Bad analogies of the Trinity abound (i.e. Father-Son-Brother; Ice-water-vapor). Richard Baxter found the Trinity under every rock. Though see God everywhere can be a good thing, drawing uncritical analogies is another. Often the oneness, or indwelling of thre persons of the Trinity is missed in analogies. This aspect was called perichoresis by early church fathers. Although the earlier Cappadocian fathers discussed the Trinity in perichoretic terms, it was a 7th century theologian, John of Damascus, who coined the term perichoresis (peri = “around”) and (choresis = “dance”). Perichoresis refers to the mutual indwelling of the Father and the Son and the Holy Spirit within the blessed Trinity. The idea is that the three persons relate to each other in fluid, dance-like motion, each person critical for the dance to be executed properly. The image also gets at the joy of the Trinity, a God who dances.
Perhaps a better analogy would be that of Song. Consider a three piece band– each person of the Godhead making music through the unique instrumentation of their personhood, without which the song would be incomplete and dissonant. The Father, the vocalist; the Son the lead guitar; and the Spirit the drums. As the vocalist, the Father gives the Song its lyrical direction and purpose. As the lead guitar, the Son’s combinations of chords musically embody the words of the Father. He is known to play stunning ‘guitar solos’. The Spirit faithfully and consistently pumps out the backbeat of the Eternal Song. He keeps its rhythm, its time, holding the song together. Without the drums, song isn’t the same; it isn’t the Song. The Trinity-as-Song reveals the harmonious, interweaving nature of the Godhead, each “instrument” a crucial part of the Song. If one person is removed, there is no song. The melodies, harmonies and sounds that comprise the Song produce music that thrills the human soul. The eternal Song can not be dislodged from its prominent, everlasting spot as number one, top of the cosmic charts. It is the Song of all songs.