Fascinating question. The Trinity (Father, Son, Spirit) and the hypostatic union are a mysteries and therefore the answer is probably is incomprehensible to some degree. Nevertheless, I'll take a vain attempt to answer.
Given: God is of ONE ESSENCE. What is said of the Father, applies equally to the Son and Spirit. All are omnipresent, omnipotent, omnithis, omnithat, etc.
THE SECOND HELVETIC CONFESSION CHAPTER III - Of God, His Unity and Trinity
GOD IS ONE. God is one in essence or nature, subsisting in himself, all sufficient in himself, invisible, incorporeal, immense, eternal, Creator of all things both visible and invisible, the greatest good, living, quickening and preserving all things, omnipotent and supremely wise, kind and merciful, just and true.
GOD IS THREE. Notwithstanding we believe and teach that the same immense, one and indivisible God is in person inseparably and without confusion distinguished as Father, Son and Holy Spirit so, as the Father has begotten the Son from eternity, the Son is begotten (eternally generated) by an ineffable generation, and the Holy Spirit truly proceeds from them both, and the same from eternity and is to be worshipped with both.
Thus there are not three gods, but three persons, co-substantial, coeternal, and coequal; distinct with respect to hypostases, and with respect to order, the one preceding the other yet without any inequality. For according to the nature or essence they are so joined together that they are one God, and the divine nature is common to the Father, Son and Holy Spirit.
What do we know ...
We know God is Spirit ... what does that mean ... well, we don't know exactly. We know by negation that this means He has no material organization, that he has neither body nor members of the body such as we have, neither shape or form, neither passions nor limitations, but only a spiritual nature. Therefore, His Divine nature is not visible and therefore it cannot be seen.
We know God's divine nature is omnipresent. This means He is everywhere. Man is not capable of seeing anything that is everywhere because God is too big to see. Furthermore, because He is present with every atom, He is in a sense too small too see.
Conclusion 1: God is not visible so man cannot see Him
Premise 1: Christ is visible.
Premise 2: Christ is God (John 20:28)
Conclusion2: God is visible so man can see Him
Conclusion1 and conclusion2 contradict. One must be wrong .... or something is missing ....
THE SECOND HELVETIC CONFESSION
A RATIONAL SOUL IN CHRIST. Moreover, our Lord Jesus Christ did not have a soul bereft of sense and reason, nor flesh without a soul, but a soul with its reason, and flesh with its senses, by which in the time of his passion he sustained real bodily pain, as himself testified when he said: "My soul is very sorrowful, even to death" (Matthew 26:38). And, "Now is my soul troubled" (John 12:27).
TWO NATURES IN CHRIST. We therefore acknowledge two natures or substances, the divine and the human, in one and the same Jesus Christ our Lord (Hebrews 2). And we say that these are bound and united with one another in such a way that they are not absorbed, or confused, or mixed, but are united or joined together in one person the properties of the natures being unimpaired and permanent.
NOT TWO BUT ONE CHRIST. Thus we worship not two but one Christ the Lord. We repeat: one true God and man. With respect to his divine nature he is consubstantial with the Father, and with respect to the human nature he is consubstantial with us men, and like us in all things, sin excepted (Hebrews 4:15).
THE DIVINE NATURE OF CHRIST IS NOT PASSIBLE, AND THE HUMAN NATURE IS NOT EVERYWHERE. Therefore, we do not in any way teach that the divine nature in Christ has suffered or that Christ according to his human nature is still in this world and thus is everywhere. For neither do we think or teach that the body of Christ ceased to be a true body after his glorification, or was deified, and deified in such a way that it laid aside its properties as regards body and soul, and changed entirely into a divine nature and began to be merely one substance.
Note1: Arminians, in general, believe the divine nature is PASSIBLE
Note2: This is referred to as the hypostatic union by theologians: the union of man and God. This is a mystery to some degree. Christ was called the Son of God (divine nature) and son of man (human nature). The human nature of Christ was not all knowing (Mark 13:32 ... Muslims use that as a cudgel), while the divine nature of Christ is all-knowing. The human nature cried and got tired, while the divine nature does neither (Arminians might disagree as they affirm a passible God).
As Christ is a hypostatic union of God and man, so are the Old Testament Theophanies, some of which were mentioned in posts above. Often (always?) the name "The Angel of God" refers to a theophany. Joshau 5:13-15; Genesis 22:15-17; Judges 2:2; Hagar, Abraham and Isaac, burning bush, pillar of fire, Jacob's wrestling match, yahda, yahda. All these material beings were mysteriously "united with the divine in such a way that they are not absorbed, or confused, or mixed, but are united or joined together in one person the properties of the natures being unimpaired and permanent." (Second Helvetic Confession)
So, the apparent contradiction above proving (a) God can be seen and (b) yet cannot be seen is explained, to some degree, by the doctrine of the hypostatic union.
Aside: Did all the theophanies involve the Second Person and human flesh (or bush or pillar of fire) .... I don't know. Many theologians believe so. I would like to see proof. Since all 3 persons have the same essence, any could form a union of sorts with a material thing.
Aside2: Was Melchizedek a theophaniy? Split decision from what I have read. Just to pick side, I say yes.
