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.