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adedios SuperPoster
Joined: 06 Jul 2005 Posts: 5060 Location: Angel C. de Dios

Posted: Mon Apr 24, 2006 8:19 am Post subject: (Math) Polygons and Polyhedra: Papercraft Polyhedra 


Science News Online
Week of April 22, 2006; Vol. 169, No. 16
Papercraft Polyhedra
Ivars Peterson
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, crystallike 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://web.aanet.com.au/robertw/Stella.html

Check out Ivars Peterson's MathTrek blog at http://blog.sciencenews.org/
References:
Peterson, I. 2005. A cabinet of mathematical curiosities. Science News Online (Dec. 24). Available at http://www.sciencenews.org/art.....thtrek.asp
______. 2001. Polyhedron man. Science News 160(Dec. 22 & 29):396397. Available at http://www.sciencenews.org/art...../bob13.asp
______. 2000. Plato's molecule. Science News Online (Sept. 16). Available at http://www.sciencenews.org/art.....thtrek.asp
______. 1999. Art of the tetrahedron. Science News Online (Nov. 6). Available at http://www.sciencenews.org/pag.....thland.htm
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/virt.....nardo.html
**********
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
http://www.sciencenews.org/art.....thtrek.asp
From Science News, Vol. 169, No. 16, April 22, 2006
Copyright (c) 2006 Science Service. All rights reserved.
*************************************************************
Questions to explore further this topic:
What are polygons?
http://www.mathleague.com/help.....lygons.htm
http://library.thinkquest.org/.....ygons.html
http://www.punahou.edu/acad/sa.....ygons.html
http://www.eduplace.com/math/m.....index.html
http://www.mathsteacher.com.au.....ns/pol.htm
http://mathworld.wolfram.com/Polygon.html
How are polygons named?
http://mathforum.org/dr.math/f.....names.html
What are regular polygons?
http://www.ul.ie/~cahird/polyh.....egular.htm
http://www.math.rutgers.edu/~e.....ygons.html
A calculator for regular polygons
http://www.projects.ex.ac.uk/t.....lpolyg.htm
What are irregular polygons?
http://www.ul.ie/~cahird/polyh.....regula.htm
The properties of polygons
http://www.bbc.co.uk/schools/g.....rev1.shtml
http://matcmadison.edu/ald/lab.....angles.htm
http://www.scienceu.com/geomet.....index.html
A lesson on polygons
http://www.learner.org/channel.....index.html
http://illuminations.nctm.org/.....px?id=L277
Videos on polygons
http://www.learner.org/channel.....video.html
Calculating the perimeter of a polygon
http://www.mathgoodies.com/les.....meter.html
Calculating the area of a polygon
http://www.homeworkhotline.com.....lygons.htm
http://www.mathreference.com/geo,area.html
Constructing polygons from triangles
http://jwilson.coe.uga.edu/EMT.....ygons.html
What is the Sierpinski triangle?
http://math.rice.edu/~lanius/f.....rjava.html
What is a polyhedron?
http://www.coolmath4kids.com/polyhedra.html
http://www.ul.ie/~cahird/polyh.....terest.htm
http://mathworld.wolfram.com/Polyhedron.html
http://mathforum.org/sum95/mat.....hedra.html
http://homepage.mac.com/efithi.....ty14.html
What is a uniform polyhedron?
http://www.mathconsult.ch/showroom/unipoly/
What are regular polyhedra?
http://members.aol.com/Polycell/regs.html
What are the Platonic solids?
http://amath.colorado.edu/alum.....tonic.html
http://www.math.utah.edu/~pa/m.....hedra.html
http://www.enchantedlearning.c.....ry/solids/
http://www.geom.uiuc.edu/docs/.....ode56.html
Tetrahedron
http://amath.colorado.edu/alum.....tetra.html
Hexahedron
http://amath.colorado.edu/alum...../cube.html
Octahedron
http://amath.colorado.edu/alum...../octa.html
Icosahedron
http://amath.colorado.edu/alum...../icos.html
Dodecahedron
http://amath.colorado.edu/alum.....dodec.html
What are prisms?
http://amath.colorado.edu/alum.....risms.html
What are antiprisms?
http://amath.colorado.edu/alum.....risms.html
What are Archimedean polyhedra?
http://www.uwgb.edu/dutchs/symmetry/archpol.htm
http://mathworld.wolfram.com/ArchimedeanSolid.html
What are the Johnson solids?
http://mathworld.wolfram.com/JohnsonSolid.html
Paper models of polyhedra
http://www.korthalsaltes.com/
http://plus.maths.org/issue27/.....index.html
Images of polyhedra
http://www.georgehart.com/virt.....ra/vp.html
http://www.physics.orst.edu/~bulatov/polyhedra/
Animations of polyhedra?
http://www.atractor.pt/mat/Polied/poliedrose.htm
A polyhedra applet
http://www.toonz.com/personal/.....pplet.html
Packing of polyhedra
http://www.queenhill.demon.co......f/five.htm
Building your own polyhedra
http://astronomy.swin.edu.au/~...../polycuts/
Models of polyhedra
http://phobos.spaceports.com/~adoskey/polyhedra/
Lesson on polyhedra
http://www.thirteen.org/edonli.....aproc.html
Polyhedra activities
http://torina.fe.unilj.si/~iz.....hedra.html
GAMES
http://www.beaconlearningcente.....efault.htm
http://www.coe.tamu.edu/~strad.....gonLesson/
http://www.mathcats.com/explore/polygons.html
Last edited by adedios on Sat Jan 27, 2007 3:12 pm; edited 1 time in total 

