In Perfect Shape

In Perfect Shape

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In Perfect Shape

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1. In Perfect Shape The Platonic Solids

2. Going Greek?

3. The Greeks were very fond of symmetry • Art • Architecture • MATH!!!

4. The most symmetric polygons are the regular ones Polygons with all sides and all angles congruent

5. But to do this…we gotta know the facts…

6. Fact of Geometry: There are only a few regular polyhedra that exist! (Contrast to regular polygons which can have any number of sides.)

7. Five different types of polyhedra

8. Tetrahedron: 4 faces (triangles)

9. Hexahedron: 6 faces (squares)

10. Octahedron: 8 faces (triangles)

11. Dodecahedron: 12 faces (pentagons)

12. Icosahedron: 20 faces (triangles)

13. Hey Mike… I’m puzzled….why are there only five regular polyhedra?

14. Don’t be a pinhead…IT’S SIMPLE!!!!

15. Think about it this way • A point or “peak” is formed by at least three polygonal faces that meet at any vertex of the polyhedron • Since the polyhedron is regular, the situation at any vertex is the same as at any other. • To make a peak, the sum of all the face angles at the vertex must be less that 360 degrees. If they add up to 360 degrees, they would make a flat surface. • Since all the faces are congruent, the angle sum at a vertex must be divided up equally among them.

16. Earth, Air, Water, and Fire Earth (Hexahedron) Air (Octahedron) Water (Icosahedron) Fire (Tetrahedron)

17. The Fifth Element

18. If interested in playing… • http://www.csd.uwo.ca/~morey/archimedean.html

19. Uses/Occurrences Dice: Crystal structures in nature. In meteorology and climatology, global numerical models of atmospheric flow are of increasing interest which employ grids that are based on an icosahedron (refined by triangulation) instead of the more commonly used longitude/latitude grid. This has the advantage of evenly distributed spatial resolution without singularities (i.e. the poles) at the expense of somewhat greater numerical difficulty. (Wikipedia)

20. Archimedean Solids Truncated Platonic Solids http://home.comcast.net/~tpgettys/trplato.html The others http://home.comcast.net/~tpgettys/archimed.html

21. Time Line • ~ 400 BCE The Greeks: Plato and Platonic Solids • ~250 BCE Archimedes and Archimedean Solids • ~1400-1500’s AD Renaissance: Rediscovery of Archimedean Solids • ~1600 AD Kepler and Planetary Motion • Today: Games, molecular structure, modeling

22. Works Cited • Appel, Rudiger. 3Quarks-GIF Animations-Platonic Solids. 21 May 2000. 21 Nov. 2005 <http://www.3quarks.com/GIF-Animations/PlatonicSolids/>. • Berlinghoff, William, and Fernando Gouvea. Math Through The Ages: A Gentle History for Teachers and Others. Farmington: Oxen House Publishers, 2002. • Google Image Search. 2005. Google. 21 Nov. 2005 <http://www.google.com/imghp?hl=en&tab=wi&client=firefox-a&rls=org.mozilla:en-US:official_s&q=>. • Greaves, David. "What do viruses look like?." My Virion Home Page. 1 Jul 1997. 21 Nov. 2005 <http://www.path.ox.ac.uk/dg/vstructure.html>. • O'Connor, Aidrian. "Musings On Sacred Geometry." Nature's Word. 2003. 21 Nov. 2005 <http://www.unitone.org/naturesword/sacred_geometry/platonics/introduction/>. • "The Fifth Element Pictures." The MovieWeb Movie VAult. MovieWeb. 21 Nov. 2005 <http://movieweb.com/movies/galleries.php?film=1227&id=622>.