Transcript Download Shape of Space PowerPoint (Computer Session)

How many sides to a piece
of paper?
Or,
Is the Universe infinite or
finite?
The Strange World of Topology
David Brown
Mathematics Faculty
Ithaca College
Infinite vs. Finite
Ultra Deep Space
• Small portion of space past
the handle of the Big Dipper
• Each galaxy – 100 billion stars
• This patch contains 100 billion galaxies
Moral: Universe is BIG!!!
BUT, is it infinite?
Infinite vs. Finite
The Edge of the Universe
Medieval art depicting the edge of the universe
Infinite vs. Finite
Boundary versus Infinite
Background
fluctuations are
faint - space is
homogenuous
Evidence that space
may likely be finite
The Wilkinson Microwave Anisotropy Probe (WMAP) map of
cosmic microwave background temperatures.
Topology
Study of Shape
Which shapes are intrinsically different
How shapes transform into other shapes
Topology - Examples
We use examples of topological equivalence
in everyday life.
The world really isn’t flat, but our maps are!
Topology - Examples
Google maps - understand the
interconnections between the different roads
and destinations
Topology - Examples
Planning - We use topological networks in our
planning, to show important connections.
Topology - Mobius
A piece of paper has two sides. If I make it
into a cylinder, it still has two sides, an
inside and an outside.
Topology - Mobius
Cut a rectangle 2 cm wide, but give it a twist
before you join the ends.
You have made a Mobius band!
Topology - Mobius
Draw a line along the center of your cylinder
parallel to one of its edges.
Now do the same on your Mobius band
How many sides to your cylinder? How
many boundaries (edges)?
How many sides to your Mobius band? How
many boundaries?
Flat Models of Surfaces - Gluing
Cylinder
Mobius
Flat Models of Surfaces
Torus – surface of donut
Flat Models of Surfaces
What is a significant difference between the
Mobius band and Torus?
No boundary on the torus!
What do you see if light travels on the
around on the torus?
Flat Models of Surfaces
Topology & Universe
Topology Games
– Torus, what does the fish see?
– Klein bottle, what does the fish see?
– Tic-Tac-Toe
– 3D analog – Cubes and the shape of
space
Play Games and Have fun Discovering Other Universes!
Topology & Universe
Topology & Universe
Topology & Universe
Latest possible model for a finite, boundless universe.
Topology & Universe
Resources
– This Powerpoint:
http://www.ithaca.edu/hs/depts/math/do
cs/mathday/mathday14/
– Great YouTube Video:
https://www.youtube.com/watch?v=5u7h
FQy9Mt0