Analogies for the Trinity are usually problematic. However, I think it is possible to distinguish two kinds of analogies. The first kind seek to help us to understand the three and the one by comparison with other examples of ‘three-in-oneness’. Examples may be water/ice/gas or mind as composing memory/understanding/well (Augustine). These analogies are usually problematic, because they tend to make us think of the Trinity heretically (e.g. modalistically). I am not going to discuss them any further here. However, there is a second kind of analogy which does not seek to illustrate or explain ‘three-in-oneness’ but seeks to illustrate or explain that it makes sense for the Trinity to escape our full understanding. The doctrine of the Trinity is hard to understand. The first kind of analogy seeks to help us make sense of the doctrine. The second kind seeks to make sense of our lack of understanding. In my view, the second kind is preferable to the first, because the first tends to reduce the Trinity down to our understanding.
What we are talking about here is the limitations of logic. And on this, Robert Letham has some stimulating ideas, using analogies that are more of the second kind than the first. The context is a discussion about the understanding of the Trinity in Eastern Orthodoxy and in the Christianities of Western Europe (Roman Catholicism & Reformed). I have quoted Letham at length, as what he has written speaks for itself. On the basis of his brief discussion in this book, his much lengthier treatment of the Trinity in [source] looks well worth reading.
Towards a resolution of the problems of East and West
Where do we go from here? It is clear that we need simultaneously to preserve both the unity and identity of the one indivisible being of God and, at the same time, the irreducible differences between the three persons. Here Gregory of Nazianzen is brilliantly helpful. In the passage in which Calvin found vast delight he says:
This I give you to share, and to defend all your life, the one Godhead and power, found in the three in unity, and comprising the three separately; not unequal, in substances or natures, neither increased nor diminished by superiorities or inferiorities; in every respect equal, in every respect the same; just as the beauty and the greatness of the heavens is one; the infinite conjunction of three infinite ones, each God when considered in himself; as the Father, so the Son; as the Son so the Holy Spirit; the three one God when contemplated together; each God because consubstantial; one God because of the monarchia. No sooner do I conceive of the one than I am illumined by the splendour of the three; no sooner do I distinguish them than I am carried back to the one. When I think of any of the three I think of him as the whole, and my eyes are filled, and the greater part of what I am thinking escapes me. I cannot grasp the greatness of that one so as to attribute a greater greatness to the rest. When I contemplate the three together, I see but one torch, and cannot divide or measure out the undivided light.
Gregory oscillates back and forth from the one to the three. When he considers the one he is illumined by the splendor of the three. When he distinguishes them he is carried back to the one. Gregory points to the danger of building our doctrine of the trinity on either the one being God in isolation, or on the three persons (or any one of them) in isolation. These dangers are demonstrated thoroughly in the subsequent history of the church.
Gregory’s hermeneutic, as he expresses it in this passage, is strikingly modern. Physicists working at the atomic level oscillate in thought between waves and particles… A parallel to this is the field of gestalt psychology, from which we learn that as we gain a grasp of the whole we tend to lose connection with the parts, while if we focus on the parts we lose our grasp on the whole.
Another way of putting it is to think of what happens when you focus your gaze on a particular object. Try doing it. Notice that when you look intently at this or that, the rest of your field of vision becomes blurred and indistinct. Then if you look away from the object of your former gaze and attend to the background, which now comes into clear focus, your former object of attention becomes a blur.
In this connection, the limitations of logic are apparent. James Loder and the late W. Jim Niedhardt have pointed out…that, while logic is of value in everyday life or in ‘trivialities’ as the call them, when we approach the boundaries of the universe it breaks down. They point to a wide range of areas where creation is not reducible to neat laws of thought. Among other things, physics, mathematics, psychology, and human development all yield this feature… If a reductionist elevation of logic is impermissible in dealing with matters of creation, how much less is it to be followed in relation to the holy trinity? T.F.Torrance insists that the proper course in seeking to know is to submit our minds to the object of knowledge so as to allow it to disclose itself on its own terms. It follows that in science knowledge is to be based on the reality of what is. Logical deductions from premises are good within certain parameters but, if absolutized, can prevent us knowing. In theology, this means we must faithfully submit ourselves to God’s revelation and allow our thoughts to proceed from the basis of who he discloses himself to be, recognizing at the same time that he infinitely transcends the capacities of our minds.
 Letham, Robert (2007) Through Western Eyes: Eastern Orthodoxy: a Reformed Perspective (Fearn, Ross-shire: Mentor). Quotes from pp.239-242
 Of course, as Mark Heath ably demonstrates, some analogies of the Trinity do not seek to help us understand anything but make us wallow in our incomprehension!!
 Letham has argued previously in the chapter that the West has historically begun with the oneness of God, and the East with the Father.
 I have omitted Letham’s development of this point, where he gives the example of the double slit experiment.
 Letham then gives the example of the Mobius strip
 in the foreword to Loder, James E. & Neidhart, W. Jim (1992) The Knight’s Move: The Relational Logic of the Spirit in Theology and Science (Colorado Springs: Helmers & Howard)