The best analogy for God would be the family. You have the father, mother, and from their union, their child. The father is a person. The mother is a person, and the child is a person.
For example, you can look at the father and mother becoming one is sort of like God. They are, and can become one, but the mother is still a distinct person from the father. The offspring also, for a time, spends time physically inside the mother and depends on the mother for nursing, say.
So, there is a strong interrelationship amongst all the persons, but even so, the child is not the mother or the father. Together, they form a family.
Although, the Trinity is considered to be a mystery. There may be some aspects of it we may never understand.
The Trinity is a communion of Persons. There is real distinction in their relationships to one another, but because the Persons are in communion, they are One God. Much like the many members of the Church make up the One Mystical Body of Christ.
You are confusing parts with essence. You know God is just one essence. The three persons are all in the one essence. They also have distinctness as persons, as you also know. However, those human definitions do not equal parts of GOD. The single essence is ONE part only. How they can have distinct persons within ONE essence is exactly what the mystery of the Trinity is! Remember, we use the word persons in a theological sense not a human sense, to convey that distinction. But there is NO WAY to define in human language how that can be. We must believe IN FAITH, that which Jesus told us is true. The Trinity was revealed to us by Jesus. The Church has tried to define the Trinity for believers as set down in the Athanasian Creed. But there is no way to make it seem humanly logical.
This is so hard to understand that Jews and Muslims think Christians are polytheists! But we are monotheists, just like them, but believe in the 3 distinct persons in the one essence of GOD. We didn’t make this up. It was revealed to us by GOD Himself! Why would anyone think the being of GOD would be something simple for humans to understand?
I see you say that you are of the Christian faith,
Javid, I find that in order to understand the
Mystery of the Trinity, you have to have Humility,
that’s what Augustine said when asked how
he was able to get such great insights into
the teachings of the Church, he replied:
“It’s all, humility, humility and humility”!
In other words, Javid, you have to approach
God in humility before His throne of mercy
and ask, no beg, recognising that w/o His
illuminating Spirit, you are just a dumb beast!
Good answer … any analogy we can - with our human limitations - use to describe the Trinity will fall short … it is the most profound mystery of the faith …
As family - we are like the Trinity - the Holy Trinity is what real family is …
That said - the analogy that always helped me is Water … the Holy Trinity is like Water - H2O …
*]When water is equal to or below 0 degrees celcius (32 degress fahrenheit) - it is a solid - Ice - H2O
*]Water is above 0 degrees celcius and below 100 degrees celcius - it is a liquid - water - H2O
*]When water is equal to or above 100 degrees celcius (212 degrees fahrenheit) - it is a vapor - Steam - H2O