The translation of the Latin phrase, "Iacta alea est" refers to
a die, the singular of dice. Not "dye". The phrase is not, "The dye
is cast." It is, "The die is cast."
The original meaning of this phrase nothing to do with dice. it
refers to the time when Julius Caesar took his army into Rome. Once
he crossed the Rubicon with his forces there was no turning back as
it was forbidden to enter Rome with ones Legion and he was now
classed as an invader under Roman Law. Caesar said Jacta Alea
Est
literally meaning Let the dye be cast. The phrase refers adding
dye or ink to water. Once mixed you cannot get it back out.
"The die (singular of dice) has been cast" basically means one
has committed to a decision.
It literally means, "The die has (dice have) been rolled." The
decision can't be reversed beyond this point. It is believed to
have been said first by Julius Caesar when he committed cross the
Rubicon river (a related saying, crossing the Rubicon also means
committed to a decision that now can't be taken back). To protect
the state from it's own military, it was forbidden to bring a
legion of Roman soldiers across the Rubicon river into Italy. It
was an act of war against the state. When he cross that line he is
rumored to have said, "Iacta alea est (The die has been cast)."
Note: The correct word is "die" (singular of dice) and not "dye"
(substance used to stain or color something, like hair, clothing or
shoes).