The DS likes their dates and high holy days.
Head on a swivel everybody.
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While the winter solstice peaks on one day, the "solstice" really lasts for three days. When graphing the "length of days" v. "day of the year" these three days are the bottom of a sine wave, which is basically a straight line (close enough), at least by the measurements of previous cultures.
These three days are the days when the Sun is dead. The day after the third day is Dec. 25th. Dec. 25th is the day when the Sun starts to stay longer in the sky. It is considered by these bygone cultures to be the day the Sun Rises Again, i.e. the birth of the Sun.
Of course because of how we tell time, leap years, precession of the Earth around the Sun, etc. the solstice (absolute minimum) changes from year to year, so the end of the three day period (according to our calendar) doesn't always end on the 25th, but that is the origin of Dec. 25th being the "Birth of the Sun."