What are those three fasting days?
Yom Kippur, you probably know .... the holiest day in the Jewish calendar when God decides everyones' fate, the only day on which the High Priest would utter the true name of God, Y_H_V_H (with _ being vowels which have been lost). It is a chilling holiday, particularly:
Unatanah Tokef wrote:On Rosh Hashanah will be inscribed and on Yom Kippur will be sealed how many will pass from the earth and how many will be created; who will live and who will die; who will die at his predestined time and who before his time; who by water and who by fire, who by sword, who by beast, who by famine, who by thirst, who by storm, who by plague, who by strangulation, and who by stoning. Who will rest and who will wander, who will live in harmony and who will be harried, who will enjoy tranquillity and who will suffer, who will be impoverished and who will be enriched, who will be degraded and who will be exalted.
Tisha B'Av (9th of Av) commemorates the destruction of the first temple by the Babylonians and 586 BCE, and the destruction of the second temple (and the end of the Jewish polity) in 70 CE. Some have suggested that this should be reduced to a half-day fast because of the recreation of Israel, but I haven't made up my mind yet.
The first two here are full fast days, 24+ hours no food or water.
The third one that I observe is
Taanit B'chorot ("Fast of the Firstborns"). Basically, since I am a first born male, the day before Passover I have to fast to commemorate the Egyptian firstborns who were slain and how I was spared. This one only starts at dawn though, and breaks at sundown. The sorrow at the deaths of our enemies gives way to the joy of Passover.
The fast days I don't observe are:
Tenth of Tevet - commemorates the day on which Nebuchadnezzar began his siege of Jerusalem prior to the destruction of the Temple in 586 BCE.
Seventeenth of Tammuz - the day the Romans breached the walls of Jerusalem after a three week siege, prior to the destruction of the Temple.
Tzom Gedalia - commemorates the assassination of Gedaliah, the Jewish Babylonian-appointed Governor of Judah, the last Jewish leader of Israel before the exile to Babylon.
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Muslim, as you can see, besides Yom Kippur, they all have a highly nationalist character to them which I find interesting. They aren't religious so much as commemorating tragic historic events.
Does Islam have other fast days?