Sandra Lach Arlinghaus, Ph.D. (Mathematical
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
|To see a World in a grain
And a heaven in a wildflower,
Hold infinity in the palm of your hand
And eternity in an hour.
William Blake, Auguries of Innocence
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" (http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_spacetime.html).
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
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: http://www.plantext.bf.umich.edu/planner/sculpture/central/cube.htm.
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).
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.
University Linear Accelerator Laboratory.
Einstein's theory of special relativity results from two statements -- the two basic postulates of special relativity:
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)
"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 (http://www.jyi.org/), 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
Coxeter, H. S. M. Non-Euclidean
Geometry. Toronto: University of Toronto Press, 1961 (fourth
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).
All rights reserved worldwide, by IMaGe and by the authors.
Please contact an appropriate party concerning citation of this article: firstname.lastname@example.org