Why does the Holy Spirit not take the form of a human being just like Jesus did?

Revelation 19:

6 Then I heard what sounded like a great multitude, like the roar of rushing waters and like loud peals of thunder, shouting:

“Hallelujah! For our Lord God Almighty reigns. 7 Let us rejoice and be glad and give him glory! For the wedding of the Lamb has come, and his bride has made herself ready. 8 Fine linen, bright and clean, was given her to wear.” (Fine linen stands for the righteous acts of God’s holy people.)

9 Then the angel said to me, “Write this: Blessed are those who are invited to the wedding supper of the Lamb!” And he added, “These are the true words of God.”

Perhaps the Bride of Christ, the Church, is the body of the Holy Spirit?

(See John 15, Romans 12 and 1 Corinthians for extended analogies comparing the Church to a single body.)

That's a very good question and I'd like to think that we can find an answer in John 16. Why did Jesus ever say the following words:

7 Nevertheless I tell you the truth. It is to your advantage that I go away; for if I do not go away, the Helper will not come to you; but if I depart, I will send Him to you. - Verse 7 (NKJV)

Since Jesus, the Son of God became flesh, He could only be in one place at a time. What a sacrifice, that God the Son would be confined and restricted in and by a human body so as to redeem man. Think about it! Is it possible that " ... it is to [our] advantage ... " that he Holy Spirit did not become flesh. That way He can be everywhere at the same time:

8 And when He has come, He will convict the world of sin, and of righteousness, and of judgment:

9 of sin, because they do not believe in Me;

10 of righteousness, because I go to My Father and you see Me no more;

11 of judgment, because the ruler of this world is judged. Verses 8 - 11

The member of The Church of Jesus Christ of Latter-day Saints believe this:

The Father has a body of flesh and bones as tangible as man’s; the Son also; but the Holy Ghost has not a body of flesh and bones, but is a personage of Spirit. Were it not so, the Holy Ghost could not dwell in us. (Doctrine & Covenants 130:22)

According to Trinity theology, there are three persons in God and they are one.

1. Father
2. The Word or the Son
3. Holy Spirit

They are all spirit according to John 4:24.

God is spirit, and his worshipers must worship in the Spirit and in truth." (John 4:24, NIV)

Though there are three persons, only one of them took the form of flesh.

The Word became flesh and made his dwelling among us. We have seen his glory, the glory of the one and only Son, who came from the Father, full of grace and truth. (John 1:14, NIV)

The Bible doesn't tell us whether all three of them can become flesh, but there is no reason why they cannot do so, because God is All-powerful. All we know is that only The Word became flesh and is called the Son of God. There is a strong reason why The Word became flesh and it was a necessity. It was all planned by God that His Son should become human and die on the Cross. It was the will of the Father that the Son should born from a woman and die for all mankind. Since it was necessary for the Son to die as human on the cross, the Son/Word became flesh.

All inhabitants of the earth will worship the beast--all whose names have not been written in the Lamb's book of life, the Lamb who was slain from the creation of the world. (Revelation 13:8, NIV)

Praise be to the God and Father of our Lord Jesus Christ, who has blessed us in the heavenly realms with every spiritual blessing in Christ. For he chose us in him before the creation of the world to be holy and blameless in his sight. In love he predestined us for adoption to sonship through Jesus Christ, in accordance with his pleasure and will— to the praise of his glorious grace, which he has freely given us in the One he loves. (Ephesians 1:3-6, NIV)

The three persons in the Trinity have different roles. The Father plays the role of a father, the Son plays the role of intercession between God and men, the Holy Spirit dwells inside the believers and help them to live a holy life.

But the Advocate, the Holy Spirit, whom the Father will send in my name, will teach you all things and will remind you of everything I have said to you. (John 14:26, NIV)

But very truly I tell you, it is for your good that I am going away. Unless I go away, the Advocate will not come to you; but if I go, I will send him to you. When he comes, he will prove the world to be in the wrong about sin and righteousness and judgment (John 16:7-8, NIV)

God can live inside the believers in the form of the Holy Spirit. Though the Father and the Son are in Heaven, the Holy Spirit is in this world to help the believers.

Do you not know that you are God’s temple and that God’s Spirit dwells in you? (1 Corinthians 3:16, ESV)