Update index.html

Author: Gmac4247

index.html

@@ -2517,12 +2517,12 @@ <h4>Archimedes and the Illusion of Limits</h4>
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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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-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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 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: