|The Trinity: the Father represented by the eye in the dome;|
The Son by the crucifix directly below;
the Spirit by the dove affixed to the front of the canopy in between and proceeding toward the congregation.
Now none of this addresses the substance/person issue, and since every person we meet is an individual substance it is difficult to grasp the logical possibility that it might not be so for every substance. (A substance, or ousia, is a whole thing, not the modern chemical notion. Thus, TOF is a substance, with a substantial form.) Logically, you cannot have one person composed of three substances; but it might be possible for three persons to be a single substance.
|TOF the trinity|
It's only an analogy, and so incomplete, but if we up the whole thing by a notch, it provides a handle of sorts.
A more intellective approach is given by Aquinas (as you knew it would be). God, considered as Prime Mover, is the source of being of all powers, including the intellect and will of rational beings. Since one cannot give what he does not have, there must be something in God that is analogous to intellect and will. God both Knows and Desires (and is therefore something analogous to a rational being). The proper object of the intellect is the True, and of the will, the Good. In particular, God knows himself and, since God is supremely good, He also desires himself. That gives us two and only two predicates as regards rational being.
As the Subject of both Predicates, we call God the Father.
The product of the intellect is a conception and this conception is expressed in words. As the Object of Knowing, we call God the Word, the only-conceived and by analogy the Son.
The product of the will is a desire for the Good and this desire is expressed by the will proceeding out from the lover to the beloved. As the Object of the Desiring, we call God the Spirit and say that he "proceeds" from the Father.
(This is why the Son is "conceived" and the Spirit "proceeds.")
Now since God is simple and not compounded of parts, as Aquinas demonstrated earlier in his treatise, and since he is knowing himself and desiring himself these three must be all one self. (A second analogy: the term persona originally referred to the masks worn by actors, and so the Persons can be thought of as 'the masks of God.' Again, imperfect and potentially misleading.) Since God is purely actual, as Aquinas showed earlier, there is no potentiality in God and so none of the three lack anything possessed by the other two. (That would make then three distinct beings, not simply three distinct persons, or hypostases.) But while they are the same thing (identitas secundum rem), they are not the same in thing and concept (identitas secundum rem et rationem). And that is as deep into the mystery as we can drill.
In particular, since the Father and Son are one, the Spirit logically proceeds from the Father and the Son. But since the Eastern Church was battling a particular heresy (Sabellianism, by which the Son was conceived by the Father but the Spirit proceeded from the Son), she rejected this formulation. One sometimes hears that the Greek church started with the Persons and tried to figure out the One whereas the Latins started with the One and tried to figure out the Persons. In addition, the pagan philosopher Plotinus also concluded that God had three hypostases, the One, the Intellect, and the Soul. These are discussed here, here, and here.
For Aquinas' arguments, see Summa theologica, here, #27-43
or in more detail, Summa contra gentiles, here, #1-#26
and in precis form in the Compendium theologica, here, #28 et seq.