When we speak about the Trinity we are not simply talking about our Christian understanding of the nature of God, but about the nature of humanity as well.
One aspect of our having been created in the image and likeness of God is that we are each unique individuals, and even after death we do not lose out distinctive identity. At the same time we are called to live in unity, "that they may be one as you and I are one" was the prayer of the Son to the Father.
The great difference between us and God is that while the Holy Trinity exists in constant perfect harmony, an interpenetration, that St. Gregory of Nanzianzus called perichoresis, in our human lives we experience it as constant tension.
We want to love and be loved and yet we are selfish. We know that we should work together but continue to divide ourselves by nation, language, race. The balance required is a difficult one.
Communism tried to erase the individual in favor of the unity. Our modern American individualism can undermine the essential unity, in a misguided notion of independence. Where do we find the balance? In God. By keeping our eyes constantly fixed on the Holy Trinity and striving to model every aspect of our lives on the Father, the Son, and the Holy Spirit.