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Polydrons (Beauty of Three Dimensional Polyhedra Workshop)

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This is the portion of my workshop on Polydrons (given at MathFest Washington DC 8/7/2015).
There are additional, associated slideshares.

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Polydrons (Beauty of Three Dimensional Polyhedra Workshop)

  1. 1. Building with Polydrons® with an Introduction to Platonic and Archimedean Solids Jim Olsen, Western Illinois University JR-Olsen@wiu.edu Homepage: http://faculty.wiu.edu/JR-Olsen/wiu/ Main webpage: http://wp.me/P6mrPm-E Main Prezi: http://bit.ly/1Uh8ePp Polydrons® (#3 of 5)
  2. 2. Polydrons® Polydrons* are 2D shapes (triangles, squares, pentagons, etc.) made from plastic. All pieces join together by a snap-action joint which allows pieces to hinge through 260°. One can construct a very wide range of 2D patterns (including nets and tessellations) and 3D polyhedra. * “Polydron” is the name given by the company. “Polydron” is not a mathematical term. They are available from most companies that sell math manipulatives.
  3. 3. Polydrons®
  4. 4. Polydrons® Introduction to Platonic and Archimedean Solids A Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. There are 5 Platonic solids. An Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices. There are 13 Archimedean solids. (15 if you count the left- and right-hand version of the snub cuboctahedron and the snub icosidodecahedron)
  5. 5. Polydrons ® We will make the Cubeoctahedron The cuboctahedron is a rectified* cube and also a rectified octahedron. *To rectify means to truncate ‘all the way’ to the midpoint. Can you see it? Need:  Eight triangles and six squares
  6. 6. Polydrons® Key Quantities for Polyhedra Certainly V, E, and F are important (number of vertices, edges and faces) and the associated Descartes-Euler polyhedral formula: 𝑉 − 𝐸 + 𝐹 = 2 Two other useful quantities: Vertex Degree – I like to use K. Number of sides on a face – I like to use N for the Platonics. For the Archimedeans, N1, N2, etc., for the various face types.
  7. 7. Other (perhaps more intuitive) Formulas for Polyhedra Given1: A polyhedron is made up of 20 triangles. Question1: How many edges are there in the polyhedron? Generalize. Given2: A polyhedron is made up of 12 pentagons and 3 faces meet at each vertex. Question2: How many vertices are there in the polyhedron? Generalize. (These can be extended to the Archimedean solids.)
  8. 8. Polydrons® The Archimedean Solids from the Platonic Solids for polyhedra See the (interactive) Prezi http://bit.ly/1Uh8ePp The Prezi shows the 13 Archimedean solids coming from the 5 Platonic solids: 5 by truncation 2 by rectification (truncating all the way) 2 by expansion (pulling each edge apart to make a square) 2 by snubification (pulling edges apart but making triangles) 2 by truncating the 2 rectified solids. (There are other ways to form the Archimedeans.)
  9. 9. Polydrons® Comment The Cuboctahedron and the Icosadodecahedron have extra symmetry: Every edge is the same (always get the same two faces meeting there). Every edge is on a Great Circle.
  10. 10. Polydrons® The Icosadodecahedron - Every edge is on a Great Circle. The Hoberman Sphere (pictures next slide)
  11. 11. The Hoberman Sphere
  12. 12. Polydrons® Comment Wikipedia has many nice graphics for the Platonic and Archimedean solids and a nice table of information. https://en.wikipedia.org/wiki/Archimedean_solid
  13. 13. Polydrons® (Page with links) Jim Olsen, Western Illinois University JR-Olsen@wiu.edu Homepage: http://faculty.wiu.edu/JR-Olsen/wiu/ Main webpage: http://wp.me/P6mrPm-E Main Prezi: http://bit.ly/1Uh8ePp (#3 of 5)

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