The "Do Balrogs have wings or no?"-eternal discussion, finalized for good by such a wonderful mathematical theorem? Excellent, Witch-King! Excellent! I really agree with your mathematical/logical solution !
Now, to be honest, I have not believed too much that those devillish creatures had wings. Why?
- from the episodes where they appear in the story - fighting, fleeing or attacking - I do not recall them using wings. Fire and flame-whips - this is their "trade-mark", IMO. But wings?... I don't think so.
- they lived under the ground, in the depths of the earth - so, what would they need wings for?
- let's look at this:
Quote:
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For of the Maiar many were drawn to his splendour in the days of his greatness, and remained in that allegiance down into his darkness; and others he corrupted afterwards to his service with lies and treacherous gifts. Dreadful among these spirits were the Valaraukar, the scourges of fire that in Middle-earth were called the Balrogs, demons of terror.
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The Published Silmarillion, Valaquenta, Of the Enemies
What we learn right from the beginning is that the Balrogs were Maiar, and as such, their material shape was just this - a shape to fit the world of Arda, but they still could "breach" the physical laws and "behave" like spirits as they actually were. So even if a Balrog could "suddenly [it ] drew itself up to a great height " I don't consider it to be because of a pair of wings, but rather because they could, I assume, in certain moments, choose to use their Maiar-powers.
- if a Balrog had wings, will the Balrog who Glorfindel fought against fall intio the abyss? And would the Balrog who Ganfdalf fought fall into the abyss? Why not
fly up safely if they had a pair of wings available?
Just .... thinking, you know... I might be wrong, but this is my overall "impression" of those powerful and .... aren't they just fascinating!!!..... creatures!
