Polar circles
What are the polar circles?
The Arctic Circle and Antarctic Circle sit at 66.6° north and south — 90° minus Earth’s tilt. Inside them the Sun can stay up all day in summer, or stay down all day in winter.
The same tilt, counted the other way
If the Sun’s overhead point never goes farther from the equator than the tilt, then at the poles the Sun never climbs farther from the horizon than the tilt either. Walk 23.4° down from the pole and you reach the first latitude where the Sun can just graze the horizon at midwinter and just refuse to set at midsummer. That walk is the polar circle.
So 90° − 23.4° = 66.6°. One subtraction. Two circles.
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Midnight sun and polar night are not total darkness
Even when the Sun stays below the horizon, twilight can light the sky. Astronomical night — Sun 18° down — is a stricter line than “the Sun did not rise.” Near a solstice the twilight band on the map is wide at high latitudes, which is why a northern summer night is often not fully dark.