Write four numbers on the corners of a square, the active square. One step works like this: for each side of the active square, write the absolute difference of the numbers at its two ends on the side's midpoint. The four midpoints form a new square, tilted 45 degrees, and that square becomes the active square.
With nonnegative integers, the process always ends with all four corners equal to 0. In this part the corners may hold real numbers.
Call two squares similar if one of them turns into the other when you:
- turn the square around, flip it over, or both, then
- multiply all four numbers by the same positive constant A, then
- add the same constant B (any real number) to all four numbers.
If one step turns a square into a similar square, the pattern repeats at every step and the corners never all become 0.
Take a starting square whose numbers, read clockwise, are
(1, x, y, 0) with 1 > x > y > 0.
Find the real numbers x and y for which one step produces a square similar to the starting square.
Answer format. Give x and y separated by a comma, x first, e.g. 0.5, 0.25. Each value may be an exact expression or a decimal correct to at least 7 significant figures.