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4.5 Platonic Solids

4.5 Platonic Solids. Wednesday, February 25, 2009. Symmetry in 3-D. Sphere – looks the same from any vantage point Other symmetric solids? CONSIDER REGULAR POLYGONS. Start in The Plane. Two-dimensional symmetry Circle is most symmetrical

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4.5 Platonic Solids

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  1. 4.5 Platonic Solids Wednesday, February 25, 2009

  2. Symmetry in 3-D • Sphere – looks the same from any vantage point • Other symmetric solids? • CONSIDER REGULAR POLYGONS

  3. Start in The Plane • Two-dimensional symmetry • Circle is most symmetrical • Regular polygons – most symmetrical with straight sides

  4. 2D to 3D • Planes to solids • Sphere – same from all directions • Platonic solids • Made up of flat sides to be as symmetric as possible • Faces are identical regular polygons • Number of edges coming out of any vertex should be the same for all vertices

  5. Five Platonic Solids • Cube • Most familiar • Tetrahedron • Octahedron • Dodecahedron • Icosahedron

  6. Powerful? • Named after Plato • Euclid wrote about them • Pythagoreans held them in awe

  7. Some Relationships • Faces of cube = Vertices of Octahedron • Vertices of cube = Faces of Octahedron

  8. Duality • Process of creating one solid from another • Faces - - - Vertices

  9. Euler's polyhedron theorem • V + F - E = 2

  10. Archimedean Solids • Allow more than one kind of regular polygon to be used for the faces • 13 Archimedean Solids (semiregular solids) • Seven of the Archimedean solids are derived from the Platonic solids by the process of "truncation", literally cutting off the corners • All are roughly ball-shaped

  11. Truncated Cube

  12. Archimedean Solids

  13. Soccer Ball – 12 pentagons, 20 hexagons

  14. Some Relationships • New F = Old F + Old V • New E = Old E + Old V x number of faces that meet at a vertex • New V = Old V x number of faces that meet at a vertex

  15. Stellating • Stellation is a process that allows us to derive a new polyhedron from an existing one by extending the faces until they re-intersect

  16. Two Dimensions: The Pentagon

  17. Octagon

  18. How Many Stellations? • Triangle and Square • Pentagon and Hexagon • Heptagon and Octagon • N-gon?

  19. Problem of the Day • How can a woman living in New Jersey legally marry 3 men, without ever getting a divorce, be widowed, or becoming legally separated?

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