1. One of the Most Charming Topics in Geometry
Polyhedra
Olivia Sandoval & Ping-Hsiu Lee
Rice University
Math Leadership Institute
June 28, 2007

2. Goal
To develop a deeper understanding of polyhedra and be able to apply the knowledge into the classroom setting.

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

4. 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.

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

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

7. 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.

8. 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.

9. 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

10. The Answer is Yes

11. 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

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

13. 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

14. 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

15. 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.

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

17. 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

18. 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….)

19. 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.

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

21. 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

22. 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

23. 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.

24. References

25. 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)

26. References http://home.btconnect.com/shapemakingclub/

27. 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.

28. 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

29. Stellating the Dodecahedron