Transcript A typical vendor of tiles…

Geometry and the Imagination
http://math.dartmouth.edu/~doyle/docs/gi/gi.pdf
Yana Mohanty
Palomar College
[email protected]
A point of view
• Geometry is a way of analyzing and classifying
objects around us.
• Examples:
– Wallpaper or floor tile patterns
– Knots
– Surfaces of 3-dimensional objects
• Change perceptions of what math is and feelings
about math.
• Mathophobes and mathophiles can coexist…in one
classroom!
History of the course
• Title taken from Geometry and the Imagination by
Hilbert and Cohn Vossen.
• Developed at the Geometry Center at the University
of Minnesota by John H. Conway, Peter Doyle, Jane
Gilman and Bill Thurston.
• Peter Doyle taught it at UCSD, Dartmouth College to
classes consisting of math majors and non-math
majors.
• I taught parts of it to students in grades 5-12 at the
San Diego Math Circle
Course materials and methods of evaluation
• Texts:
– The Shape of Space by Jeff Weeks
– The Knot Book by Colin Adams
– Course notes
http://math.dartmouth.edu/~doyle/docs/gi/gi.pdf
• Grading:
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Homework problems from the above
Short, low pressure quizzes testing students on the basics
Journal assignments
Projects
• vary in extent depending on student level
• ideas can be found the course notes
A possible topic:
classify all wallpaper or tile patterns.
Example: brick wall
2*22
2 mirrors cross
mirrors
Gyration of order 2
Rough idea : find smallest repeating piece and described how
such pieces are glued together
A vendor of tiles…
Specialists in the Installation, Restoration and Supply of Decorative Victorian Tiles and Period Mosaics
Pattern 1
4
Detail of pattern 1
mirrors
2
*442
4
Pattern 2
Detail of pattern 2
3*3
zz
order 3 gyration point
z
z
z
z
z
Identical points where 3 mirrors meet
17 patterns possible
Another topic: study of knots
Associate a polynomial to a knot in a clever way:
Polynomial of this
Polynomial of this
and this
and this
must be 1
must be equal
Example of what you can get out of this
Polynomial of this
Polynomial of this
is
x 4  x12  x16
is x 4  x 12  x 16
Moral: you can’t make the green knot into the pink one without tearing it
open and regluing!
Features of this topic
• Practice manipulating polynomials and rational functions
• Associate above with tangible objects
• Introduce a question that has stumped some of the smartest
people in the world:
Suppose the polynomial of something like this
is 1. Does that mean it is not tangled?
• Exercise in logic: untangled implies polynomial is 1
polynomial is 1 does NOT imply
that knot is untangled
Surface: the “skin” of a 3-D object
without any breaks
Examples of surfaces
Classifying surfaces
Terminology:
VERTEX
EDGE
FACE
We are also permitted to have…
“bi-gon”
VERTEX
EDGE
FACE
Define the Euler Characteristic, c
c=V-E+F
Where
V = number of vertices
E = number of edges
F = number of faces
Leohard Euler, 1700’s
Examples
Euler numbers of some surfaces.
SURFACE
tetrahedron
icosahedron
box
soccer ball
torus triangle
hoberman sphere
starfruit
V
E
F
Euler number=V-E+F
What the Euler number can tell you
• As long as your surface is not too strange (does not
include Mobius bands) the Euler number tells you
exactly what it is!
• The Euler number is insensitive to any deformations
(as long as there are no tears) or the cell
decomposition you choose.