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Author: Gmac4247

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@@ -3171,10 +3171,14 @@ <h4 style="margin:12px">Archimedes and the Polygonal Trap</h4>
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 This narrowing gap was key. Archimedes likely believed that as the number of sides increased, the difference between the perimeters of the inscribed and circumscribed polygons would converge toward zero, approaching the circumference of the circle.
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-He assumed that more sides mean closer resemblance to a circle. That was backed by the isoperimetric inequality theory, which states that a circle maximizes area for a given perimeter. That idea likely emerged from observing simple polygons: the triangle has the smallest area, the square is larger, and so on. From this pattern, it was assumed that the trend continues indefinitely — that a polygon with an infinite number of sides would resemble a circle perfectly, with its area approaching from below.
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+He assumed that more sides mean closer resemblance to a circle. He or those who later formalized his reasoning — believed that the circle maximizes area for a given perimeter. This seems obvious when comparing a triangle or a square to a circle. An isoperimetric triangle has the smallest area, the square is larger, and so on. From this pattern, it was assumed that the trend continues indefinitely — that a polygon with an infinite number of sides would resemble a circle perfectly, with its area approaching from below.
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-But that assumption ignores a crucial geometric reality: as the number of sides increases, the internal angles of the polygon approach 180° — it is 180° - 360° / 96 = 176.25° in the case of a 96-gon —, nearing a straight line rather than a curve. In contrast, polygons with internal angles in the range between 150° and 160°, such as the 13- to 16-gon, preserve a meaningful bend that better reflects circularity.
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+But that assumption ignores a crucial geometric reality: as the number of sides increases the internal angles flatten toward 180°, — it is 180° - 360° / 96 = 176.25° in the case of a 96-gon —, nearing a straight line rather than a curve, and the polygon no longer reflects the circle’s curvature. When the circle’s area 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. 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 14 sides, the comparison underestimates the circle.
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+In contrast, polygons with internal angles in the range between 150° and 160°, such as the 13- to 16-gon, preserve a meaningful bend that better reflects circularity.
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 Archimedes pushed his method far beyond this curve-aligned threshold — and the result was a recursive underestimate. The perimeter of the circumscribed polygon that he believed to be an overestimate of the circumference was practically an underestimate of it.