Boxes and fractions
BY Joshua Bowman
Summary
The article “Boxes and Fractions” explains a beautiful connection between geometry and fractions using a simple rectangle-cutting process. Starting with a fraction such as ($30/13$), we draw a rectangle with side lengths equal to the numerator and denominator, then repeatedly remove the largest possible squares (a process related to the Euclidean algorithm). The numbers of squares of each size form the terms of a continued fraction, which reconstructs the original fraction.
The author shows examples like ($30/13$) and ($25/7$), explains that the method works because square cutting mirrors Euclidean division, and connects the idea to the golden ratio, where the process continues forever with squares of decreasing size. The post also shows how the same idea approximates irrational numbers like ($\pi$), revealing a deep relationship between geometry, algorithms, and number representation.
ARTICLE
BY Joshua Bowman
Summary
The article “Boxes and Fractions” explains a beautiful connection between geometry and fractions using a simple rectangle-cutting process. Starting with a fraction such as ($30/13$), we draw a rectangle with side lengths equal to the numerator and denominator, then repeatedly remove the largest possible squares (a process related to the Euclidean algorithm). The numbers of squares of each size form the terms of a continued fraction, which reconstructs the original fraction.
The author shows examples like ($30/13$) and ($25/7$), explains that the method works because square cutting mirrors Euclidean division, and connects the idea to the golden ratio, where the process continues forever with squares of decreasing size. The post also shows how the same idea approximates irrational numbers like ($\pi$), revealing a deep relationship between geometry, algorithms, and number representation.
ARTICLE
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