Section 5.1Rubber 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 5.2The Band That Wouldn’t Stop PlayingExperimenting 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 5.3Circuit 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 5.4Feeling 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

Number of Edges

Number of Faces

V-E+F

Tetrahedron

Cube

Octahedron

Dodecahedron

Icosahedron

Five Platonic Solids

There are only five regular solids.

Question: Could there be a regular solid that we have not thought of?

Section 5.5Knots 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 5.6Fixed Points, Hot Loops, and Rainy Days

How the Certainty of Fixed PointsImplies 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 TheoremSuppose 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.