The illustrations must not be taken as perfect and all-revealing. Each has weaknesses. But they can still help.
1.. Is there a relational illustration of the Trinity?
Nothing is more relatable to us humans than love.
Augustine (354-430 AD) comes up with this deep analogy: the lover, the beloved, and love.
At the baptism of Jesus, the Father says that he loves the Son, and at that very moment, the Spirit descends and rests on Jesus (Matt. 3:16-17). Augustine seems to say that the Father loves the Son, who receives his love and returns it, for example, in his willingness to obey the Father and die on the cross, while the Spirit communicates the love between them.
This image of a triad of love expresses how the Father, the Son, and the Holy Spirit existed before the creation of time and the universe, and how the three persons will forever exist. It is into this love that the Trinity welcomes all believers and all who have received the love of God, through Christ and by the power of the Holy Spirit. (The Trinity Books VIII.14; IX.2, and XV.10)
The verses “God is love” (1 John 4:8 and 16) now take on a much, much deeper meaning.
In contrast, the strict unity of God seen in Islam or Judaism, for example, does not make sense in terms of the love of God, because divine love amounts to self-love. Inadequate and deficient.
Back to Augustine. His analogy is profound, because we have all experienced love from a lover, and we have returned the same love. But it still does not adequately explain how the one God shares this divine attribute of love and all the other divine attributes in three persons. Our human love is but a poor reflection of divine love.
2.. What about an image to illustrate the Triunity?
Grudem produces this one (p. 255):
Note how the three persons Father (F), Son (S), and Holy Spirit (HS) are located in one circle—God. The dotted lines merely indicate their personal relationship, not a division in God.
Here is an ancient representation of the Trinity (Moody p. 207):
Note that God and the three persons are within the triangle, which symbolizes their unity. God is in the center, which symbolizes their one essence. It says the Father is not the Son, and the Son is not the Spirit, and the Spirit is not the Father, and so on. But all three persons are one God in the center. The image would have been better if God was the triangle itself because the three persons are within God, but the teaching still communicates some truths,
Geisler suggests a simple triangle, a simultaneous threeness in its oneness (p. 551).
4.. Would God really exist in a complicated way like this?
This Q&A is repeated in several posts on the Trinity.
God is knowable as far as he has revealed himself through Jesus Christ, the Scriptures, the Holy Spirit who lives in us, and other avenues like the creation. However, God in his pure essence is unknowable. No one has seen God in all of his fullness, except Jesus (John 1:18). We humans down here on earth are limited by our five senses and our finite minds, so how can we figure out and calculate the pure nature of God?
C.. S. Lewis alludes to this in his book Mere Christianity, in the chapter “The Invasion” which discusses the Incarnation (God the Son becoming man), which can also apply to the Trinity. He writes:
It is no good asking for a simple religion. After all, real things are not simple. They look simple, but they are not. The table I am sitting at looks simple: but ask a scientist to tell you what it is really made of—all about atoms and how the light waves rebound from them and hit my eye and what they do to the optic nerve and what it does to my brain—and, of course, you find that what we call “seeing a table” lands in mysteries and complications which you can hardly get to the end of.
If we keep our knowledge of him simple, then we are not bothered by the deeper doctrines. But once we probe more deeply, we discover that God is ultimately a mystery in his pure essence.
I simply have to accept Scripture as it is, even when I don’t understand all of it.
I trust God in the teachings I do understand (e.g. God loves me and saved me), and that leads me to trust him in the ones that I don’t understand completely (the Trinity).
It is a blessing, rather, that I can experience the fullness of the God the Son and God the Spirit in all their deity. I can worship and pray to all three members of the Trinity and have the fullest experience with God. Wonderful!
Click on the Trinity: What Do Theologians Say? for their main point that the Trinity is ultimately a mystery.
So how does knowing about the Trinity help me know God better?
There is an entire ten-point post that answers that question, here:
Written by James Malcolm
ARTICLES IN THE TRIUNITY SERIES
The Trinity: What Are Some Illustrations?
Augustine, The Trinity, trans. Edmund Hill ((New City, 1991).
C. S. Lewis, Mere Christianity (Touchstone, orig. 1943 and then 1980 and 1996).