Eye of Sauron

[Inziladun]

Originally Posted by Galadriel

I do, however, also feel that it is important to note that it is one eye and not two. It seems as if it is a singular vision that can only see one path of life, and makes no exceptions for anything or anyone.

That's an interesting idea. Also, as a device, one Eye works better than two, and could also have simply been another reminder to his troops that Sauron was a singular being, as a god-king unique in the world.

Originally Posted by Galadriel

What I do not understand, however, is why it is a flaming eyeball. It seems to me to have little relevance, but then again I haven't given it much thought.

I don't think it actually was "flaming"; that's more a Peter Jackson visualisation.
When Frodo saw the Eye in the Mirror of Galadriel it was:

.....rimmed with fire, but was itself glazed, yellow as a cats', watchful and intent, and the black slit of its pupil opened on a pit, a window into nothing.

FOTR

However, the association with flame might have its origin with Sauron's apparent fascination with Mt. Doom. Or, alternatively, there's a quote from the UT essay on the Istari that I think interesting.

....for [Gandalf] was the enemy of Sauron, opposing the fire that devours and wastes with the fire that kindles, and succours in wanhope and distress.

Sauron had become a servant of that first type of fire, and possibly the fire symbolism was also shown by the Eye being red in its depictions.