Download Adventures in Flatland

Adventures in Flatland
Tom McNamara
Department of Mathematics, SWOSU
April 9, 2014
Overview
1
Introduction
2
First Journey
3
Universal Consequences
4
The Second Journey
5
Other Possible Models
6
Mathematical Studies
Definitions – Don’t Get Too Philosophical
The word universe will be used throughout this talk.
When we say universe, we mean the totality of all things that
have physical existence.
Introducing A. Square
About A. Square
Lived in the 2-D Universe, Flatland
Everyone knew Flatland was a plane – Euclidean 2-space
One physicist theorized Flatland might by a“Hyper-Circle”
Everyone in Flatland stayed near Flatsburgh
About A. Square
Lived in the 2-D Universe, Flatland
Everyone knew Flatland was a plane – Euclidean 2-space
One physicist theorized Flatland might by a“Hyper-Circle”
Everyone in Flatland stayed near Flatsburgh
About A. Square
Lived in the 2-D Universe, Flatland
Everyone knew Flatland was a plane – Euclidean 2-space
One physicist theorized Flatland might by a“Hyper-Circle”
Everyone in Flatland stayed near Flatsburgh
About A. Square
Lived in the 2-D Universe, Flatland
Everyone knew Flatland was a plane – Euclidean 2-space
One physicist theorized Flatland might by a“Hyper-Circle”
Everyone in Flatland stayed near Flatsburgh
Overview
1
Introduction
2
First Journey
3
Universal Consequences
4
The Second Journey
5
Other Possible Models
6
Mathematical Studies
The Adventure Begins. . .
A. Square decided to explore his universe
A. Pentagon and A. Hexagon decide to accompany him (so
A. Square doesn’t get himself into trouble)
“We’ll make him turn back after a month.”
Bought all the red thread they could, so they could find
their way back
Leave Flatsburgh, heading West
Much to A. Pentagon and A. Hexagon’s relief, they are back in
Flatsburgh after three weeks, and approached from the East.
Why Did Flatlanders Have Trouble?
Everyone assumed that the group had inadvertently traveled in
a large circular route instead of staying on a straight line.
It didn’t help that Flatlanders used the words “Euclidean
2-space” and “Flatland” interchangeably.
Anyone talking about Flatland having a different configuration
would be stuck having to say “Flatland is not Flatland.”
Overview
1
Introduction
2
First Journey
3
Universal Consequences
4
The Second Journey
5
Other Possible Models
6
Mathematical Studies
Sphere as a Possible Shape for Flatland
The idea of a “Hyper-Circle” – what we call a sphere – was very
hard for Flatlanders to understand.
Try to visualize making a sphere by taking these two disks and
gluing them together along their boundaries.
Flatlanders’ Trouble
Flatlanders can’t understand “picking them up” of the screen.
To them, UP means NORTH. A Flatlander has difficulty with “off
the screen” because the screen is their universe.
Helping the Flatlanders
The idea of identifying boundary points might help these
creatures understand a sphere. Imagine a Flatlander in the red
disk. When he reaches the boundary, he does not reach a
barrier. Rather, he reappears on the boundary of the blue disk
at the corresponding boundary point.
Sympathy for the Flatlanders
As we can see, it is difficult for Flatlanders to visualize the
sphere, S 2 .
To give you some sympathy with our friend, lets try to imagine
S 3 , the 3-sphere.
Imagine two balls in 3-space (spheres with the middle filled in).
Take the boundary of one and glue it to the boundary of the
other. Further, you need to do this without cutting or tearing the
balls you started with.
Overview
1
Introduction
2
First Journey
3
Universal Consequences
4
The Second Journey
5
Other Possible Models
6
Mathematical Studies
A Second Journey
A. Square leaves Flatland, heading North
Marks his path with blue thread
Returns from the South after 2 weeks
Never found the red thread
Both threads were re-traced, and found to be intact
Thus, the Mystery of the Non-Intersecting Threads began.
Consequences for the Shape of Flatland
If Flatland were Euclidean 2-space, or a sphere, the threads
would need to intersect.
Thus, for Flatlanders to understand their universe, they would
need to conceive of some different possibilities.
Mathematically, the possible shapes for Flatland are called
surfaces.
For any mathematicians, we mean compact 2-manifolds without
boundary.
Overview
1
Introduction
2
First Journey
3
Universal Consequences
4
The Second Journey
5
Other Possible Models
6
Mathematical Studies
Help From the 1980s
Tied up by threads
We can resolve the Mystery of the Non-Intersecting Threads by
using a torus (the inner tube). How could the Flatlanders
visualize this?
bc
bc
bc
Glue edges of the
same color together.
bc
More Sympathy for Flatlanders
As before, Flatlanders would have great difficulty visualizing a
torus without using identifications.
To help you see their trouble, try to imagine creating a 3-torus.
Start with this room. Glue the left wall to the right wall. Now
glue the ceiling to the floor.
You have a thickened up inner tube. Now glue the inside to the
outside, and you have T 3 .
Overview
1
Introduction
2
First Journey
3
Universal Consequences
4
The Second Journey
5
Other Possible Models
6
Mathematical Studies
Determining Possible Shapes for Flatland
Figuring out what shapes Flatland could have falls under the
mathematical field of topology.
Locally Flatland is a plane, i.e., Flatland is a 2-manifold
(w/o boundary)
2-Manifolds are often called surfaces
The Surface Classification Theorem is an important result
2-D and 3-D Universes
We have discussed 3 possibilities for the shape of Flatland:
1
Euclidean 2-space
2
S2
3
T 2.
There are analogous possibilities for our universe:
1
Euclidean 3-space
2
S3
3
T 3.
Our Universe
Locally, it behaves like Euclidean 3-Space
We’ve never really traveled far from home, so globally, we
don’t know how the universe looks
Possible “shapes” are 3-manifolds
Thurston’s Geometrization Conjecture says (essentially)
there are 8 possible geometries for 3-manifolds
Reading Ideas
E. A BBOTT, Flatland, Dover, 1992
J. W EEKS, The Shape of Space, CRC Press, 2001
R. RUCKER, Geometry, Relativity and the Fourth
Dimension, Dover, 1977
M. S TARBIRD, A. B ERGER, The Heart of Mathematics,
Springer, 2000