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adedios SuperPoster
Joined: 06 Jul 2005 Posts: 5060 Location: Angel C. de Dios

Posted: Sun May 28, 2006 12:37 pm Post subject: The art of paper folding 


The art of paper folding, like the art of papermaking, began in China, but it spread to Japan by the sixth century A.D. Over the centuries, it became an integral part of Japanese culture. Paper butterflies symbolized the bride and groom at weddings; folding a thousand paper cranes became a traditional way to ensure a long and healthy life. Animals were an especially popular theme, as "folders" developed more and more ingenious ways to make a little diamond of folded paper sprout legs or wings—and even to make it float, hop, or fly. Even so, Japanese origami evolved very slowly. According to American origami expert Peter Engel, Japanese folders invented about 150 traditional origami figures in a millennium of folding.
The full article:
http://www.exploratorium.edu/e.....index.html 

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adedios SuperPoster
Joined: 06 Jul 2005 Posts: 5060 Location: Angel C. de Dios

Posted: Fri Oct 20, 2006 10:27 pm Post subject: Swirling Seas, Crystal Balls 


Week of Oct. 21, 2006; Vol. 170, No. 17 , p. 266
Swirling Seas, Crystal Balls
Spirals of triangles crinkle into intricate structures
Ivars Peterson
A field of triangles crumples and twists into a wavy crystalline sea. A crystal ball sprouts spiraling, labyrinthine passages. Faceted bricks stack snugly into a tidy, compact structure. Underlying each of these objects is a remarkable geometric shape made up of a sequence of triangles—a spiral polygon that resembles a seahorse's tail.
For the full article:
http://sciencenews.org/articles/20061021/bob11.asp 

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adedios SuperPoster
Joined: 06 Jul 2005 Posts: 5060 Location: Angel C. de Dios

Posted: Sat Jan 13, 2007 7:52 am Post subject: Art of the Tetrahedron, Revisited 


Art of the Tetrahedron, Revisited
Ivars Peterson
13 January 2007
The tetrahedron is the simplest of all polyhedra. Any four points in space that are not all on the same plane mark the corners of four triangles. The triangles in turn are the faces of a tetrahedron.
For more than 30 years, Arthur Silverman of New Orleans has created artworks arising out of explorations of this angular form. Many of his sculptures are on display in public spaces and various buildings in New Orleans and other cities from Florida to California.
For the full article:
http://sciencenews.org/article.....thtrek.asp 

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adedios SuperPoster
Joined: 06 Jul 2005 Posts: 5060 Location: Angel C. de Dios

Posted: Sun Apr 08, 2007 7:40 am Post subject: Can't Knock It Down 


Week of April 7, 2007; Vol. 171, No. 14
Can't Knock It Down
Julie J. Rehmeyer
The "Comeback Kid" is a wooden toy with an intriguing property: No matter which way you set it down—on its head, for example, or on its side—it turns itself upright. Two factors account for this: the object's shape, and the fact that the bottom of the toy is heavier than the top.
Give mathematicians such a toy, and they're liable to turn it into a math problem.
Mathematicians Gábor Domokos of the Budapest Institute of Technology and Economics and Péter Várkonyi of Princeton University wondered if they could make an improved version that wouldn't require the weight at the bottom to right itself. Could the shape of the object alone be enough to pull it upright?
For the full article:
http://sciencenews.org/article.....thtrek.asp 

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adedios SuperPoster
Joined: 06 Jul 2005 Posts: 5060 Location: Angel C. de Dios

Posted: Sat May 26, 2007 2:07 pm Post subject: Covering New Ground with Polygons 


Week of May 26, 2007; Vol. 171, No. 21
Covering New Ground with Polygons
Julie J. Rehmeyer
Grab a pen and draw a figure. Follow a few rules: keep your lines straight, don't pick up your pen, don't cross the lines, and finish at the spot where you starUted. You'll have a polygon.
Polygons are among the simplest mathematical objects in existence. Even so, they hold mysteries. Here's one: What is the polygon with the largest area that has n sides and fixed diameter?
Mathematicians still don't know. Michael Mossinghoff of Davidson (N.C.) College has made some recent advances on the question, however. In January, he presented them at the 2007 Joint Mathematics Meetings in New Orleans. Mathematicians had already tackled the problem for some types of polygons, but Mossinghoff broke new ground for polygons with an even number of sides numbering 10 or more.
For the full article:
http://sciencenews.org/article.....thtrek.asp 

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