Daniel 3: when it is a sin to apologize

Nebuchadnezzar made a great statue (27 meters high) to be adored, mainly by the leaders of his reign. The friends of Daniel were summoned to adhore the statue, but they refused to do it:

16-18 Shadrach, Meshach, and Abednego answered King Nebuchadnezzar, “Your threat means nothing to us. If you throw us in the fire, the God we serve can rescue us from your roaring furnace and anything else you might cook up, O king. But even if he doesn’t, it wouldn’t make a bit of difference, O king. We still wouldn’t serve your gods or worship the gold statue you set up.”

The Bible says to submit to the authorities of this world when they are not demanding anything against the Lord.
They refused to even apologize their refusal because in this case, they would sin.
The three friends of Daniel were thrown in the furnace, but the Lord saved them.

Who is to be adored?
Our hearts have a strong inclination to adore false idols (money); however in this case, the false idol was to be adored by intimidation, by a stronger power.
The friends of Daniel knew that this power could kill them, but ultimately, it didn't matter. Their lives were dedicated to the Lord and not to any one who thought to have their lives in hand.