# minimal oblique rectangle to sample wallpaper

From a rectangular wallpaper pattern with periods $$(w,0)$$ and $$(0,h)$$, you must cut a rectangle with aspect ratio $$m$$, to include every part of the pattern-unit; in other words, you must cover the (flat) torus. You may rotate this sample-piece at any angle relative to the pattern. How do you choose the rotation to allow the smallest such rectangle?

A sub-problem is: for a given rotation angle, what is the smallest $$m:1$$ rectangle that covers?

Partial answer, 2019 Feb 13. I think it necessary and sufficient that, for each point $$(x,y)$$ on the boundary of the (scaled and rotated) ‘sample’ rectangle, there exist integers $$j,k$$, not both zero, such that $$(x+jw,y+kh)$$ is within the (closed) rectangle.

The solution, then, will be to minimize (over rotation) the maximum (over the set of boundary points) of the minimum (over $$j,k$$) of a family of trig functions which express the minimum scale factor.

• It seems the edit has turned the question into a different question? – joriki Jun 16 '18 at 1:26
• @joriki The is the question I wanted to ask; if you think that's a different question, you confirm that the edit was necessary! – Anton Sherwood Jun 16 '18 at 5:51