C.S. Lewis's Cube Analogy

In Chapter 24 (entitled 'The Three-Personal God') of C.S. Lewis's brilliant book, 'Mere Christianity', Lewis attempts to show that it is not irrational to believe in the Trinity by using the following argument:

'You know that in space you can move in three ways - to left or right, backwards or forwards, up or down. Every direction is either one of these three or a compromise between them. They are called the three Dimensions. Now notice this. If you are using only one dimension, you could draw only a straight line. If you are using two; you could draw a figure: say, a square. And a square is made up of four straight lines. Now a step further. If you have three dimensions, you can then build what we call a solid body: say, a cube - a thing like a dice or a lump of sugar. And a cube is made up of six squares.
'Do you see the point? A world of one dimension would be a straight line. In a two-dimensional world, you still get straight lines, but many lines make one figure. In a three-dimensional world, you still get figures but many figures make one solid body. In other words, as you advance to more real and more complicated levels, you do not leave behind you the things you found on the simpler levels: you still have them, but combined in new ways - in ways you could not imagine if you knew only the simpler levels.
'Now the Christian account of God involves just the same principle. The human level is a simple and rather empty level. On the human level one person is one being, and any two persons are two separate beings - just as, in two dimensions (say on a flat sheet of paper) one square is one figure, and any two squares are two separate figures. On the Divine level you still find personalities; but up there you find them combined in new ways which we, who do not live on that level, cannot imagine. In God's dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. But we can get a sort of faint notion of it. And when we do, we are then, for the first time in our lives, getting some positive idea, however faint, of something super-personal - something more than a person. It is something we could never have guessed, and yet, once we have been told, one almost feels one ought to have been able to guess it because it fits in so well with all the things we know already.' (End quote)

I have come across Lewis's cube analogy used as a defense of the doctrine of the Trinity (DOT) many times, including in this talk by well known Christian apologist Ravi Zacharias, and in this YouTube video which purports to explain the DOT 'with reason' by using the cube analogy.
Now if Lewis's point about people in a two dimensional world not being able to grasp the concept of a three-dimensional shape is merely intended to illustrate the impossibility of us mortals ever being able to fully comprehend the infinite God, then I have no argument with him. In that case he is effectively saying that while he believes in and teaches the DOT, he accepts that there is no way the doctrine can ever make sense to the human mind and it must be accepted on faith in spite of its apparent incoherence.
If, however, his depiction of a cube appearing, to the inhabitants of a two-dimensional world, to be six separate squares when it is actually one single cube in the three-dimensional world, is intended as an actual analogy by which to explain the DOT, then it fails dismally. This is because, even in the three dimensional world, none of the six squares that comprise the cube actually is the cube. Since the squares would be analogous to the Divine Persons and the cube would be analogous to the Divine Being, this would mean that each of the Divine Persons is only a part (one third, presumably) of the Divine Being. Consequently, we would have three Persons, each of whom is a part of God but none of whom actually is God. Since the DOT requires that each of the Persons is fully God, C.S. Lewis's cube analogy sadly fails to shed any light on how the doctrine can be understood in a coherent and non-contradictory way.

One God in three presences

Martin Luther once said, "to try to deny the Trinity is to endanger your salvation. To try to comprehend the Trinity is to endanger your sanity." The reformer chose not to risk losing his marbles but some of us Christians are less willing to simply accept such a central tenet of the faith as an incomprehensible mystery.
Over the years I have heard and read many attempts to explain the doctrine of the Trinity in a way that is both coherent and does not stray from the bounds of orthodox Christian theology. I have also spent many hours trying to figure out for myself how to make sense of the idea of three Divine Persons in one Divine Being. None of the explanations that I have heard, read, or attempted to devise, have ever stood up to rigorous scrutiny no matter how subtle, complex or seemingly clever they were or how sophisticated the mental gymnastics involved.
Ultimately, all attempts to explicate the doctrine of the Trinity come up against the following impasse: If each of the Persons is no less than God (and this must be the case, for orthodox theology requires that each is fully God, not partly God or part of God) and each of the Persons is no more than God (and they cannot be, for Christian theology teaches that each of the Persons has existed from the beginning, and in the beginning there was only God - nothing which was not God could yet exist, for God is the creator of all things) and if there is only one God then it follows logically from these premises that there can be absolutely no distinction between the Persons - each just is the other; in fact, there must really be only one Person. Yet Trinitarian theology teaches that each of the Persons is really distinct; though there is only one God, He exists eternally as three Persons, each of whom is not the other two.
It seems clear to me that there is no way around this conundrum and I have come to the conclusion (perhaps conclusion is the wrong word as my views on this subject are constantly under development) that since it is internally incoherent and logically impossible (not merely impossible in the sense of being beyond human experience – that, of course, would not be a problem where God is concerned - but impossible in the sense of being self contradictory; for a thing cannot be both x and not x simultaneously, neither can there be both three of something and only one of the same thing simultaneously, no matter how many mind-bending theories one posits about ‘three whos and one what’, relative identity or different senses of the word 'is') then it must be the case that not all of the tenets of the doctrine are literally true. In the words of Sir Arthur Conan Doyle, "once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth."
It seems, then, that the only reasonable understanding of the doctrine of the Trinity must be a symbolic, figurative, parabolic or poetic (as opposed to literal) one. In other words, it must be the case that either the Persons are not literally distinct from each other; or that none - or only one - of the Persons is literally God; or that God is not One in a literal (mathematical) sense.
Although I contend that the doctrine of the Trinity cannot be true in the literal sense, it is important to remember that just because something is not literally true does not mean that it is not actually true - it can still be 100 percent true but in a sense other than that implied by the straightforward, 'face value' meaning of the words.
My own understanding of the doctrine of the Trinity is that it expresses and symbolises the idea that, though there is only one God, Christians specifically encounter that God in three main ways; firstly, as the omnipotent, omnipresent creator of the universe and spiritual parent of all human beings; secondly, we encounter God in the life and teachings of Jesus Christ, who for all Christians is the ultimate revelation of God's character and disposition towards humanity; thirdly, we meet God in the life of the Church and in the lives of individual believers as they live out their faith, empowered by God's spiritual presence.
Another way of saying this is that God, our Divine Parent, is fully present in and revealed by Jesus (who is, in a special sense, God's Son) through the agency of God's Spirit - which is not merely a blind force but a personal Presence of love and compassion. This view of the Trinity could, perhaps, be described as one God in three presences (rather than Persons).
There are, of course, other ways one could understand the doctrine of the Trinity non-literally but, for me, this is the interpretation that makes the most sense in terms of reflecting both the structure of the Trinitarian formula (albeit in a figurative way) and my actual experience of the Christian faith.