Beyond the Shadow

Sandra Lach Arlinghaus, Ph.D. (Mathematical Geography)*
Adjunct Professor of Mathematical Geography and Population-Environment Dynamics
School of Natural Resources and Environment
The University of Michigan, Ann Arbor, Michigan

William Charles Arlinghaus, Ph.D. (Mathematics)*
Professor of Mathematics
Department of Mathematics and Computer Science
Lawrence Technological University
Southfield, Michigan

To see a World in a grain of sand
And a heaven in a wildflower,
Hold infinity in the palm of your hand
And eternity in an hour.
  William Blake, Auguries of Innocence

Far more than mere dark recesses, shadows have long served as toolsl to aid scientific communication, explanation, and calculation.  Herodotus noted that Thales of Miletus systematically forecast an eclipse in 585 B. C.  Kepler used the shadows of protruberances on the moon to calculate their elevation above base level (  Eratosthenes of Alexandria used the shadow of an obelisk to apply a theorem of Euclid to measure, with remarkable accuracy, the circumference of the Earth (, Ebook on Spatial Synthesis).

More generally, a shadow is a projection of a 3-dimensional object into a 2-dimensional space (and even more generally, of an n dimensional space into and n-1 dimensional space).  Sometimes one focuses only on the shadows, as in the case of the eclipse.  Sometimes one focuses only on the object itself.  When the system is taken together, however, both shadow and what casts the shadow, it is then that understanding arises.  As Minkowski noted:  "Henceforth Space by itself, and Time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality" (

Consider the following example.  On the campus of The University of Michigan there is an interesting sculpture of a large cube standing on one of its vertices (Figure 1).  Even a gentle push of the cube will cause it to rotate smoothly on its vertical axis.

Figure 1.  Cube sculpture, "Endover," The University of Michigan Regents' Plaza, created by Bernard Rosenthal and installed in 1968.  There are similar sculptures elsewhere in the world, including one in New York City, by I. M. Pei, outside the Marine Midland Bank building.  The length of a side of this cube is 8 feet (approximately 2.438 meters).  Original source of link to this image of the sculpture is:
On cloudy days, the shadow cast by the cube is diffuse, as it is in Figure 1.  Here the focus is on the cube, itself.  On a clear day, in mid-afternoon, the shadow of the cube becomes unobstructed by surrounding buildings and is sharply traced on the surrounding sidewalk (Figure 2).

Figure 2.  Cube and shadow.  Photograph taken 2:00 p.m. September 12, 2005, S. Arlinghaus using an Olympus C-50 Digital camera mounted on a tripod (latitude approximately 42.27 North)Click here to see a virtual reality version of the cube spinning (without shadows).
Clearly, one reconizes the shadow on the ground to the left of the cube, in Figure 2, as the shadow of the cube and not as the shadow of a tree or of something else.  However, the cube itself is clearly present in the picture when one looks at the shadow.  If one were to look first at only shadows, would one know that the cube produced them?  Consider the following set of two images (Figure 3.a and Figure 3.b).  In the top frame of Figure 3 (Figure 3.a) one sees a four-sided shadow on the ground; in the bottom frame of Figure 3 (Figure 3.b) one sees a six-sided shadow on the ground.  These two shadows occupy approximately the same space (referenced to cracks and spots on the sidewalk).  That seems contrary to intuition when one sees only part of the picture.

Figure 3a.  Four-sided shadow.

Figure 3b.  Six-sided shadow.

Figure 4 (composed of an animated sequence of static frames as a Quicktime Movie) explains, of course, how this is possible:  the cube spins, it is a moving body.  Thus, different shadows emerge in approximately the same location and and at approximately the same time depending only on the position of the cube.  The cube is a constant body that controls what one sees in different static frames: the cube is the constant body in this simple, easy to visualize, system.  As does Blake's poetic art, this sculptor's physical art communicates the essence of a larger system.

