1. The Shape of the Universe
Topology, Geometry, and Curvature

2. The “Big BANG”
About 14 billion years ago, the Universe was very hot and very dense
Since then, it has been expanding and cooling (the Big BANG does not mean an explosion)
After about 4 hundred thousand years, the Universe cooled enough for photons (light) to be released
The Universe was lit up!

3. The Cosmic Microwave Background (CMB)
Discovered by radio-astronomers, Wilson and Penzias
Relic of the Big Bang
These microwaves have been traveling through the Universe since light was first released almost 14 billion years ago
10% of TV “snow” is CMB

4. CMB showing temperature variations

5. CMB Today
Cooled to almost uniform 2.7 Kelvin (about -455 degrees Fahrenheit)
Interest in CMB lies in its variations in temperature of one ten-thousandth of a degree
Indicates differences in density of early Universe
Could hold the key to how galaxies were formed
Denser regions indicate cooler temperatures where galaxies eventually formed

6. Last Scattering Surface (LSS)
Theoretical spherical surface centered at the Earth where the CMB originates
Before light was released, photons were scattered by the free ions, electrons and protons, hence the name LSS
The radius of the LSS is estimated at 47 billion light years

7. Topology of the Universe
Discovery of the CMB stimulated interest in the shape of the Universe
Is it flat, spherical, donut shaped, saddled shaped?
And, is it finite or infinite?
If it’s finite, how big?
If it’s infinite, well

8. Topology Topology is the study of shapes of objects or spaces
Two spaces are equivalent if one can be deformed into the other without breaking or tearing
Distance is not important to topology
Donut = Coffee Cup

9. Donut (Solid Torus) & Cup

10. CMB and Topology Imagine a finite 2-dimensional Universe
It could be spherical, or shaped like the surface of a donut (torus), or the surface of a donut with several holes

11. 2 Dimensional Sphere (2-sphere) and 2 Dimensional Torus (2-torus)

12. Mathematician’s and Cosmologist’s Flat Torus: glue together or identify opposite sides of the rectangle

13. A 2-torus cannot be constructed in the Euclidean plane
If you were a 2-dimensional person in a 2-dimensional Universe, you could not see a complete torus!

14. Torus Games developed by Jeff Weeks, Freelance Mathematician
Open Internet Explorer
Go to
Download “Torus Games”
Open zipped file and click icon
Extract files
Click tic tac toe icon
Click on the window that opens
Play pool on a flat torus

15. 2-Torus in Euclidean 3-Space
Introduces curvature

16. Measuring Angles on a Torus
Draw two large triangles on the tube provided, one half way around, the other halfway around on the other side
Bring the ends of the tubes together so that one triangle is on the outside and the other on the inside
Secure the ends with duct tape
You now have a solid torus!
Measure the sum of the angles in each triangle

17. Curvature of Space in Two Dimensions
Positive: sum of angles in triangle is greater than 180 degrees (outside of a torus, sphere)
Zero: sum of angles in triangle equals 180 degrees (flat torus, Euclidean plane)
Negative: sum of angles in triangle is less than 180 degrees (inside of a torus, saddle, or hyperbolic space)

18. Sphere: Positive Curvature

19. Hyperbolic Plane: Negative Curvature

20. the way that mathematicians like to view the 2-torus in the plane: Tiling

21. Building a 3-Torus Start with a cube
Glue together opposite faces of the cube, analogous to the construction of a 2-torus

22. CMB and the Shape of the Universe: Circles in the Sky
If the Universe is finite and shaped like a torus, then we might be able to see the last scattering surface on the faces of the cube
Think of the LSS as a balloon on the inside of the cube
As the balloon expands it will press against the faces of the cube
We could see this as matched circles on opposite faces of the cube
Look to the east, see a circle; look to the west, see the same circle
Circles match by pointwise temperature readings

23. Last Scattering Surface in a Box

24. A 3-torus cannot be built in Euclidean 3-space, so we can view it by tiling

25. Flying a Spaceship through a Tiled 3-Torus Universe
Go to (credit to Jeff Weeks)
Download Curved Spaces
Open zipped file
Click icon
Extract all files
Open folder, click icon
Choose 3-Torus
Click on the window, fly your spaceship!
You can change orientation or direction by holding down either control or shift or both
Open Internet Explorer
Tear down or build up walls using left or right arrows

26. Is the Universe Shaped Like a Soccer Ball?
A theory, championed by French astrophysicist, J.P. Luminet
Build the Poincaré Dodecahedral Space
Glue together opposite faces (pentagons) of a dodecahedron (soccer ball)

27. You can also fly your spaceship through a tiled Poincaré Dodecahedral Space using the “Curved Space” folder on the website,