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One of the Most Charming Topics in Geometry

One of the Most Charming Topics in Geometry. Polyhedra Olivia Sandoval & Ping-Hsiu Lee Rice University Math Leadership Institute June 28, 2007. Goal. To develop a deeper understanding of polyhedra and be able to apply the knowledge into the classroom setting.

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One of the Most Charming Topics in Geometry

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  1. One of the Most Charming Topics in Geometry Polyhedra Olivia Sandoval & Ping-Hsiu Lee Rice University Math Leadership Institute June 28, 2007

  2. Goal To develop a deeper understanding of polyhedra and be able to apply the knowledge into the classroom setting.

  3. Important Things to Know About Polyhedra • Elements of Polyhedra • Platonic solids • Regularity • Archimedean Polyhedra • Kepler Poinsot Solids • Dual Solids

  4. Basic Concepts • A polygon is a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments. • A vertex of polyhedron is a point at which three of more edges meet. • An edge is a joining line segment between two vertices of a polygon. • A face of polyhedron is a polygon that serves as one side of a polyhedron. • A polyhedron is a geometric object with flat faces and straight edges.

  5. Platonic Solids Plato related them to the fundamental components that made up the world: Tetrahedron Fire Cube Earth Octahedron Air Dodecahedron Universe Icosahedron Water

  6. The Only Five Regular Solids Euler’s Rule

  7. What Have We Observed from the Platonic Solids ? Angles Vertices (points) Edges ( line segments) Faces ( polygons) Regularity All the corresponding elements( vertices, edges, angles and faces )must be congruent.

  8. What Have We Learned about Regularity from the Platonic Solids? No other figure, besides the said five figures, can be constructed by equilateral and equiangular figures equal to one another. ( a proposition have been appended by Euclid possibly in Book XI of the Elements) • The faces must be equal. (congruent). • The faces must be regular polygons.

  9. Why are There Only Five Regular Polyhedra? In order to form a solid, the sum of the interior angles where the edges meet at a vertex has to be less than 360 degrees. Are there any more regular polyhedron

  10. The Answer is Yes

  11. A Theorem to Define the Regularity of Polyhedron Let P be a convex polyhedronwhose faces are congruent regular polygons. Then the following statements about P are equivalent: • The vertices of P all lie on sphere • All the dihedral angles of P are equal • All the vertex figures are regular polygons • All the solid angles are congruent • All the vertices are surround by the same number of faces

  12. Archimedean Solids… • Archimedes said he found 13 polyhedra which can be made from a combination of polygons.

  13. Archimedean Solids Euler’s Rule

  14. Archimedean Solids Euler’s Rule

  15. The Kepler-Poinsot Solids In the Kepler-Poinsot group there are 4 shapes, these shapes were discovered by Kepler was a German mathematician and astronomer and Poinsot was a French mathematician and physicist . The Kepler-Poinsot solids are stellations of a couple of the Platonic Solids.

  16. Kepler-Poinsot Solids Euler’s Rule

  17. Platonic Solids & Archimedean Solids Platonic Solids & Archimedean Solids are convex Polyhedron. A famous formula of Euler's Let P be a convex polyhedron with V vertices, E edges, and F faces. then V - E + F = 2. Platonic Solids Archimedean Solids1 Archimedean Solids2 Kepler-Poinsot Solids

  18. Questions Which Need to be Addressed… Are there generalization that apply to all? Does face shape matter? (Polyhedra available to build new model) Can regular polyhedra be made with other regular polygons? ( square, pentagons, hexagons….)

  19. Dual Solids Duality is the process of creating one solid from another. There are connections between these two solids. The faces of one correspond to the vertices of the other. The images of dual solids of Platonic solids are shown above.

  20. Dual Platonic Solids Platonic Solids Dual Tetrahedron Tetrahedron Hexahedron Octahedron Octahedron Hexahedron Dodecahedron Icosahedron Icosahedron Dodecahedron

  21. Archimedean Duals (Catalan Solids)

  22. Archimedean Duals (Catalan Solids)

  23. Introducing Polyhedra to the Classroom • Activity: Hands-on paper folding. • Manipulatives: Transition from paper folding to manipulatives of the Platonic solids and discover the geometric relationships among the solids. • History: Show students the powerpoint presentation of the historic background of the Polyhedra. • Assessement: Students will produce a portfolio to demonstrate their understanding of Polyhedra.

  24. References

  25. References Polyhedra by Peter R. Cromwell (Paperback - Nov 15, 1999) Mathematical Models by H. M. Cundy and A. P. Rollett (Paperback - Jul 1997) Paper Square Geometry :The Mathematics of Origami by Michelle Youngs and Tamsen Lomeli (Paperback - Dec 15, 2000) Investigating Mathematics Using Polydron by Caroline Rosenbloom & Silvana Simone (Paperback - Dec 15, 1998) The Heart of Mathematics: An invitation to effective thinking by Edward B. Burger and Michael Starbird (Hardcover - Aug 18, 2004) Unfolding Mathematics with Unit Origami by Betsy Franco (Paperback - Dec 15, 1999)

  26. References http://home.btconnect.com/shapemakingclub/ http://math.rice.edu/~pcmi/sphere/gos6.html http://mathworld.wolfram.com/DualPolyhedron.html http://en.wikipedia.org/wiki/Platonic_solid http://agutie.homestead.com/files/solid/platonic_solid_1.htm http://www-history.mcs.st-and.ac.uk/~john/geometry/Lectures/L10.html http://www.halexandria.org/dward099.htm http://www.friesian.com/elements.htm

  27. Dihedral Angles in Polyhedra Every polyhedron, regular and non-regular, convex and concave, has a dihedral angle at every edge. A dihedral angle (also called the face angle) is the internal angle at which two adjacent faces meet. An angle of zero degrees means the face normal vectors are anti-parallel and the faces overlap each other (Implying part of a degenerate polyhedron). An angle of 180 degrees means the faces are parallel. An angle greater than 180 exists on concave portions of a polyhedron. Every dihedral angle in an edge transitive polyhedron has the same value. This includes the 5 Platonic solids, the 4 Kepler-Poinsot solids, the two quasiregular solids, and two quasiregular dual solids.

  28. Stellating the Dodecahedron Stand the dodecahedron on one face and imagine projecting the other faces down on to the plane of that face. Each will meet it in a line. The lines will join at the points A, B, C, D. The diagram in the plane is called the stellation diagram. If you project the faces from the plane they meet at E, forming a pentagonal pyramid standing on the face. In this way you can form a new polyhedron from the original one. Alternatively you can select areas of the stellation diagram to form the faces of the new polyhedron. The diagrams below show which areas to select to make the polyhedra shown in the row beneath them. Original

  29. Stellating the Dodecahedron

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