There are, I think, basically two 'models' or 'methods' of contemplating the Holy Trinity. One is based on the nature of the self, the other on the nature of human relationships.
The first was famously championed by Augustine. Working on the principle that man is made in the Divine Image and Likeness, and that man has capax dei, the capacity for God, he saw a triune in the faculties of the mind: intellect, memory, will.
He (he was not the first in this, but most prolific) explores the phenomena of self-realisation in the sense of the knower, the known, and the knowledge, and again in love: the lover, the loved, and the love they share.
The problem with these analogies is that they do not account for the New Testament data, in which the persons of the Trinity are actual centers of consciousness, entering into various transactions with one another: the Father sends the Son, the Son prays to the Father, the Father answers the prayers of the Son, the Father and Son together send the Spirit.
The psychological models sail close to Sabellianism, that the Trinity is three modes of the One Person, not three distinct Persons in One — Wil's analogy, unfortunately, falls into the same error, it's a progression, three different orders of things, not a unity ... nice try, Wil, it's a really tricky one.
The second model, taken from interpersonal relationships, has been called 'social trinitarianism'. The Cappadocian Fathers (Basil the Great, of Caesarea; Gregory Nazianzen, the Theologian, and Gregory of Nyssa) came at the problem the other way.
Augustine begins with the unity of God and tries to find pluralities within the unity; the Cappadocians, some three hundred years earlier, started from the three persons and sought to describe various kinds of unity among them.
Still toady, there is some stand-off between Latin and Orthodox. The Orthodox see the Latins as veering towards Monarchianism (One God in three aspects, and indeed Latin Christianity walks along this cliff edge), the Latins see the Orthodox veering towards Tritheism and, even Arianism (they, too, walk their own cliff ... I rather think Latin and Greek walk side by side, conscious of the risk to their neighbour, more than the risk to themselves.)
Vestigia Trinitatis are the marks of God’s trinitarian character found in the creation. If all of creation reflects God’s invisible nature, his power and glory, is there any way in which creation reflects the Trinity as such?
This way, we can have a field day, thinking up triunes until the cows come home, but not all are correct, and even the 'good' models need qualification, because no model is precisely like the Trinity. Some analogies come close, some analogies miss by a mile, and I see in pagan triunes, for example, not the blueprint on which Christians fabricated their doctrine – a risible notion, really – but rather that in the pagan faiths there is some truth, they are not completely without their graces.
So as long as we remember 'the Trinity is like ...' (a cataphatic statement), 'but being like does not mean the Trinity is ... (balanced by the apophatic).
One of my favourites is Satcitānanda (Sanskrit: सच्चिदानन्द) "Being, Consciousness, Bliss", as a subjective model of Brahman.
Whilst the sublime or beatific vision of the boundless, of pure consciousness, of pure love is a glimpse of ultimate reality, it is not it. Bliss, for example, is really a by-product, a side-effect, it's not 'it' at all. (In a sense, the more bliss you feel, the greater the gulf between knower and known.)
The scholars (I think) delighted in discussing such points. Do the elect in heaven experience beatitude, when they know that not all men have been brought to salvation? Our Lord seems to say the joy of heaven will not be complete, until the last soul is saved.
When it comes to the Trinity, language simply falls short — how can be otherwise? Thus the very terms we use sew the seeds of error.
Westerners say God had 'one substance' (substantia), but is not made of stuff;
Three Persons (personae), but 'person' derives from the Etruscan notion of 'masks', so we're into Modalism or Sabellianism ...
Easterners say God has 'one being' (ousia) and 'three substances' (hypostaseis), which sounds, as said above, tritheistic or Arian.
Not to mention that bloomin' iota!
There is the story attributed to Justin, that he was pondering the Trinity, walking along the sea shore when he came across a child, filling a bucket from the ocean and pouring it into a hole in the sand. Long story short, the kid says 'I'll drain all the world's oceans into this hole before you figure out the Mystery of the Holy Trinity' ...
