Figure 1. This is a flat, two-dimensional universe, with a smooth matter distribution. It extends infinitely in all directions. It contains an infinite amount of matter. This universe is steadily expanding, but maintains a fixed average density by automatic "steady-state" matter creation. This universe meets the modesty (i.e. "we were not intended") requirement of the Cosmological Principle. If "homogeneity" means "isotropy from any location", this universe is homogenous. If "homogeneity" means "smoothness", this universe is homogenous.
	Figure 2. This is a two-dimensional universe that is curved into a three-dimensional sphere, with a finite amount of matter, smoothly distributed. Space itself is expanding by the continuous enlargement of the sphere. This universe meets the CP modesty requirement. If "homogeneity" means "isotropy from any location", this universe is homogenous. If "homogeneity" means "smoothness", this universe is homogenous.
	Figure 3. This is a flat, two-dimensional universe with a finite amount of matter distributed relatively smoothly throughout a disc-shaped area. This area is expanding as the matter spreads out by simple momentum. Beyond this disc-shaped area, there is an infinite, empty void. Given that we observe isotropy from our location, this universe does not meet the CP modesty requirement, because isotropy would be observed only by viewers at or near the center of this universe. (Viewers near the edge of this universe would see no distant matter in half of their sky.) If "homogeneity" means "isotropy from any location", this universe is not homogenous. If "homogeneity" means "smoothness", this universe is homogenous — but only within the disc-shaped area in which all matter resides.
	Figure 4. This is a flat, two-dimensional universe, with a lumpy matter distribution. It extends infinitely in all directions. It contains an infinite amount of matter. This universe is steadily expanding, but maintains a fixed average density by automatic "steady-state" matter creation. This universe meets the CP modesty requirement. If "homogeneity" means "isotropy from any location", this universe is homogenous. If "homogeneity" means "smoothness", this universe is not homogenous.
	Figure 5. This is a two-dimensional universe that is curved into a three-dimensional sphere, with a finite amount of matter, distributed in a lumpy manner. Space itself is expanding by the continuous enlargement of the sphere. This universe meets the CP modesty requirement. If "homogeneity" means "isotropy from any location", this universe is homogenous. If "homogeneity" means "smoothness", this universe is not homogenous.
	Figure 6. This is a flat, two-dimensional universe with a finite amount of matter distributed in a lumpy manner throughout a disc-shaped area. This area is expanding as the matter spreads out by simple momentum. Beyond this disc-shaped area, there is an infinite, empty void. Given that we observe isotropy from our location, this universe does not meet the CP modesty requirement — see Figure 3 comments. If "homogeneity" means "isotropy from any location", this universe is not homogenous. If "homogeneity" means "smoothness", this universe is not homogenous — not even within the disc-shaped area in which all matter resides.

©2004 Darel R. Finley