06-15-2026, 09:30 AM
Summary
The author, while trying to explain recursion in programming, devised a geometric algorithm that recursively covers a half-circle with triangles — starting from a large central triangle and repeatedly subdividing the remaining curved regions with smaller and smaller triangles.
This process, which turns out to be equivalent to the classical polygon-approximation method dating back to Archimedes, converges to the area of the half-circle (π/2). By carefully tracking the areas and dimensions of each generation of triangles, the author derives two new theorems: a recursive formula whose limit equals π/2, and a striking infinite series of nested square roots that sums to π²/4 — essentially an additive counterpart to Viète's famous infinite product formula for π.
The article includes full proofs using the Pythagorean theorem, basic trigonometry, and an inductive lemma about nested square roots of 2, and concludes by noting the approach's pedagogical value for teaching mathematics to younger audiences.
ARTICLE
The author, while trying to explain recursion in programming, devised a geometric algorithm that recursively covers a half-circle with triangles — starting from a large central triangle and repeatedly subdividing the remaining curved regions with smaller and smaller triangles.
This process, which turns out to be equivalent to the classical polygon-approximation method dating back to Archimedes, converges to the area of the half-circle (π/2). By carefully tracking the areas and dimensions of each generation of triangles, the author derives two new theorems: a recursive formula whose limit equals π/2, and a striking infinite series of nested square roots that sums to π²/4 — essentially an additive counterpart to Viète's famous infinite product formula for π.
The article includes full proofs using the Pythagorean theorem, basic trigonometry, and an inductive lemma about nested square roots of 2, and concludes by noting the approach's pedagogical value for teaching mathematics to younger audiences.
ARTICLE
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│ KONSTANTINOS MICHAILIDIS │
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│ KONSTANTINOS MICHAILIDIS │
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