A couple of months back, Ross Cameron offered a really nice example in favor of relative identity. I want to adapt Cameron's story, and use it as a putative analogy for the Trinity. So, Bob decides to make a non-representational statue out of clay, the statue being in the shape of a dodecahedron. Sara then makes a statue of Bob's statue. She makes her statue out of a lump of clay, and is so true to her subject that her statue is intrinsically just like Bob's. But of course her statue differs in representational properties from Bob's: Bob's statue is non-representational while Sara's statue represents Bob's statue.
Jacob decides to get all the fame, and notes that Bob and Sara's statues look exactly the same so one can save some clay. So he takes a single lump of clay and simultaneously makes three statues, each being made out of the whole lump: Statue1 is a beautiful non-representational dodecahedron, Statue2 is a statue of Statue1 and Statue3 is a statue of what Statue1 and Statue2 have in common. Physically, what we have is one dodecahedral lump of clay. But there are three statues that this lump of clay is: it is a non-representational dodecahedral statue, a representational statue of the non-representational statue, and a more abstract representational statue of what the first two statues have in common. They differ precisely in representational properties (and in the aesthetic properties that depend on these).
This analogy seems to work moderately well as an illustration of the Trinity. It really does seem that Statue1 is the whole lump, that Statue2 is also the whole lump, and that so is Statue3, but that they are nonetheless the same statue. What's kind of neat about this example is that the distinctions between the three statues are grounded precisely in the mutual relations in which they stand—in this case, the representational relations. It is even the case that in some sense these relations are origination relations and yet they do not imply any temporal priority, again just as in the case of the Trinity.
On my own view of material constitution, this is an impossible example. For I think there really is at most one object made out of a lump of matter, and artifacts don't exist except in special cases (a hammock could be both an artifact and a snake[note 1], in which case the artifact exists because it is a snake). So on my view there is neither a lump, nor three statues. However, the analogy is still a useful one. For it analogizes the Trinity to what Jacob's work of art would be on metaphysical theories that support the above analysis. And one can do that even if one rejects these metaphysical theories, as long as one thinks that they are not flagrantly self-contradictory. (Moreover, I am partial to the view that something like relative identity holds in the case of God but only the case of God, for Thomistic reasons that are not, I think, ad hoc.)