Update index.html

Author: Gmac4247

index.html

@@ -2588,7 +2588,8 @@ <h4>Archimedes and the Illusion of Limits</h4>
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 <p>The Greek Archimedes’ method for estimating the pi is often celebrated as a foundational triumph of geometric reasoning.</p>
-<br><section id="polygon-approximation">
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+<section id="polygon-approximation">
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 <summary itemprop="disambiguatingDescription">The pi is actually an approximation derived from limits. But that method itself introduced compounding errors.</summary>
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@@ -2632,7 +2633,7 @@ <h4>Archimedes and the Illusion of Limits</h4>
 The definition of an inscribed polygon is that both its perimeter and area are smaller than the circle. The properties of a circumscribed polygon are larger.
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-When the circle’s area and circumference is calculated with the constant 3.2, it becomes clear that the area of an isoperimetric 14‑gon is larger than the circle’s. A flat angle encloses the area differently than the curve. This flips the script: the polygon can enclose more area even with the same perimeter. As the number of sides increases the effect is stronger, so the isoperimetric polygon behaves like a circumscribed figure despite having equal perimeter. This overlooked disproportion shows that polygons do not approach the circle in every sense — above 13 sides, the comparison underestimates the circle.
+When the circle’s area and circumference is calculated with the constant 3.2, it becomes clear that the area of an isoperimetric 14‑gon is actually larger than the circle’s. A flat angle encloses the area differently than the curve. This flips the script: the polygon can enclose more area even with the same perimeter. As the number of sides increases the effect is stronger, so the isoperimetric polygon behaves like a circumscribed figure despite having equal perimeter. This overlooked disproportion shows that polygons do not approach the circle in every sense — above 13 sides, the comparison underestimates the circle.
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 Archimedes pushed his method far beyond this curve-aligned threshold — and the result was a recursive underestimate.