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(Math) Polygons and Polyhedra: Papercraft Polyhedra

 
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adedios
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PostPosted: Mon Apr 24, 2006 8:19 am    Post subject: (Math) Polygons and Polyhedra: Papercraft Polyhedra Reply with quote






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, 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://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):396-397. 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.....ty-14.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/poliedros-e.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.uni-lj.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
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PostPosted: Sun May 28, 2006 12:37 pm    Post subject: The art of paper folding Reply with quote

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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PostPosted: Fri Oct 20, 2006 10:27 pm    Post subject: Swirling Seas, Crystal Balls Reply with quote

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
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PostPosted: Sat Jan 13, 2007 7:52 am    Post subject: Art of the Tetrahedron, Revisited Reply with quote

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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PostPosted: Sun Apr 08, 2007 7:40 am    Post subject: Can't Knock It Down Reply with quote

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

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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