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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. - PowerPoint PPT Presentation
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One of the Most Charming Topics in Geometry
PolyhedraOlivia Sandoval & Ping-Hsiu Lee
Rice UniversityMath 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 SolidsFaces Edges Vertices
Tetrahedron 4 6 4
Hexahedron (Cube)
6 12 8
Octahedron 8 12 6
Dodecahedron 12 30 20
Icosahedron 20 30 12
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
Faces Edges Vertices
Truncated tetrahedron 8 18 12
Cub-octahedron 14 24 12
Truncated octahedron 14 30 28
Truncated cube 14 36 24
Rhomb-cub-octahedron 26 48 24
Great rhomb-cub-octahedron
26 72 48
Euler’s Rule
Archimedean SolidsFaces Edges Vertices
Icosi-dodecahedron 32 60 30
Snub-cube 38 60 24
Truncated Dodecahedron 32 90 60
Truncated Icosahedron 32 90 60
Rhombicosidodecahedron 62 120 60
Truncated Icosidodecahedron 62 180 120
Snub Dodecahedron 92 150 60
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 12
Great Stellated Dodecahedron
12 30 20
Great Dodecahedron 12 30 12
Great Icosahedron 20 30 12
Euler’s Rule
Platonic Solids & Archimedean Solids
Platonic Solids & Archimedean Solids are convex Polyhedron.A famous formula of Euler'sLet P be a convex polyhedron with V vertices, Eedges, 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 Polydronby 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.htmlhttp://mathworld.wolfram.com/DualPolyhedron.htmlhttp://en.wikipedia.org/wiki/Platonic_solidhttp://agutie.homestead.com/files/solid/platonic_solid_1.htmhttp://www-history.mcs.st-and.ac.uk/~john/geometry/Lectures/L10.htmlhttp://www.halexandria.org/dward099.htmhttp://www.friesian.com/elements.htm
Dihedral Angles in PolyhedraEvery polyhedron, regular and non-regular, convex and concave, has adihedral 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 degreesmeans the faces are parallel. An angle greater than 180 exists onconcave portions of a polyhedron. Every dihedral angle in an edgetransitive polyhedron has the same value. This includes the 5 Platonicsolids, the 4 Kepler-Poinsot solids, the two quasiregular solids, and twoquasiregular dual solids.
Stellating the Dodecahedron
Stand the dodecahedron on one face and imagine projecting theother faces down on to the plane of that face. Each will meet it in aline. 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 apentagonal pyramid standing on the face. In this way you can forma new polyhedron from the original one.Alternatively you can select areas of the stellation diagram to formthe faces of the new polyhedron.The diagrams below show which areas to select to make thepolyhedra shown in the row beneath them.Original
Stellating the Dodecahedron