sense has been wrong before. To illustrate a fallacy of common sense, Einstein gave the example of a chair being pushed across a stage. Common sense says that the pushing moves the chair, because when the pushing stops, the chair stops. Cause and effect could not be clearer. But one could give the chair a little shove such that it continues moving when no one is pushing. This small counterexample posed a problem for common sense....

  8. Appendix
    (pp. 119-120)
  9. Notes
    (pp. 121-124)
  10. Bibliography
    (pp. 125-128)
  11. Index
    (pp. 129-137)