When it comes to analogies of the Trinity, we must take care to remember that anything we can think of or encounter will not be exhaustively sufficient to explain the triune nature of God. That’s why they’re called analogies. If any analogy were to describe the Trinity perfectly, it would cease to be an analogy and it would become the Trinity itself. Yet, this does not mean we should not endeavor to understand who God is and how He functions. Thus we strive to find better ways in which our finite minds might grasp the infinite complexities of God.
There are many analogies that float around that attempt to describe certain aspects of the Trinity. The example of the egg is often used. The shell is the Father, the egg white is the Son, and the yolk is the Holy Spirit. There is also the example of the 3 physical forms of material, such as water. Ice, water and steam are all water, yet they serve different purposes. These are great examples for many reasons, and they do wonders when speaking with certain audiences. However, they are but analogies, and as such they each contain notable flaws. The three parts of an egg do indeed make up the totality of an egg. An egg is not a complete egg without these three parts. In this way it explains the Trinity. However, the egg analogy lacks the critical insight in showing that God is One while being three distinct persons. They are inseparable. You cannot take one person of the Trinity in a vacuum. If you could, we could not say that God is One. If you take a shell away from an egg, you still have the other two parts. It’s three parts are distinct and separable. An egg white is not an egg. Nor can an egg shell claim to constitute and entire egg. However, Jesus is God is the Holy Spirit. They are distinct persons, but inseparable in essence.
In a different way, the analogy of the physical states of water has its own shortcomings. Unlike the egg, water is ice is steam. It is the same substance. It is the same essential stuff. However, they cannot physically coexist in the same environment over time. The state of water, as we well know, necessarily depends on the external ambient temperature in which it exists. Yes, it is possible to have ice and water and steam present in a general vicinity at the same time, but those three phases of water will each necessarily have at least one different property than the other two. Assuming Earth’s atmospheric pressure, we know that the ice must be below 0 degree centigrade, the water must be between 0 and 100 C, and the steam must be over 100 C. If this were not the case, they would not be ice, water or steam respectively. It temperature is a defining characteristic of each phase of water. This means that at 40 degrees centigrade, only water can ultimately exist. Yet the Trinity coexists as One God at all times in all situations. God is not compartmentalized by space, temperature, time or any other modes of measurement. He is infinite, and that is the problem that most every analogy cannot overcome.
One way to circumvent the problem of the infinite is to consider something that is not finite. This brings us to the spatial dimension analogy that has been discussed elsewhere. Mathematics and logic are special in that they are not necessarily governed by the existing universe. It is my own personal opinion (one that I would be glad to back up with a plethora of arguments. ) that the physical universe is not infinite in any of its contents or dimensions. However, in a very real sense mathematics and logic do not abide by the rules of this universe. Rather, the universe abides by the laws of logic. For example, imagine that there are exactly 1 trillion “things” in existence, be they quarks, protons, donuts or elephants. We have 1 trillion fundamental things and not a unit more. If that were true, could we still do that mathematical problem 1 trillion + 1 trillion = 2 trillion? Of course we could! But no one would know what 2 trillion things looks like! It exists only in our mind. Similarly the laws of logic are transcendent. The statement “truth exists” must necessarily be true at all times at all places. We know this because the statement “truth does not exist,” if true, would be necessarily false. Thus truth must exist in and outside of any and all dimensions. Logic is transcendent. (This is a great argument for an atheistic friend who wants to use logic to refute the existence of a transcendent being of any kind. It’s a self-refuting position.)
This transcendence is extremely helpful for the human mind to grow in its understanding of its creator. Though I can make a strong argument against the physical existence of the infinite, I would just as vehemently assert that we can dwell upon the concept of the infinite. We can imagine the infinite and often use it in mathematical modeling. (side note: this is another wonderful brain teaser to set before a skeptical friend, especially if they are of the engineering/science/math makeup. They can easily do a limit problem by conceiving the idea of the infinite in their mind. Yet, ask them if they can think of a color that they have never seen! No one will be able to do it! They cannot think of something which they have never experienced or known. So why is it, then, that we can so readily think of and dwell upon the infinite? GREAT rhetorical question! It clearly seems to imply that the human mind has experienced the infinite in some form or fashion) Whereas the other analogies failed in that they were not infinite in nature, a purely theoretical, easily understood, mathematical model would not have to suffer from such a deficiency.
This brings us to the spatial dimension analogy of the Trinity. The first dimension is but a straight, infinite like. There is no up or down, nor is there any forward or back. There is just side to side. Yet this one line must exist for all other things to possibly exist. We cannot even imagine a world in which a first dimension did not exist. It is unfathomable, and thus the first dimension is foundational and necessary for all other spatial entities to exist. In a very real sense, it is the Father of all physical space.
Once we add the second dimension to the first, we begin to see things as an infinite plane. In mathematics, this is usually described by the infinite x and y axis that make up the Cartesian plane. Both the x-axis and the y-axis are single dimensions. They are both infinite lines that are essentially the same. But the second single dimension allows one to see the world around them. We see everything in two dimensions. It is the way we were designed. Even 3D images are seen by our brain as 2 dimensional images. We may interpret them differently, but their appearance to us is always 2 dimensional. It was Jesus who said to Phillip, “Don’t you know me, Philip, even after I have been among you such a long time? Anyone who has seen me has seen the Father. How can you say, ‘Show us the Father’?” (John 14:9). We see God in the Son.
Lastly, we can add the infinite z-axis that penetrates our 2 dimensional Cartesian plane. This addition of another infinite one dimensional line, essentially the same as the first two, allows us to experience things such as depth. What was only a square that we could see on a plane before can now become a cube that we can lift and touch. Indeed, each of exists in three dimensional space. This is the spatial dimension in which we will live out and experience our entire lives. We cannot be constrained to 1 or 2, and we cannot fathom a fourth. Yet, the Scriptures tell us clearly that the Holy Spirit lives within us as believers (1 Corinthians 2:10). We should not confuse our theology at this point and claim that the Holy Spirit is confined to spatial dimensions. That would infer that God is bound by his own physical creation, a logical absurdity. Rather, we are noting that we experience life in its totality in the third dimension, and that includes how we experience the indwelling of the Holy Spirit. We do not simply see Him or acknowledge Him as being foundational. We experience Him.
Thus we see that we have an infinite analogy in which three equivalent things work together in an inseparable way, each to serve its own purpose. Remove one, and space ceases to exist. Yet the only thing that differentiates the x, y and z axis are the letters that we arbitrarily labeled them with. God is Jesus is the Holy Spirit. Essentially the same while serving different, distinct purposes.
Lastly, this analogy also proves useful in that it is eternal. Time is the fourth dimension that governs the physical universe, but it is by its very nature independent of three dimensional space. A unit cube will be the same unit cube yesterday, today and tomorrow. It will never change, or at least not until the universe ceases to be. But at that time, we won’t need an analogy to explain the Trinity. We will see for ourselves.
(We would do well to remember that this is still just an analogy, and it will never be more than that. God is not confined to spatial dimensions. He created space, and thus He cannot be confined within it. However, for the purpsoses stated, the model serves its purposes, and is still one of the best analogies that I have yet come accross when dwelling on the Trinity.) Perhaps God wanted us to see this analogy from the very beginning, as the Psalmist writes:
The heavens declare the glory of God;
the skies proclaim the work of his hands. – Psalm 19:1