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I have found there is no illustration that does the trinity justice. Examples such as ice, water and steam do not illustrate the Trinty but modalism. Examples such as the alleged use of the shamrock by St Patrick likewise show a defective Trinity.
Part of the Athanasius creed is probably the most simple
So the Father is God;
The Son is God;
And the Holy Spirit is God.
Yet there are not three Gods but one.
Appreciate the replies so far! I like the egg one; I worry a bit that it sort of presents a picture of God as being a composite made up of three parts, which isn't quite right -- though of course, it's an analogy, and isn't supposed to be perfect. Maybe comparing to the Senate, or the two-headed monster, is more concrete -- there are multiple persons but only one composite whole (and as you pointed out, the crucial difference is that the persons of the Trinity never disagree.
I'd quibble with the one about water because water isn't simultaneously all three of its phases, and thus the analogy seems to point more to a modalist conception of God, where there is one God who simply changes forms from one to the other.
Thanks for the great thoughts so far.