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