The Major Heresies
Historically, the Church’s elaborated doctrine about the Trinity developed by the meditation of the saints on the Bible, especially the Gospels and St. Paul. This reflection gave rise to many errors which the Church rejected in the early Councils and in the various decrees of the Successor of St. Peter, the Pope in Rome.
Heresies and errors about the Trinity can be classified in the following way: 1) the unity of the divine nature is denied so that you have some form of tritheism, that is, three gods joined together is some kind of moral unity.
2) Or, a real trinity of persons is denied. This comes in two forms: a) Monarchianism or Modalism: those holding these positions deny a real personal distinction between Father, Son and Holy Spirit. They say that there is only a “rational distinction” between them, or they call them “modes.”
b) Subordinationism: those holding this position deny implicitly or explicitly the true divinity of the Second and Third Persons, who are creatures and are subordinated to the first person who alone is really God.
These errors come from the difficulty of the mystery of the Trinity and from the confusion between nature and person. They are also related to false ideas about God deriving from the doctrines of the Stoics and Neo-Platonists about the aeons and the emanations from God. The Gnostics tried to adapt these ideas to Christian dogma. And of course, errors made here about the inner nature of God affect one’s view of who Jesus Christ is and so have an impact on Christology.
Monarchianism takes two forms: a) Adoptionist Monarchianism which says that Christ was a mere man, but was adopted by God as his Son at his Baptism in the Jordan; b) Modalist Monarchianism which says that there is only one Person in God who manifests himself variously as Father, Son and Holy Spirit. So these names, for them, all refer to the one Person in God — they are synonyms for the one Person in God. A chief proponent of this was Sabellius about 220 A.D. Thus, it is often called Sabellianism.
Subordinationism accepts three Persons in God, but denies the consubstantiality of the Son and the Holy Spirit with the Father, and therefore their true divinity. They are exalted creatures produced by the First Person. One view says that the Father created the Son, and the Son created the Holy Spirit. The main proponents of this view were the priest Arius and his followers in the 4th century; the Council of Nicaea in 325 A.D., the first ecumenical council, was called to combat this error. Those who denied the divinity of the Holy Spirit in the second half of the 4th century were called Macedonians after Bishop Macedonius who was deposed in 360.
Tritheism means that one holds that there are three gods. Those who were accused of this were Roscelin of Compiegne (+ about 1120), Gilbert of Poitiers (+1154), and Abbot Joachim of Fiore (+1202).
(From ICU Courses at International Catholic University. I quote here for you, so as to gently suggest to any lurkers and readers that your metaphor would be consistent with Modalistic Monarchianism rather than Trinitarian orthodoxy. I understand from your referenced article that you do not accept the inspiration of the Bible.)
Jesus, in His John 14 explanation, said that all the disciple could see of the Father was where The Father intersected their spacetime limits: "I am in the Father and The Father is in Me. If you have seen Me, you have seen the Father."
Our spacetime coordinate system limits our capabilities to see very real dimensional reality no further away than the length of you forearm, as illustrated in Daniel 5 and elsewhere.
The third member of the Trinity of the One God is the Holy Spirit Whom Jesus told His disciples HE would send after His Ascension. Thus God is proven for the faithful to be expressing His Oneness in three distinct manifestations while being One.