Transcript Section 3.1 Beyond Numbers What Does Infinity Mean?

Section 5.1
Rubber Sheet Geometry
Discovering the Topological Idea of Equivalence by Distortion.
“The whole of mathematics is nothing more than a
refinement of everyday thinking.”
Albert Einstein
Question of the Day
Is Earth a ball or a donut?
Equivalence by Distortion
Two objects are equivalent by distortion if
we can stretch, shrinking, bend, or twist
one, without cutting or gluing, and deform
in into the other.
What is a Torus?
A torus is the boundary of a doughnut.
Is a torus a sphere?
Why or why not?
Can you prove they are, or are not,
equivalent by distortion?
Section 5.2
The Band That Wouldn’t Stop Playing
Experimenting with the Mobius Band and Klein Bottle
Make guesses!
Question of the Day
Take a strip of paper and tape the short
ends together to make a loop. How many
pieces do you get if you cut the loop down
the middle?
The Mobius Band.
How do you make a mobius band?
How many sides does a mobius
band have?
Trace along the center of the band with a
pencil. What do you notice?
How many edges does a mobius
band have?
Trace along the edge of the band with a
pencil. What do you notice?
Other mobius band explorations!
Cut lengthwise down the center core of the
band. What do you see?
Other mobius band explorations!
• Make another mobius band and cut by
staying close (about 1/3 of the way) to the
right edge. What do you see?
The Klein Bottle
The Klein Bottle is a one sided surface.
How is a Klein Bottle made?
Section 5.3
Circuit Training
From the Konigsberg Bridge Challenge to Graphs.
Simplify whenever possible.
Question of the Day
What is the Konigsberg Bridge
Challenge?
Is it possible to walk a path in such a way that each bridge is crossed
only once?
Euler’s Circuit Theorem
A connected graph has an Euler circuit if
and only if every vertex appears an even
number of times as an end of an edge in
the list of edges.
Map Coloring
What is the minimum number of colors that always
suffice to color any potential world map?
Section 5.4
Feeling Edgy?
Exploring Relationships Among Vertices, Edges, and Faces
Insight into difficult challenges often
comes by first looking at easy cases.
Question of the Day
Can I read into your psyche?
The Euler Characteristic
Theorem
For any connected graph in the plane,
V – E + F = 2,
where V is the number of vertices, E is the
number of edges, and F is the number of
regions.
Going back to the Five Platonic Solids…
Number of
Vertices
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Number of
Edges
Number of
Faces
V-E+F
Five Platonic Solids
There are only five regular solids.
Question: Could there be a regular solid
that we have not thought of?
Section 5.5
Knots and Links
Untangling Ropes and Rings
Experiment to discover new insights.
Question of the Day
When is a knot not a knot?
The Gordian Knot
Knots you may know…
Links and Chains
Chain – An object that is constructed from
some number of closed loops that may be
knotted either individually or about one
another.
Link – a collection of loops.
The Linking Challenge
Is it possible to link three rings together in
such a way that they are indeed linked yet
if we remove any one of the rings, the
other two remaining rings become
unlinked?
The Borromean Rings
Do any of these look familiar? 
Section 5.6
Fixed Points, Hot Loops, and Rainy Days
How the Certainty of Fixed Points
Implies Certain Weather Phenomena
Act locally, think globally.
Question of the Day
What is the temperature on the other side of
the world?
The Brouwer Fixed Point Theorem
Suppose two disks of the same size, one red and one blue, are initially
placed so that the red disk is exactly on top of the blue disk.
If the red disk is stretched, shrunken, rotated, folded, or distored in any
way without cutting and then placed back on top of the blue disk in
such a manner that it does not hang off the blue disk, then there
must be at least one point on the red disk that is fixed.
That is, there must be at least one point on the red disk that is in the
exact same position as it was when the red disk was originally on
top of the blue one.
The Meteorology Theorem
At every instant, there are two diametrically
opposite places on Earth with identical
temperatures and identical barometric
pressures.
The Hot Loop Theorem
If we have a circle of variably heated wire,
then there is a pair of opposite points at
which the temperatures are exactly the
same.