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about.html

@@ -326,35 +326,38 @@ <h1 style="font-size:160%;margin:7px;">How Accurate Are The Conventional Geometr
 <p style="margin:12px;"><b>Archimedes and the Polygonal Trap</b>
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-The Greek Archimedes’ method for estimating the π is often celebrated as a triumph of geometric reasoning. 
+The Greek Archimedes’ method for estimating the π is often celebrated as a foundational triumph of geometric reasoning.
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-He began with a circle bounded by an inscribed and a circumscribed hexagon—figures whose properties can be described with exactness. From there, he increased the number of sides to 96, using trigonometry to approximate the perimeter.
-The polygons can be divided into triangles. The ratio between the legs of the triangles and their hypotenuses can be measured linearly.
+But that method contains critical flaws.
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-But this method rests on a flawed assumption: that a circle maximizes area for a given perimeter. This is not universally true, and it introduces a logical error into the foundation of the approximation. Moreover, calculating the properties of a 96-gon involves rounding infinite fractions—errors that are multiplied 96 times, amplifying the imprecision.
+Archimedes approximated the circle using inscribed and circumscribed polygons. 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.
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-Calling the inscribed hexagon a “lower bound” is already questionable. Calling the 96-gon an “upper bound” is even more so. 
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+He assumed that more sides mean closer resemblance to a circle. That was backed by the isoperimetric 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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-  <summary><strong style="margin:12px;">The theoretical upper bound would be a polygon with the number of sides approaching infinity...</strong></summary>
-  <p style="margin:12px;">In that case the angles between the side and the diagonals approach a right angle. They never reach a right angle as the diagonals converge towards the center. 
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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°, which is far from the curvature of a circle. In contrast, polygons with internal angles in the range of ~150°–160°, such as the 13- to 16-gon, preserve a meaningful bend that better reflects circularity.
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-If we relate the arc of a corresponding slice of an isoperimetric circle, the length of the arc equals the side in question. So the chord related to the arc is shorter than the side. If we want to place the arc with the chord so that it touches both diagonals, it has to be within the polygon. With the curvature of the arc becoming decreasingly distinctive, it doesn't bulge beyond the side. Eventually it will not even touch the side. 
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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. 
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+Hence his final value of 3.14... lies between two underestimates. The method itself introduced compounding errors. These include:
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+- Misapplied isoperimetric logic beyond its valid range
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+- Possible trigonometric inaccuracies in calculating the properties of the 96-gon
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-Hence the polygon with the same number of sides, which circumscribed the circle is smaller, so its perimeter is shorter than the circle.
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-<p style="margin:12px;">The perimeter of the circumscribed polygon that was believed to be an overestimate of the circumference was practically an underestimate of it. 
+- Rounding errors of infinite fractions, multiplied 96 times
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-Hence the value of the π lies between two underestimates. What we’re left with is not a proof, but a layered approximation.
+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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 Similarly, the area formula A = πr² is not a direct result of calculus. It’s reverse-engineered by multiplying the circumference formula C = 2πr by half the radius—treating the area as the sum of infinitesimal rings. While the method is algebraically valid, it bypasses the geometric logic that defines area: the comparison to a square.