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@@ -2523,81 +2523,18 @@ <h3 itemprop="name" style="margin:7px">Calculate the Circumference of a Circle</
 <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>
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-<section id="polygon-approximation">
-<details>
-<summary>The pi is actually an approximation derived from limits. But that method itself introduced compounding errors.</summary>
-<br>
-<p>Archimedes approximated the circumference 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. By bisecting the angles (splitting them in half), he turned the hexagons into 12‑gons, then 24‑gons, all the way to 96‑sided shapes. This allowed him to calculate the perimeter of these shapes in terms of the diameter using only straight lines and Pythagoras’ theorem.
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-<p>Archimedes approximated the circumference 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. By bisecting the angles (splitting them in half), he turned the hexagons into 12‑gons, then 24‑gons, all the way to 96‑sided shapes. This allowed him to calculate the perimeter of these shapes in terms of the diameter using only straight lines.
+The pi is actually an approximation derived from limits. But that method itself introduced compounding errors.
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+Archimedes approximated the circumference 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. By bisecting the angles (splitting them in half), he turned the hexagons into 12‑gons, then 24‑gons, all the way to 96‑sided shapes. This allowed him to calculate the perimeter of these shapes in terms of the diameter using only straight lines.
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 Observing how the difference between the two polygonal perimeters — one inside the circle, one outside — became smaller, 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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-But that method depends entirely on the assumption that the relationship between a chord and its half-angle is a fixed, linear progression that never "drifts" as the angles get smaller.
+But that method depends entirely on the assumption that the relationship between a chord and its half-angle is a fixed, linear progression that never deviates as the angles get smaller.
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 He started with the hexagon (30° half-angle), where the sine is exactly 0.5 because it's half of an equilateral triangle side. That is a solid reference point. But every step after that — 15°, 7.5°, 3.75° — relies on the Half-Angle Formula.
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-If the angle bisection formula is even slightly "off" for non-standard angles, then by the time Archimedes reached the 96-gon, that error would have compounded.
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-Furthermore, inscribed and circumscribed describe only the position of the polygon relative to the circle — vertices on the circle, or sides touching it.  
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-Traditional geometry adds the claim that the perimeter of the circumscribed polygon must be longer than the circumference.
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-But that claim depends on assumptions about curvature that fail once the polygon’s internal angles flatten toward 180°.
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-A simple physical model exposes this flaw: two rigid plates forming a narrow V, closed by a straight lid that just fits. If we bend that lid into a curve, its ends can slip lower between the plates — even if the lid becomes slightly longer.
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-The curved path fits the same angular span with a greater length. This shows that ‘lying outside’ does not uniquely determine that a path is longer than the corresponding curve.
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-As the number of sides increases, the internal angles flatten toward 180°, nearing a straight line rather than a curve, and the polygon no longer reflects the circle’s curvature.  
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-For example, a 96‑gon has angles of 176.25°.
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-A line segment, no matter how short, is fundamentally different from a curve. A circle has constant curvature. A line segment has zero curvature. Treating the two as interchangeable is a category error disguised as approximation.
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-Conventional math ignores this qualitative difference, assuming that “close enough” is the same as “equal.”  
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-It assumes that more sides mean closer resemblance to a circle, hence the circle encloses the maximum possible area for a given perimeter (isoperimetric theorem).
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-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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-Some try to prove it via area relationships based on the pi. But that is problematic if the pi itself is the quantity under investigation.
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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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-When the circle’s area and circumference are calculated with the constant 3.2, it becomes clear that a flat angle encloses the area differently than a curve. This flips the script: the polygon can enclose equal area with the same perimeter. This overlooked disproportion shows that polygons do not approach the circle in every sense.
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-Archimedes pushed his method far beyond this curve‑aligned threshold — and the result is a recursive underestimate.
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-The traditional method of polygon approximation fails due to a fundamental divergence of shape that invalidates its own geometric ordering.  
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-The polygon method attempts to define the perfect circle using imperfect, flawed limits. The basic geometric ordering that the method relies on stops being valid once the internal angles of the polygon become too flat to meaningfully approximate curvature, making it unsuitable for determining the true circumference‑to‑diameter ratio of a circle.</p>
-<details>
-  <summary><h4>What we’re left with is not a proof, but a flawed approximation — one that has shaped centuries of geometry, but now deserves a closer, more rational reexamination.</h4></summary>
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-<p>To analyze it further, my 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.
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-Another overlooked aspect of the traditional method is the assumption that as the perimeters of the polygons approach the circumference with the increase of the number of sides, the ratio of the gaps between the arc and the vertices of the circumscribed polygon, and the sides of the inscribed polygon converge toward 1:1.
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-Analyzing the gaps of an isoperimetric equilateral triangle reveals that the ratio between the gaps flips compared to the in- and circumscribed triangles.
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-While the number of sides is only 3, the perimeter is equal to the circumference, yet the ratio flipped.
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-Rather than treating inscribed and circumscribed polygons separately and relying on assumptions about how their perimeter gaps behave as the number of sides increases, we introduce a creative and grounded condition: equal distance between the polygon’s sides, vertices, and the circle’s arc.
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-We begin with a strong geometric foundation: the area of a circle is exactly 3.2r². This gives us reason to suspect that the true circumference is 6.4r, not 2r×pi. To test this, we reframe the polygon approximation method.
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-This equidistance constraint allows us to calculate perimeters for polygons of various side counts (triangle, square, hexagon, 14-gon, 96-gon), each tuned to balance deviation symmetrically. The results show that:
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-- Perimeters are not proportional to the number of sides.
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-- The 14-gon already approximates the circle remarkably well.
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-- The 96-gon converges precisely to a circumference of 6.4, confirming the area-based ratio.</p>
-</details>
-</details>
-</section>
+If the angle bisection formula is even slightly inaccurate for non-standard angles, then by the time Archimedes reached the 96-gon, that error would have compounded.
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