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Y in a Cube

Inside a unit cube, place a 2 dimensional letter "Y", such that each leg of the Y is the
same length. Maximize the length of the legs and describe the orientation
of the "Y" if
- The "top" two legs are at 90 degrees to each other, and 135 degrees to the
third.
- All legs are 120 degrees from each other. [I believe this is the same as
asking for the largest equilateral triangle you can inscribe in a cube.]

Source: 1. Wesley Rose. 2. Original extension.

Solution

Mail to Ken