One morning a few days ago, while I was still half awake, I found myself imagining a discussion of the Most Holy Trinity with a (rather rude) skeptic, and I thought of an analogy in terms of self-similar fractals, in particular the Mandelbrot set.
Notice that you see the same pattern at the very beginning (at 0:08) and three more times in this video (at 1:15, 2:29, and 5:03). Note there are other repetitions: we zoom past the same pattern repeated many times before diving into it at 0:46, only to see yet more copies of it on the inside immediately thereafter. At 1:27, we are once again zooming past more copies of this same pattern. There is also a 4-armed pattern that we see several times, at 1:01, 1:38, 2:52, 3:11, and 3:48. Each of these patterns is contained within the others and also contains the others, so whenever you see one, you know the others are also there.
My analogy is that these patterns are "like" the Divine Persons (the Father, the Son, and the Holy Ghost); the patterns are never really separated from each other, as the Divine Persons are not separated from each other. Yet these patterns together form a unity, as there is only one God in three Persons. Also, presumably the first pattern would correspond by analogy to God the Father, because although the other two Persons are co-eternal with Him, the Father is in some sense their origin, as He is neither begotten nor proceeding.
There are two other nice things about this analogy. Firstly, the Mandelbrot set is beautiful, which is one of the reasons videos like the one above are produced. Likewise, God is beautiful, or if you prefer, God is the ultimate perfection of beauty. Secondly, although the Mandelbrot set seems infinitely complicated in such a video, in fact it is actually quite simple in its definition. This might make philosophical statements about the simplicity of God a bit easier to accept.
Please understand that this is only an analogy, and all analogies have their limitations. A much more careful explanation of what the Church actually teaches about the Trinity can be found in the Athanasian Creed.