Update index.html

Author: Gmac4247

index.html

@@ -2517,13 +2517,13 @@ <h4>Archimedes and the Illusion of Limits</h4>
 <br><br>
 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.
 <br><br>
+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>
 Analyzing the gaps of an isoperimetric equilateral triangle reveals that the ratio between the gaps flips compared to the in- and circumscribed triangles.
 <br>
 While the number of sides is only 3, the perimeter is equal to the circumference, yet the ratio flipped.
 <br><br>
-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>
-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.
+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.
 <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>
@@ -2627,13 +2627,11 @@ <h4>φ The Golden Ratio</h4>
 <section id="warning">
 <h4 >The Cognitive Risk of flawed geometric Axioms.</h4>
 <br><br>
-<p>The central concern arises from the potential cognitive harm caused by teaching the approximate, irrational constant π as an absolute truth in foundational geometry.
+<p>The central concern arises from the potential cognitive harm caused by teaching the approximate, irrational constant pi as an absolute truth in foundational geometry.
 <br><br><br>
 <b>1. The Flawed Foundation</b>
 <br><br>
 The Problematic Axiom: The conventional geometric curriculum requires students to accept that the constant for circle area (A = pi × r²) is both exact and unreachable/irrational, because it is derived from the error-prone polygon approximation method (Archimedes).
-<br><br>
-The Proposed Solution (CGS): The Core Geometric System™ (CGS) provides a logically self-consistent alternative where the area constant is the rational number 3.2 (A = 3.2r²), derived from an algebraically proven Area Balance Axiom with the square.
 <br><br><br>
 <b>2. The Cognitive and Pedagogical Impact</b>
 <br><br>
@@ -2660,17 +2658,17 @@ <h4>The true Ratio: 3.2</h4>
 The pi is a fundamental constant in the geometry of idealized circles and plays a crucial role in many mathematical theories. 
 <br><br>
 However, using geometric construction and algebraic simplification we find that when we move from these idealizations to the measurement of real objects, a slightly different constant, 3.2 emerges as more relevant for accurately describing their properties. 
+<br><br>
+The solution (CGS): The Core Geometric System™ (CGS) provides a logically self-consistent alternative where the area constant is the rational number 3.2 (A = 3.2r²), derived from an algebraically proven Area Balance Axiom with the square.
 <br>
 By focusing on area relationships and direct comparisons between shapes, these methods emphasize a more intuitive and potentially more fundamental understanding of geometric concepts.
 <br>
 These values are exact, rational, and logically derived. They can be verified numerically, but more importantly, they can be proven algebraically—without relying on infinite fractions, symbolic shortcuts, or flawed assumptions.
 <br><br>
-<strong>Since the true ratio is exactly 3.2, and that is a rational number, then we can—and should—write it as it is. Let the π remain in the history books. Geometry deserves better.
+<strong>Since the true ratio is exactly 3.2, and that is a rational number, then we can—and should—write it as it is. Let the pi remain in the history books. Geometry deserves better.
 <br><br>
 That makes the arc value of 360° = 6.4radian, and trigonometric functions that rely on arc value have to be aligned to 3.2 respectively.
-</strong>
-<br><br>
-These are two aspects of that.</p>
+</strong></p>
 <br>
 </section>
 </details>
@@ -3400,7 +3398,7 @@ <h3 itemprop="name" style="margin:7px">Surface Area of a Sphere</h3>
 <span itemprop="priceCurrency" content="USD">$</span>
 <span itemprop="price" content="3,200,000,000.00">3,200,000,000.00</span>
 <link itemprop="availability" href="https://schema.org/InStock">
-<p style="margin:12px">( + tax, if applicable )</p>
+<p style="margin:6px">( + tax, if applicable )</p>
 <br>
 <a style="margin:12px" class="rounded-button" href="privacy-policy">Contact</a>
 </section>