Packing rectangles: Does rotation ever help?

Question

Dominic van der Zypen posed an interesting Box stacking problem. This is a spin-off question.

Let a collection of rectangles $r_1,\ldots,r_n$ be given by their side lengths in $\mathbb{R}$. Let $R$ be a rectangle of minimum area enclosing the rectangles arranged in the plane without overlap (i.e., with disjoint interiors).

Q. Is there an example where not all the rectangles have sides aligned with the sides of $R$?

In other words, where at least one rectangle's sides are not parallel to the sides of $R$? Is it ever advantageous to "tilt" one or more rectangles to achieve a minimal area?

Answer 1

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          ^{@YosemiteStan's example.}

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          ^{Detail: Tilt angle $=\sin ^{-1}\left(5 \sqrt{\frac{2}{61}}\right) \approx 65^\circ$.}

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Answer 2

The classic answer to this is a paper of Erdos and Graham 'On packing squares with equal squares'. Given a square of side $n+\varepsilon$, where $0<\varepsilon<1$, we can obviously fit in $n^2$ unit squares, and it's fairly trivial to check that if the squares are axis aligned this is best possible (count the squares intersecting vertical lines on the integers). Obviously, this means about $2\varepsilon n$ area is going to waste.

But Erdos and Graham show one can cover asymptotically all but $n^{7/11}$ area, using skew angles - this is maybe more surprising than Yosemite Stan's example (which also works perfectly well).