Cube to square answer for Wednesday


Answer for this week:
If the longest line that can fit into a unit square is the diagonal, with a length of sqrt(2), and the largest square that can fit into a unit cube has an area of 9/8 and a side of 3/4 * sqrt(2), then what’s the largest cube that can fit into a tesseract?

First, the largest square that can fit into a cube, AKA Prince Rupert’s Cube, is going to have corners that are exactly 1/4 of the distance from the corner of the cube. For a unit cube, the length of the side of the square is going to be sqrt((3/4)^2 + (3/4)^2) = sqrt(9/16 + 9/16) = 3/4 * sqrt(2). Then the area of that square is (3/4 * sqrt(2))^2 = 9/16 * 2 = 9/8.

The answer for the largest cube that can fit in a tesseract can be found in the Generalizations section for the Prince Rupert’s Cube wiki entry, which was found by Kay R. Devicci, and was also mentioned by Gardner in the addendum to this chapter.

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