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- Grades from the fourth exam are available on WebCT.
- Observing for the rest of the semester, until December 9th, will count for the fourth exam.

- Expansion of the Universe
- Geometry
- Non-Euclidean geometry
- Geometry of the Universe

distance = velocity time velocity = 0.5 cm/s

time = distance / velocity = 3 cm/(0.5 cm/s) = 6 s

Velocity =

= 500 km/s

= 0.508 Mpc/Gyr

When were galaxies in the same place?

time = distance / velocity = 7 Mpc/(0.508 Mpc/Gyr)

= 13.8 Gyr ago

Time = distance/velocity

= d/H0d

= 1/H0

= 1/(71 km/s/Mpc)

= 13.8 Gyr

- most of the galaxies approaching her.
- the same Hubble’s law that we see.
- about equal numbers of red and blue shifted galaxies.
- everything rushing away from a point near the Milky Way galaxy.

- Only two directions: up/down and left/right
: north/south and east/west

- All motion of particles, light confined to two dimensions
- Examples: black board, piece of paper, surface of sphere, surface of donut, surface of saddle

- How are the diameter and circumference of a circle related?
- What is the sum of all of the angles in a triangle?

- Circumference = 2 radius
= 3.1415926… 2 = 6.28…

- The sum of the angles in a triangle is 180º

- How about in a curved space?
- Do demo 8C10.50

- Circumference < 2 radius
- The sum of the angles in a triangle > 180º

- Finite
- Unbounded
- No center

How do the properties of positively curved space differ from flat space?

- Circumference > 2 radius
- The sum of the angles in a triangle < 180º

- Suppose it was discovered that the Hubble constant is twice the value that it is currently thought to be, how would this change our estimate of the age of the Universe?
- Compare a flat universe with a positively curved universe as to whether they are finite, bounded, and have a center.