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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.
Important Things to Know About
Polyhedra
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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
4
Hexahedron
6
12
8
Octahedron
8
12
6
Dodecahedron
12
30
20
Icosahedron
20
30
12
(Cube)
Euler’s Rule
What Have We Observed from the
Platonic Solids ?
Angles
Edges ( line segments)
Vertices (points)
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:
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–
–
–
–
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-cuboctahedron
26
72
48
Euler’s Rule
Archimedean Solids
Faces
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 KeplerPoinsot 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'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