Great Discussion guys!
Before I say anything I want to remind us that God is God... He is infinite. He is Omniscient. He is Omni-present. He is Omnipotent. He is a being that exists and functions outside the confines of what we know and understand and so in so many ways He is mysterious too. I say all this first of all to help us get in the right frame of mind; that in all of our attempts to describe and understand this amazing being we will never fully grasp and understand God in His totality (not in this life and world at least).
So, our attempts to understand this Triune God will be feeble at best. We will at some point have to accept that the mystery is part of what makes Him this great God and we are not. How do we truly understand One God composed of three persons... Weird and mind boggling.
Yes the Egg gives us insight to 3 distinct parts making up 1 egg. Each with their own qualities and yet each still egg. Different roles and functions. One part is no more egg than the other and missing one of the parts makes the egg less.
Another, but more limited analogy, is water. Water is manifest in 3 forms... liquid, gas, and solid... all still water, but three different manifestations of one substance.
It may help to understand the trinity through the earthly representation of God... People! Mankind is created in the likeness and image of God. God is a Triune God. Mankind is a triune being too. Man is composed of Spirit, Soul, and Body (1Thessalonians 5).
Each part has a different function and the truth is each part is eternal. Yes the body will die, but every human beings body will be resurrected and rejoined with their soul and spirit.
You can check out this subject further in a study I did on it for Associated Content. http://www.associatedcontent.com/article/446538/doctrine_of_trichotomy_of_man_a_biblical.html?cat=34
Fact is we see the triune God referenced all throughout the Bible. We see the Spirit working, the Father working, and the Son working (sometimes all at the same time, like in the baptism of Jesus).
I've heard many different thoughts on it, like the Father is the Architect, the Son is the construction manager, and the Spirit is the builder. I don't know that I like that, but people are always coming up with ways to try to understand this awesome triune God.
I hope this helps.