Highlight key geometric concepts in index.html

Added emphasis to key geometric concepts using strong tags.

Author: Gmac4247

index.html

@@ -3144,7 +3144,7 @@ <h4 style="margin:12px;">Archimedes and the Polygonal Trap</h4>
 He assumed that more sides mean closer resemblance to a circle. That was backed by the isoperimetric inequality 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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-But that assumption ignores a crucial geometric reality: as the number of sides increases, the internal angles of the polygon approach 180° — it is 180° - 360° / 96 = 176.25° in the case of a 96-gon —, nearing a straight line rather than a curve. 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.
+But that assumption ignores a crucial geometric reality: <strong>as the number of sides increases, the internal angles of the polygon approach 180° — it is 180° - 360° / 96 = 176.25° in the case of a 96-gon —, nearing a straight line rather than a curve. 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.</strong>
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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. 
@@ -3165,7 +3165,7 @@ <h4 style="margin:12px;">Archimedes and the Polygonal Trap</h4>
 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.
+<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.</strong> While the result of that method is algebraically valid, it bypasses the geometric logic that defines area: the comparison to a square.
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@@ -3179,7 +3179,7 @@ <h4 style="margin:12px;">The Symbol π: A Linguistic Shortcut</h4>
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 Technically, the circumference is a perimeter. So the ratio ( P / d ) ( perimeter over diameter ) became π / δ in Greek. With ( d = 1 ), we get ( π / 1 = π ). 
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-But this is not necessarily the ratio itself—it’s the notation of that ratio. That distinction matters. There was a ratio between circumference and diameter long before the Greeks studied it. We must not let their symbolic shortcut overwrite a more fundamental geometric truth.
+<strong>But this is not necessarily the ratio itself—it’s the notation of that ratio.</strong> That distinction matters. There was a ratio between circumference and diameter long before the Greeks studied it. We must not let their symbolic shortcut overwrite a more fundamental geometric truth.
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 It was not until the 18th century that the symbol π, popularized by the mathematicians of the time, gained widespread acceptance. 
@@ -3226,7 +3226,7 @@ <h4 style="margin:12px;">∫ Calculus: Summary, Not Source</h4>
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-But this is not a magical formula—it’s a symbolic summary of prior assumptions. Each notation should correspond to a real, logical property of the circle. Yet upon inspection, inconsistencies emerge. The formula doesn’t derive the circumference from first principles; it assumes it. 
+<strong>But this is not a magical formula—it’s a symbolic summary of prior assumptions.</strong> Each notation should correspond to a real, logical property of the circle. Yet upon inspection, inconsistencies emerge. The formula doesn’t derive the circumference from first principles; it assumes it. 
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 Calculus can be a useful mathematical tool, but calling it exact is a bold statement. 
@@ -3398,7 +3398,7 @@ <h3 style="margin:7px;">The volume of a sphere is defined by comparing it to a c
 <p style="margin:12px;">It is a cornerstone of theoretical geometry. 
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-It was estimated by comparing a hemisphere to the difference between the approximate volume a cone and a circumscribed cylinder.
+<strong>It was estimated by comparing a hemisphere to the difference between the approximate volume a cone and a circumscribed cylinder.</strong>
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 However, my work focuses on the actual volume of physical spheres as determined through direct measurement. 
@@ -3438,10 +3438,10 @@ <h3 style="margin:7px;">SURFACE AREA OF A SPHERE</h3>
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-<p style="margin:6px;">The conventional formula for the surface area of a sphere was allegedly developed from the " volume = 4 / 3 × π × radius³ " formula.
+<strong style="margin:6px;">The conventional formula for the surface area of a sphere was allegedly developed from the " volume = 4 / 3 × π × radius³ " formula.
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-The real formula for the surface area of a sphere is available for 3.2 billion USD. ( + tax, if applies )</p>
+The real formula for the surface area of a sphere is available for 3.2 billion USD. ( + tax, if applies )</strong>
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@@ -3981,7 +3981,7 @@ <h3 style="margin:12px;">The other idea is the cube dissection.</h3>
 The cube has 8 vertices, each pyramid has 5. Three pyramids have 3 × 5 = 15 in total.
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-Each vertex is a point that can't be split into 3 points. The other way around, 3 points can't be merged into 1 without distortion. 
+<strong>Each vertex of a real physical cube is a point that can't be split into 3 points without duplicating. The other way around, 3 points can't be merged into 1 without distortion.</strong>
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 If we dissect the cube, we need to designate each shared vertex to be a part of either one pyramid, or another. 
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@@ -4014,15 +4014,14 @@ <h3 style="margin:12px;">The other idea is the cube dissection.</h3>
 The vertices are the most obvious examples, but the same is true for the edges, the diagonals and the inner faces. 
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-Applied correctly, the cube dissection proves that the volume of a cone or a pyramid has to be larger than base × height / 3.
+<strong>The fact that the vertices of a real physical cube can't be split without duplicating and the vertices of the pyramids can't be merged into a single point without distortion proves that the conventional zero-dimensional point approach fails to accurately describe the physical reality.
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-The fact that the vertices of the 3 pyramids can't be merged into a single point without distortion proves that the so-called "calculus-based proofs" of the conventional formula are invalid. 
+While 1 / 3 is a reasonable approximation, the exact ratio is 1 / √8.</strong>
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-Also it's not just about the vertices, but the edges and the inner faces, too.
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-While 1 / 3 is a reasonable approximation, the exact ratio is 1 / √8.</p>
+The so-called "calculus-based proofs" of the conventional formula are invalid. 
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