Update index.html

Added expandable callapse blocks.

Author: Gmac4247

index.html

@@ -950,15 +950,12 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 <p style="margin:12px;">(2 × 10³) + (0 × 10²) + (2 × 10¹) + (5 × 10⁰) = 2025</p>	
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-<div>
-<p style="margin:12px;"><strong>Disclaimer</strong>
-<br>
+<details>
+  <summary><strong>Disclaimer</strong></summary>
+<p style="margin:12px;">Basic Geometry uses Arab numbers and the decimal system, but there are other numbers and numeral systems in use.</p>
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-Basic Geometry uses Arab numbers and the decimal system, but there are other numbers and numeral systems in use.</p>
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-<br>
-<div>
-<p style="margin:12px;">Example #2 - The year 2025 in the binary system:</p>
+Example #2 - The year 2025 in the binary system:</p>
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@@ -1020,13 +1017,14 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
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 <p style="margin:12px;">(1 × 2¹⁰) + (1 × 2⁹) + (1 × 2⁸) + (1 × 2⁷) + (1 × 2⁶) + (1 × 2⁵) + (0 × 2⁴) + (1 × 2³) + (0 × 2²) + (0 × 2¹) + (1 × 2⁰) = 2025</p>	
-</div>
-</div>
+</details>
 </section>
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+<br>
+<br>
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 <p style="font-size:160%;font-weight:bold;margin:7px">Trigonometry</p>
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@@ -2021,10 +2019,9 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 <strong style="font-size:160%;margin:7px">Area and circumference of a circle</strong>
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-<section>
-<p style="margin:12px;">For centuries, the circle has been a symbol of mathematical elegance—and π its most iconic constant. But beneath the surface of tradition lies a deeper question: Are the formulas we use truly derived from geometric logic, or are they inherited approximations dressed in symbolic authority?
-<br>
-<br>
+<details>
+  <summary><strong>For centuries, the circle has been a symbol of mathematical elegance—and π its most iconic constant. But beneath the surface of tradition lies a deeper question: Are the formulas we use truly derived from geometric logic, or are they inherited approximations dressed in symbolic authority?</strong></summary>
+<p style="margin:12px;">
 This article revisits the foundations of circle geometry, challenging long-held assumptions and offering a more exact, algebraic alternative.
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@@ -2215,7 +2212,7 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 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.
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-</section>
+</details>
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@@ -2405,8 +2402,8 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
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-<section>
-<strong style="margin:12px;">Proof</strong>
+<details>
+  <summary><strong>Proof</strong></summary>
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@@ -2777,7 +2774,7 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
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 The quadrant method proves that the area of a circle equals exactly 3.2 × radius², thus ruling out the validity of the π.
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-</section>
+</details>
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@@ -3060,11 +3057,9 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
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-<section>
-<p style="margin:12px;"><strong>Proof</strong>
-<br>
-<br>
-Expand the term (r - x)²:
+<details>
+  <summary><strong>Proof</strong></summary>
+<p style="margin:12px;">Expand the term (r - x)²:
 </p>
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 <math style="margin:12px;" xmlns="http://www.w3.org/1998/Math/MathML">
@@ -3251,11 +3246,9 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
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 Irrational or not, with an infinitesimally small thickness the circumference practically equals 6.4 × radius.
-</p>
-</section>
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-<p style="margin:12px;"><b>Conclusion: Time to Move On</b>
+<b>Conclusion: Time to Move On</b>
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 The π has served its symbolic purpose. But in geometry, clarity matters more than tradition. 
@@ -3269,6 +3262,7 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
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 These are two aspects of that.
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+</details>
 </section>
 </section>
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@@ -3354,12 +3348,12 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 <p style="margin:12px;" id="sphere-volume"></p>
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-<p style="margin:12px;">The edge length of the cube, which has the same volume as the sphere, equals the square root of the area of the square that has the same area as the sphere's cross section.
-<br>
-<br>
-The " V = 4 / 3 × π × radius³ "  estimate is widely used for the volume of a sphere, and it's a cornerstone of theoretical geometry. 
+<p style="margin:12px;">The edge length of the cube, which has the same volume as the sphere, equals the square root of the area of the square that has the same area as the sphere's cross section.</p>
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+<details>
+  <summary><strong>The " V = 4 / 3 × π × radius³ "  estimate is widely used for the volume of a sphere, and it's a cornerstone of theoretical geometry.</strong></summary>
+<p style="margin:12px;">
 It was estimated by comparing a hemisphere to the difference between the approximate volume a cone and a circumscribed cylinder.
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@@ -3378,6 +3372,7 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
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 If you're trying to calculate the volume of a physical ball or sphere for a practical purpose – whether it's for a science experiment, engineering, or any other real-world application – my empirically derived V = " cubic value of ( √( 3.2 ) × radius ) " formula offers a result that aligns more closely with what you would measure in the lab.
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+</details>
 </section>
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@@ -3778,6 +3773,7 @@ <h2 style="margin:6px;">SURFACE AREA OF A SPHERE</h2>
 </script>
 <p style="margin:12px;" id="cone-volume"></p>
 </div>
+</section>
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@@ -3952,6 +3948,7 @@ <h2 style="margin:6px;">SURFACE AREA OF A SPHERE</h2>
 </script>
 <p style="margin:12px;" id="frustum-cone-volume"></p>
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+</section>
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@@ -4029,6 +4026,7 @@ <h2 style="margin:6px;">SURFACE AREA OF A SPHERE</h2>
 </script>
 <p style="margin:12px;" id="cone-surface"></p>
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+</section>
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@@ -4642,8 +4640,10 @@ <h2 style="margin:6px;">SURFACE AREA OF A SPHERE</h2>
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-<section>
-<p style="margin:12px;">The commonly used base × height / 3 approximation for the volume of a pyramid was likely estimated based on two observations. 
+<details>
+  <summary><strong>
+The volume of a cone or pyramid is conventionally approximated as base × height / 3.</strong></summary>
+<p style="margin:12px;">The conventional approximation was likely estimated based on two observations. 
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 One is that the area of the mid-height cross section of a regular pyramid — of which's apex can be connected to the midpoint of the base with a perpendicular line — is exactly a quarter of a circumscribed solid's with the same base and height. 
@@ -4745,12 +4745,10 @@ <h2 style="margin:6px;">SURFACE AREA OF A SPHERE</h2>
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 No. Because it's not true in case of most other shapes.
-</p>
-</section>
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-<section>
-<p style="margin:12px;">The other idea is the cube dissection. 
+<br>
+The other idea is the cube dissection. 
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@@ -4848,7 +4846,7 @@ <h2 style="margin:6px;">SURFACE AREA OF A SPHERE</h2>
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 Also it's not just about the vertices, but the edges and the inner faces, too.
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-</section>
+</details>
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