What is the technical understanding of the assertion of Jesus dying on the cross? In technical terms, using the Nicea creed as the basis of understand, how can you explain the assertion of Jesus dying on the cross?

This is what I understand so far:

Co-existence
Co-equal
Co-Substantial
Jesus is begotten, not made


When you say Jesus died on the cross, is this referring to Jesus the 3rd person in Trinity?  I understand that Jesus is understood as being divided into 2 parts: Human and Divine.  So did the divine part die on the cross or the human?

The idea is that original sin has to be accounted for, so does this necessitate the Divine or Human aspect to be sacrificed and die?  

Depending on the answer, which aspect/nature of Jesus is said to have risen on the 3rd day?
First off, I'm going to answer according to traditional Catholic Christology. This includes the Fourth Ecumenical Council, which took place in AD451, and issued the definition on Christology that most churches follow today. (The Oriental Orthodox form the most significant body that does not.) It also includes the Fifth Ecumenical Council of AD553.

You say:


  When you say Jesus died on the cross, is this referring to Jesus the 3rd person in Trinity? I understand that Jesus is understood as being divided into 2 parts: Human and Divine. So did the divine part die on the cross or the human?


First, the Son (existing before all worlds, only-begotten of the Father) is the second person of the Trinity. The Son became incarnate in Jesus Christ.

Second, Jesus is not divided in two parts, as you say. Traditional (Chalcedonian) Christology says that Jesus has both a human nature and a divine nature, but that he is one person. He is not half-and-half human and divine, but fully human and fully divine. This is called the hypostatic union of Christ. He is one human, with two natures.

Because Christ is only one person, there were no times when the divine nature was absent and no times when the human nature was absent. The two natures were fully united in Christ. As Catechism of the Catholic Church says quoting the Fifth Ecumenical Council:


  Everything in Christ's human nature is to be attributed to his divine person as its proper subject, not only his miracles but also his sufferings and even his death: "He who was crucified in the flesh, our Lord Jesus Christ, is true God, Lord of glory, and one of the Holy Trinity."


That is to say, the person Jesus Christ, both God and man, suffered and died upon the Cross.

As Thomas Aquinas writes in Summa Theologica:


  as before death Christ's flesh was united personally and hypostatically with the Word of God, it remained so after His death, so that the hypostasis of the Word of God was not different from that of Christ's flesh after death


You go on to ask which nature had to be sacrificed and which was raised on the third day. The answer by now should be obvious and is the same in both cases: it is the one person Jesus Christ, true God and true man, who was sacrificed, died and rose again.