Aside3: Can the essence of the Son, Father or Spirit be separated from one another ... that would be 3 Gods.
My 2 cents worth ....
Given: God is of ONE ESSENCE. What is said of the Father, applies equally to the Son and Spirit. All are omnipresent, omnipotent, omnithis, omnithat, etc.
THE SECOND HELVETIC CONFESSION CHAPTER III - Of God, His Unity and Trinity
GOD IS ONE. God is one in essence or nature, subsisting in himself, all sufficient in himself, invisible, incorporeal, immense, eternal, Creator of all things both visible and invisible, the greatest good, living, quickening and preserving all things, omnipotent and supremely wise, kind and merciful, just and true.
GOD IS THREE. Notwithstanding we believe and teach that the same immense, one and indivisible God is in person inseparably and without confusion distinguished as Father, Son and Holy Spirit so, as the Father has begotten the Son from eternity, the Son is begotten (eternally generated) by an ineffable generation, and the Holy Spirit truly proceeds from them both, and the same from eternity and is to be worshipped with both.
Thus there are not three gods, but three persons, co-substantial, coeternal, and coequal; distinct with respect to hypostases, and with respect to order, the one preceding the other yet without any inequality. For according to the nature or essence they are so joined together that they are one God, and the divine nature is common to the Father, Son and Holy Spirit.
What do we know ...
We know God is Spirit ... what does that mean ... well, we don't know exactly. We know by negation that this means He has no material organization, that he has neither body nor members of the body such as we have, neither shape or form, neither passions nor limitations, but only a spiritual nature. Therefore, His Divine nature is not visible and therefore it cannot be seen.
We know God's divine nature is omnipresent. This means He is everywhere. Man is not capable of seeing anything that is everywhere because God is too big to see. Furthermore, because He is present with every atom, He is in a sense too small too see.
Conclusion 1: God is not visible so man cannot see Him
Premise 1: Christ is visible.
Premise 2: Christ is God (John 20:28)
Conclusion2: God is visible so man can see Him
Conclusion1 and conclusion2 contradict. One must be wrong .... or something is missing ....
THE SECOND HELVETIC CONFESSION
A RATIONAL SOUL IN CHRIST. Moreover, our Lord Jesus Christ did not have a soul bereft of sense and reason, nor flesh without a soul, but a soul with its reason, and flesh with its senses, by which in the time of his passion he sustained real bodily pain, as himself testified when he said: "My soul is very sorrowful, even to death" (Matthew 26:38). And, "Now is my soul troubled" (John 12:27).
TWO NATURES IN CHRIST. We therefore acknowledge two natures or substances, the divine and the human, in one and the same Jesus Christ our Lord (Hebrews 2). And we say that these are bound and united with one another in such a way that they are not absorbed, or confused, or mixed, but are united or joined together in one person the properties of the natures being unimpaired and permanent.
NOT TWO BUT ONE CHRIST. Thus we worship not two but one Christ the Lord. We repeat: one true God and man. With respect to his divine nature he is consubstantial with the Father, and with respect to the human nature he is consubstantial with us men, and like us in all things, sin excepted (Hebrews 4:15).
THE DIVINE NATURE OF CHRIST IS NOT PASSIBLE, AND THE HUMAN NATURE IS NOT EVERYWHERE. Therefore, we do not in any way teach that the divine nature in Christ has suffered or that Christ according to his human nature is still in this world and thus is everywhere. For neither do we think or teach that the body of Christ ceased to be a true body after his glorification, or was deified, and deified in such a way that it laid aside its properties as regards body and soul, and changed entirely into a divine nature and began to be merely one substance.
Note1: Arminians, in general, believe the divine nature is PASSIBLE
Note2: This is referred to as the hypostatic union by theologians: the union of man and God. This is a mystery to some degree. Christ was called the Son of God (divine nature) and son of man (human nature). The human nature of Christ was not all knowing (Mark 13:32 ... Muslims use that as a cudgel), while the divine nature of Christ is all-knowing. The human nature cried and got tired, while the divine nature does neither (Arminians might disagree as they affirm a passible God).
As Christ is a hypostatic union of God and man, so are the Old Testament Theophanies, some of which were mentioned in posts above. Often (always?) the name "The Angel of God" refers to a theophany. Joshau 5:13-15; Genesis 22:15-17; Judges 2:2; Hagar, Abraham and Isaac, burning bush, pillar of fire, Jacob's wrestling match, yahda, yahda. All these material beings were mysteriously "united with the divine in such a way that they are not absorbed, or confused, or mixed, but are united or joined together in one person the properties of the natures being unimpaired and permanent." (Second Helvetic Confession)
So, the apparent contradiction above proving (a) God can be seen and (b) yet cannot be seen is explained, to some degree, by the doctrine of the hypostatic union.
Aside: Did all the theophanies involve the Second Person and human flesh (or bush or pillar of fire) .... I don't know. Many theologians believe so. I would like to see proof. Since all 3 persons have the same essence, any could form a union of sorts with a material thing.
Aside2: Was Melchizedek a theophaniy? Split decision from what I have read. Just to pick side, I say yes.
Aside3: Can the essence of the Son, Father or Spirit be separated from one another ... that would be 3 Gods.
My 2 cents worth ....