Why is the Devil also known as Lucifer (and other names) three-faced?
To understand this symbolic meaning of Dante in his Canto XXXIV, we have to have a basic understanding of the beliefs of the Catholic Church.
First of all, the devil was a murderer and the father of lies from the beginning.
Ye are of your father the devil, and the lusts of your father ye will do. He was a murderer from the beginning, and abode not in the truth, because there is no truth in him. When he speaketh a lie, he speaketh of his own: for he is a liar, and the father of it. - John 8:44
We have all heard of the expression about someone being two-faced:
Definition of two–faced for English Language Learners
not honest or sincere
saying different things to different people in order to get their approval instead of speaking and behaving honestly
Far worse is the Devil to Dante than someone simply two-faced, thus he imagines the Devil as having three faces.
Another aspect of this symbolic meaning is Lucifer's desire to be like God and to be worshiped as such. The God of Catholics is known as the Holy Trinity composed of the Father, Son and Holy Spirit, or "one God in three Divine Persons".
- In this three-face appearance of Satan, Dante is showing us the Evil One's desire to mock the very image of God Himself as if he could take his place!
Whether you call him Satan, Lucifer, or Mephistopholes, he’s a beast with even more faces than he has names. Over the past five centuries, artists have variously depicted the devil as a fanged, horned demon; as an armored, Apollo-like army leader; and as a tailor of Nazi uniforms. - The Changing Face Of Satan, From 1500 To Today
An analysis of Canto XXXIV reveals a few tidbits of Dante's thinking in regards to Judas, Brutus, and Cassius:
One who betrays one’s benefactor comes closest to betraying God directly. Thus, the ultimate sinner, Judas Iscariot, was a man who betrayed both simultaneously, for his benefactor was Jesus Christ.
Since spiritual concerns must, in the end, outweigh temporal ones, Judas has committed the greater sin, and his head, rather than his legs, feels the constant chewing of Lucifer’s teeth. However, the fact that Brutus and Cassius suffer a punishment only slightly less harsh demonstrates Dante’s belief that church and state play equally important roles, each in its own sphere. Throughout Inferno, Dante has expressed the view that church and state should remain separate but equal. Now, Dante finds an arrangement for the final circle of Hell that both completes his vision of the moral hierarchy and makes one last, vivid assertion of his politics. Analysis: Canto XXXIV
Satan tricephalos-tonguesticker, with fiery crown or hair, holding the Serpent,
on the amazing 12th century façade at Tuscania (Viterbo).