Update index.html

Author: Gmac4247

index.html

@@ -205,7 +205,7 @@
 },
 "dateCreated": "2019-01-11",
 "datePublished": "2020-01-11",
-"dateModified": "2025-11-20",
+"dateModified": "2025-11-24",
 "description": "Introducing the best-established and most accurate framework to calculate area and volume.",
 "disambiguatingDescription": "Introducing exact, empirically grounded and logically consistent formulas over the flawed conventional approximations.",
 "headline": "Introducing the Core Geometric System ™",
@@ -3148,12 +3148,24 @@ <h4 style="margin:12px">Archimedes and the Polygonal Trap</h4>
 <br>
 <br>
 <p style="margin:12px">The Greek Archimedes’ method for estimating the π is often celebrated as a foundational triumph of geometric reasoning.
+</p>
 <br>
+<div>
+<details>
+  <summary style="margin:12px">But that method itself introduced compounding errors. These include:
 <br>
-But that method contains critical flaws.
 <br>
+- Misapplied isoperimetric logic beyond its valid range
+<br>
+<br>
+- Possible inaccuracies in calculating the properties of the 96-gon via angle bisecton
 <br>
-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. 
+<br>
+- Rounding errors of infinite fractions, and any other inaccuracies amplified over 96 iterations 
+  </summary>
+<br>
+<br>
+<p style="margin:12px">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. 
 <br>
 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.
 <br>
@@ -3166,22 +3178,39 @@ <h4 style="margin:12px">Archimedes and the Polygonal Trap</h4>
 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. 
 <br>
 <br>
-Thus his final result of 3.14... lies between two underestimates. The method itself introduced compounding errors. These include:
+Thus his final result of 3.14... lies between two underestimates. <br>
 <br>
 <br>
-- Misapplied isoperimetric logic beyond its valid range
+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.
 <br>
 <br>
-- Possible inaccuracies in calculating the properties of the 96-gon via angle bisecton
+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.
 <br>
 <br>
-- Rounding errors of infinite fractions, amplified over 96 iterations 
+Instead, 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 2πr. To test this, we reframe the polygon approximation method.
 <br>
 <br>
-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.
+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.
+<br>
+<br>
+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:
+<br>
+<br>
+- Perimeters are not proportional to the number of sides.
+<br>
+<br>
+- The 14-gon already approximates the circle remarkably well.
 <br>
 <br>
-<strong>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 result of that method is algebraically valid, it bypasses the geometric logic that defines area: the comparison to a square.</strong>
+- The 96-gon converges precisely to a circumference of 6.4, confirming the area-based ratio.
+<br>
+<br>
+This method upgrades the classical approach by replacing inherited assumptions with geometric conditions — and aligns the approximation process with the true nature of the circle.
+</p>
+<br>
+</details>
+</div>
+<p style="margin:12px"><strong>Archimedes' 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 result of that method is algebraically valid, it bypasses the geometric logic that defines area: the comparison to a square.</strong>
 </p>
 </section>
 <br>