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Ivars Peterson's MathTrek |
Drawing and constructing polyhedra is a pastime that goes back to the Renaissance and perhaps even earlier times. Leonardo da Vinci (1452–1519), for one, created illustrations of various polyhedra for a 1509 book on the divine proportion by Luca Pacioli (1445–1517).
These immensely varied, crystal-like shapes, with regular features and flat faces (plane polygons), come in all sorts of configurations. Many people know of the five regular polyhedra: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. But the realm of polyhedra encompasses all sorts of additional forms: spiky stellated polyhedra, intricate, interlocked shapes, buckyballs and their cousins, and many more.
Fr. Magnus J. Wenninger, a mathematician and philosopher at Saint John's Abbey in Collegeville, Minn., has been painstakingly and meticulously constructing polyhedra since 1961. His colorful, precise models, fashioned from paper, reflect the broad range of shapes that symmetrical polyhedra can take on.
Over the years, Father Magnus has written books and articles about how to construct accurate models of various types of polyhedra. Some of his many papercraft models, which are typically 30 to 40 centimeters in diameter, are now available for purchase from Saint John's Abbey (see http://www.saintjohnsabbey.org/store/ and click on polyhedrons).
Creating such models is no simple task. Several years ago, I had a chance to observe Father Magnus quietly at work. His patience, care, and skill were clearly evident. And the results were awesome.
In recent years, Father Magnus has worked with other polyhedron experts to develop design software for creating polyhedral forms.
If you're interested in templates and patterns for such forms, software such as Stella, developed by Robert Webb, provides a good starting point. For information about Stella, go to http://www.software3d.com/Stella.php.
Check out Ivars Peterson's MathTrek blog at http://blog.sciencenews.org/.
References:
Peterson, I. 2005. A cabinet of mathematical curiosities. MAA Online (Dec. 26).
______. 2001. Polyhedron man. Science News 160(Dec. 22 & 29):396-397. Available at http://www.sciencenews.org/articles/20011222/bob13.asp.
______. 2000. Plato's molecule. MAA Online (Sept. 18).
______. 1999. Art of the tetrahedron . MAA Online (Nov. 8).
Wenninger, M. 1999. Spherical Models. New York: Dover.
______. 1983. Dual Models. New York: Cambridge University Press.
______. 1971. Polyhedron Models. New York: Cambridge University Press.
Fr. Magnus Wenninger has a Web site at http://employees.csbsju.edu/mwenninger/. To purchase his models, go to http://www.saintjohnsabbey.org/store/ and click on polyhedrons.
You can learn more about Leonardo da Vinci's polyhedra at http://www.georgehart.com/virtual-polyhedra/leonardo.html.
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A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.
Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.