SMRTR Science & EngineeringJun 1, 2025John D. Cook

The circle hidden inside Pascal’s triangle

SMRTR summary

Pascal's triangle reveals an intriguing property when large rows are visualized: the logarithms of binomial coefficients form an ellipse-like shape. For row 1000, the best-fitting ellipse has parameters a = 554.2, b = 47.12, and y0 = -19.87. Although a parabolic fit is theoretically justified, the elliptical fit provides a better approximation, particularly in the tails. This phenomenon underscores the complexities of applying the central limit theorem in practice.

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