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.

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

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.

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

The Only Five Regular Solids Faces Edges Vertices Tetrahedron 4 6 Hexahedron (Cube) 12 8 Octahedron Dodecahedron 30 20 Icosahedron Euler’s Rule

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.

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.

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

The Answer is Yes

A Theorem to Define the Regularity of Polyhedron Let P be a convex polyhedron whose 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

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

Archimedean Solids 8 18 12 14 24 30 28 36 26 48 72 Faces Edges Vertices Truncated tetrahedron 8 18 12 Cub-octahedron 14 24 Truncated octahedron 30 28 Truncated cube 36 Rhomb-cub-octahedron 26 48 Great rhomb-cub-octahedron 72 Euler’s Rule

Archimedean Solids Icosi-dodecahedron 32 60 30 Snub-cube 38 24 Faces Edges Vertices Icosi-dodecahedron 32 60 30 Snub-cube 38 24 Truncated Dodecahedron 90 Truncated Icosahedron Rhombicosidodecahedron 62 120 Truncated Icosidodecahedron 180 Snub Dodecahedron 92 150 Euler’s Rule

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.

Kepler-Poinsot Solids Name: Faces Edges Vertices Small Stellated Dodecahedron 12 30 Great Stellated Dodecahedron 20 Great Dodecahedron Great Icosahedron Euler’s Rule

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

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….)

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.

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

Archimedean Duals (Catalan Solids) Name: Dual Triakis Tetrahedron Truncated Tetrahedron Triakis Octahedron Truncated Cube Tetrakis Hexahedron Truncated Octahedron Trapezoidal Icositetrahedron Rhombicuboctahedron Triakis Icosahedron Truncated Dodecahedron Trapezoidal Hexecontahedron Rhombicosidodecahedron Rhombic Tricontahedron Icosidodecahedron

Archimedean Duals (Catalan Solids) Name: Dual Rhombic Dodecahedron Cuboctahedorn Pentakis Dodecahedron Truncated Icosahedron Pentagonal Icositetrahedorn Snub Cube Pentagonal Hexecontahedron Snub Dodecahedron Hexakis Octahedron Truncated Cuboctahedron Hexakis Icosahedron Truncated Icosidodecahedron

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.

References

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)

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

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.

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

Stellating the Dodecahedron