For us, this simple system offered added insight into the basic postulates of Special Relativity Theory; thus, we wish to share it with others.  For us, the "cube" represents the constant body of knowledge embodied by the Laws of Physics.  The table below shows two statements of these Postulates.  Both were found using the Internet:  the first is from a single authoritative source and the second set is from an compilation of user views with interaction coming from web forms.
Stanford University Linear Accelerator Laboratory.
Einstein's theory of special relativity results from two statements -- the two basic postulates of special relativity: 
  • The speed of light is the same for all observers, no matter what their relative speeds. 
  • The laws of physics are the same in any inertial (that is, non-accelerated) frame of reference. This means that the laws of physics observed by a hypothetical observer traveling with a relativistic particle must be the same as those observed by an observer who is stationary in the laboratory. 
Given these two statements, Einstein showed how definitions of momentum and energy must be refined and how quantities such as length and time must change from one observer to another in order to get consistent results for physical quantities such as particle half-life.  To decide whether his postulates are a correct theory of nature, physicists test whether the predictions of Einstein's theory match observations. Indeed many such tests have been made -- and the answers Einstein gave are right every time!
(external links in text have been removed below)
Main article: Postulates of special relativity

1. First postulate (principle of relativity)

The laws of electrodynamics and optics will be valid for all frames in which the laws of mechanics hold good.
Every physical theory should look the same mathematically to every inertial observer.
The laws of physics are independent of location space or time.
2. Second postulate (invariance of c)
The speed of light in vacuum, commonly denoted c, is the same to all inertial observers, is the same in all directions, and does not depend on the velocity of the object emitting the light. When combined with the First Postulate, this Second Postulate is equivalent to stating that light does not require any medium (such as "aether") in which to propagate.

*NOTE:  A previous version of this paper was submitted to the Pirelli Relativity Challenge of 2005.   That competition causes the authors, neither one a physicist but both with strong orientations toward pure mathematics, to reflect on the endurance of high-level interest in scientific communication throughout their academic lives.  As pre-collegiate students, both participated in the experimental stages of the Physical Science Study Committee's "PSSC Physics" (Jerrold Zacharias, Massachusetts Institute of Technology, Director):  the first author at The University of Chicago Laboratory Schools, 1958-59 (Mr. Bryan Swan, teacher), and the second author at The University of Detroit (Jesuit) High School, 1959-61 (Mr. Herbert Stepaniak, teacher).  The mission of the PSSC, in developing these courses, is summarized in the quotation below.
"On Labor Day 1956, Jerrold Zacharias, an MIT physics professor, gathered together a group of prominent American scientists to change the mediocrity of physical science education. The group, called the Physical Sciences Study Committee (PSSC), consisted of Nobel Laureate physicists, MIT professors, prominent high school teachers, and industry leaders. The purpose of the committee was to evaluate the 'content of courses in physical science, hoping to find a way to make more understandable to students the world in which we live' ".  (A. J. Howes, Chemical Engineering, University of Virginia, "Sputnik:  Fellow Traveler Takes America for a Ride.  How a Russian Satellite Placed American Education Reform in the Spotlight," Journal of Young Investigators (, Featured Article, lIssue 4, January 2002.)
Almost 50 years later, as adult academic "students,"  we continue to seek systematic and interesting ways to look at the world around us.  The material above offers others outside the college town of Ann Arbor, Michigan (USA), a chance to see how a piece of sculpture, created by Bernard Rosenthal in 1968 and prominently displayed on the campus of the University of Michigan, continues to inspire us in that quest for imaginative views of science, technology, and the communication of associated ideas

Related Classical References

Coxeter, H. S. M.  Non-Euclidean Geometry.  Toronto:  University of Toronto Press, 1961 (fourth edition).
Coxeter, H. S. M.  Regular Polytopes.  New York:  Macmillan, 1963 (second edition).
Coxeter, H. S. M.  Introduction to Geometry.  New York:  John Wiley and Sons, 1965 (fourth printing).
Loeb, Arthur L.  Space Structures:  Their Harmony and Counterpoint.  Reading MA:  Addison-Wesley Advanced Book Program, 1976.
Weyl, Hermann.  Symmetry.  Princeton:  Princeton University Press, 1952.

Solstice:  An Electronic Journal of Geography and Mathematics.
Volume XVI, Number 2, 2005.
Institute of Mathematical Geography (IMaGe).
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