Continued fractions in optimal cutting of a rectangular sheet into equal smal...

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Continued fractions in optimal cutting of a rectangular sheet into equal smal...

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Continued fractions provide an elegant mathematical solution to the problem of optimally cutting a rectangular sheet into smaller equal-sized rectangles with minimal waste. This classic optimization problem arises in industrial cutting applications like wood, metal, or fabric production.

The core idea leverages the continued fraction expansion of the aspect ratio (width/height) of the original sheet. By expressing this ratio as a continued fraction, we recursively decompose the rectangle according to the convergents - the best rational approximations at each step. This approach naturally yields partitions where cuts align perfectly along integer multiples, minimizing offcuts.

For implementation, the Euclidean algorithm can generate the continued fraction sequence efficiently. Each partial quotient indicates how many times a sub-rectangle fits optimally before requiring orientation switching. The method guarantees solutions where the number of small rectangles is maximized while respecting integer dimension constraints - crucial for real-world manufacturing where fractional cuts aren't practical.

This mathematical approach outperforms naive grid partitions by adaptively choosing cutting directions (horizontal/vertical) at each recursion level based on the remainder terms in the continued fraction expansion. The algorithm scales well for large sheets and arbitrary target rectangle sizes, making it particularly valuable for mass production scenarios.