The use of analogy is fundamental for human thought and language, and in particular for theology. Derived from the Greek analogia "proportion" (ana- "upon, according to" + logos "meaning" or "word"), it refers to the way we compare one thing with another on the basis of some likeness or similarity. It is more complicated than a simile, which happens when I straightforwardly compare one thing to another ("God is like a light"). It is also more complicated than a metaphor, which is when in poetic language I simply assume the similarity in the way I describe something ("God is a light for my eyes and a path for my feet").
An analogy is built out of similes and metaphors - it extends them not just to things but to relationships between things. If a simile is like a ratio (A : B), an analogy is a ratio of ratios (A:B : C:D, or "the relationship of A to B is like the relationship of C to D"). So to form an analogy we might say, for example, "Clay is to the potter as the world is to God". But "analogy" is also used more generally to cover all the ways we compare things that are similar to each other in some respect but not others. Everything we say about God relies on metaphor and analogy, because the words we use necessarily come from the things we can see and touch.*
So God-talk has to be taken with a pinch of salt. When we talk about God we mustn't take ourselves too literally. There is an analogy here with the problem of "graven images", or the temptation to mistake the image for what it represents. But what if God talks about God? In Jesus, we believe, God spoke as a man. Just as the Incarnation gave a justification for icons, so it gave a justification for saying certain things about God and believing them to be true. Philosophy and mysticism were possible before Christ - but now there is also theology. The things in the world were always "like" God in certain ways, not just as signs of his presence and activity but as expressions of his nature, or natural symbols of him. But now they can also be sacraments and sacramentals.
All of this perhaps serves as background to the use in my book of geometrical and mathematical "analogies" to the Trinity. Thomas Aquinas was perfectly clear on the fact that the Trinity cannot be proved by anything in nature - nevertheless, once we know by revelation that God is triune, we can see traces or impressions of the Trinity everywhere. So, for example, all things (1) subsist, (2) have a definite form, and (3) are ordered to an end (echoing Father, Son and Spirit). Following Simone Weil, I wanted to show that fundamental numbers and shapes also "echo" the Trinity in this way. There is no attempted proof here, just an intellectual intuition or an aid to contemplation, but the point is that knowledge of the Trinity enables us to appreciate the beauty of creation by seeing in its ordered harmony a meaning that we could not know before. Mathematics, in its own way (and you won't hear this said too often!), is a picture of love.
* There is much talk in theology about the "analogy of being", or attempts to compare the existence of God and that of the world. For an interesting discussion of that topic go here.
The photograph is by Tom Bree