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

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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.

Polydrons®

Polydrons®Introduction to Platonic and Archimedean SolidsA 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)

Polydrons ®We will make the CubeoctahedronThe 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

Polydrons®

Key Quantities for PolyhedraCertainly V, E, and F are important (number of vertices, edges and faces) and the associated Descartes-Euler polyhedral formula:

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.

Other (perhaps more intuitive) Formulas for PolyhedraGiven1: 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.)

Polydrons®The Archimedean Solids from the Platonic Solidsfor polyhedraSee the (interactive) Prezi http://bit.ly/1Uh8ePp The Prezi shows the 13 Archimedean solids coming from the 5 Platonic solids:5 by truncation2 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.)

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.

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The Icosadodecahedron - Every edge is on a Great Circle.

The Hoberman Sphere (pictures next slide)

The Hoberman Sphere

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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

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

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