Question

Why would it appear that the sun does not rise during the winter solstice at 66 degrees latitude? (Consider the tilt of the earth during that season)

Answer 1

This is because the sun never goes beyond the Tropical of Capricorn  and the Tropical of Cancer whcih are approximately 23.5 degrees S and N respectively. The 66 degrees latitude N and S is the Arctic and Antarctic regions. Because if you add 23.5 and 66 = 90 degrees (approximately), you realize that the sun rays are unable to reach these regions, during winters, because light cannot bend over 90 degrees (light moves in straight lines and only diffracts).