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@@ -3166,7 +3166,6 @@ <h4 style="margin:12px">Archimedes and the Polygonal Trap</h4>
 - Rounding errors of infinite fractions, and any other inaccuracies amplified over 96 iterations 
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 <p style="margin:12px">He began with a circle bounded by an inscribed and a circumscribed hexagon — not the absolute minimum of 3 or 4 sides — likely because the hexagon is closer to the circle while still being easily calculable. He then increased the number of sides to 96, observing how the difference between the two polygonal perimeters — one inside the circle, one outside — became smaller. 
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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.
@@ -3175,10 +3174,10 @@ <h4 style="margin:12px">Archimedes and the Polygonal Trap</h4>
 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 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.
+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. 
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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.
+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. 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 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. The perimeter of the circumscribed polygon that he believed to be an overestimate of the circumference was practically an underestimate of it. 
@@ -3190,7 +3189,7 @@ <h4 style="margin:12px">Archimedes and the Polygonal Trap</h4>
 What we’re left with is not a proof, but a layered approximation — one that has shaped centuries of geometry, but now deserves a closer, more rational reexamination.
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-The classical polygon-based approach to approximating a circle’s circumference relies on inscribed and circumscribed polygons, calculated using trigonometric functions aligned to π. But this alignment is problematic if π itself is the quantity under investigation.
+<strong>The classical polygon-based approach to approximating a circle’s circumference relies on inscribed and circumscribed polygons, calculated using trigonometric functions aligned to π. But this alignment is problematic if π itself is the quantity under investigation.</strong>
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 The equal distance polygon method upgrades the classical approach by replacing inherited assumptions with geometric conditions — and aligns the approximation process with the true nature of the circle.