What do "homogeneity" and "isotropy" mean?

Can you please give me the definitions of homogeneity and isotropy in astronomy terms?

This is a good question, because these concepts are a bit subtle. Homogeneous is defined as "the same in all locations" while isotropic means "the same in all directions." Imagine that the whole universe is an infinitely large field with one perfectly symmetrical hill, which you are seated atop. Look around: you see an isotropic universe, since the hill is equally green and equally steep in all directions. But the universe is not homogeneous: it has a hill!

These concepts are important because most modern cosmology is based on the "cosmological principle," the assumption that, on large scales, the universe is both homogeneous and isotropic. Studies of large-scale structure in the universe and analysis of the microwave background radiation help confirm that this assumption is justified.

March 2003, Kate Becker (more by Kate Becker)

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What is the difference between homogeneity and isotropy?

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