Update dateModified and enhance π discussion

Updated the dateModified field and enhanced the content discussing the historical context and mathematical reasoning behind the value of π, including references to Archimedes and the Indiana Pi Act.

Author: Gmac4247

about.html

@@ -187,7 +187,7 @@
 },
 "dateCreated": "2024-08-31",
 "datePublished": "2024-08-31",
-"dateModified": "2025-10-16",
+"dateModified": "2025-11-08",
 "description": "About the context of the Core Geometric System ™, the best-established and most accurate framework to calculate area and volume.",
 "disambiguatingDescription": "Exact, empirically grounded and rigorously proven formulas over the conventional approximations.",
 "headline": "Introducing the Core Geometric System ™",
@@ -291,82 +291,181 @@ <h1 style="font-size:160%;margin:7px;">How Accurate Are The Conventional Geometr
 </p>
 <br>
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-<p style="margin:12px;">The constant relationship between a circle's circumference and its diameter has captivated mathematicians for millennia. 
-While its approximate value of 3.14, commonly denoted by the Greek letter π, is widely recognized today, the historical development of this concept is less understood.
-Ancient civilizations grappled with this geometric challenge, employing various methods to approximate this ratio. 
+<div>
+<p style="margin:7px;"><strong>Rethinking the Circle: A Logical Reexamination of Area and Circumference</strong>
+<br>
+<br>
+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>
-The Babylonians initially estimated it as 3, later they calculated with 3.125.
+This article revisits the foundations of circle geometry, challenging long-held assumptions and offering a more exact, algebraic alternative.
+</p>
+<br>
 <br>
+<div>
+<p style="margin:12px;"><b>Historical Approximations: Respect, Not Reverence</b>
 <br>
-A Greek mathematician is credited with refining these approximations through the method of inscribed and circumscribed polygons.
 <br>
+The constant relationship between a circle's circumference and its diameter has captivated mathematicians for millennia. While its approximate value of 3.14159…, commonly denoted by the Greek letter π, is widely recognized today, the historical development of this concept is less understood.
+Ancient civilizations grappled with this geometric challenge, employing various methods to approximate this ratio. They deserve credit for their mathematical ingenuity. But the fact that these methods were developed thousands of years ago should not shield them from scrutiny.
 <br>
-His approach was that the ratio between the perimeter and the diameter of a circle can be estimated by comparing the circumference of the circle to the perimeters of an inscribed and a circumscribed polygon. 
+<br>
+The verse of 1. Kings 7:23 in the Holy Bible suggests that some estimated it as 3. 
+<br>
+<br>
+Historical records suggest that ancient Babylonians initially calculated it as 3, later they used 3.125; Egyptians estimated it is ( 16 / 9 )² ~ 3.16.
+</p>
+</div>
+<br>
+<br>
+<div>
+<p style="margin:12px;"><b>Archimedes and the Polygonal Trap</b>
+<br>
+<br>
+The Greek Archimedes’ method for estimating the π is often celebrated as a triumph of geometric reasoning. 
+<br>
+<br>
+He began with a circle bounded by an inscribed and a circumscribed hexagon—figures whose properties can be described with exactness. From there, he increased the number of sides to 96, using trigonometry to approximate the perimeter.
 The polygons can be divided into triangles. The ratio between the legs of the triangles and their hypotenuses can be measured linearly.
 <br>
-That is where the pi divided by delta ( with delta = 1 ) π ~ 3.14 notation might originate from.
 <br>
-This method has several limitations. He tried to increase the accuracy by increasing the number of sides of the polygons. This approach cannot produce an accurate result. 
+But this method rests on a flawed assumption: that a circle maximizes area for a given perimeter. This is not universally true, and it introduces a logical error into the foundation of the approximation. Moreover, calculating the properties of a 96-gon involves rounding infinite fractions—errors that are multiplied 96 times, amplifying the imprecision.
 <br>
 <br>
-Where the perimeter-based estimation went wrong: 
+Calling the inscribed hexagon a “lower bound” is already questionable. Calling the 96-gon an “upper bound” is even more so. 
+</p>
 <br>
-- aside of that it's just an approximation instead of an exact calculation - 
+<details>
+  <summary><strong style="margin:12px;">The theoretical upper bound would be a polygon with the number of sides approaching infinity...</strong></summary>
+  <p style="margin:12px;">In that case the angles between the side and the diagonals approach a right angle. They never reach a right angle as the diagonals converge towards the center. 
 <br>
 <br>
-The in- and circumscribed polygons method seems logical, but there's a catch. It's based on the assumption that the circle maximizes the area with a given circumference. That assumption is false. It's obviously true in the case of an isoperimetric triangle and a square, but it becomes less and less obvious with the increase of the number of the polygon's sides. Until eventually it is not even true. 
+If we relate the arc of a corresponding slice of an isoperimetric circle, the length of the arc equals the side in question. So the chord related to the arc is shorter than the side. If we want to place the arc with the chord so that it touches both diagonals, it has to be within the polygon. With the curvature of the arc becoming decreasingly distinctive, it doesn't bulge beyond the side. Eventually it will not even touch the side. 
 <br>
 <br>
-Imagine a side of a polygon with a number of sides approaching infinity. 
+Hence the polygon with the same number of sides, which circumscribed the circle is smaller, so its perimeter is shorter than the circle.
+</p>
+</details>
 <br>
+<p style="margin:12px;">The perimeter of the circumscribed polygon that was believed to be an overestimate of the circumference was practically an underestimate of it. 
 <br>
-The angles between the side and the diagonals approach a right angle. They never reach a right angle as the diagonals converge towards the center.  If we relate the arc of a corresponding slice of an isoperimetric circle, the length of the arc equals the side in question. So the chord related to the arc is shorter than the side. If we want to place the arc with the chord so that it touches both diagonals, it has to be within the polygon. With the curvature of the arc becoming decreasingly distinctive, it doesn't bulge beyond the side. Eventually it will not even touch the side. Hence the polygon with the same number of sides, which circumscribed the circle is smaller, so its perimeter is shorter than the circle. 
 <br>
+Hence the value of the π lies between two underestimates. What we’re left with is not a proof, but a layered approximation.
 <br>
-The perimeter of the circumscribed polygon that was believed to be an overestimate of the circumference was practically an underestimate of it. 
 <br>
+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 is numerically valid, it bypasses the geometric logic that defines area: the comparison to a square.
+</p>
 <br>
-Hence the value of the π lies between two underestimates.
+<br>
+</div>
+<div>
+<p style="margin:12px;"><b>The Symbol π: A Linguistic Shortcut</b>
 <br>
 <br>
-The same coefficient was used to calculate the ratio between the area and the squared radius of a circle.
+The symbol π was introduced because the true ratio—approximately 3.14159…—is an infinite fraction. Since we can’t write all its digits, we needed a symbol. But this symbol has taken on a life of its own.
 <br>
 <br>
-Despite these early advances, a precise, universally accepted value of this constant remained elusive for centuries. 
+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 ). 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.
 <br>
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-Being uncertain about its numeric value and how to calculate it, it was comfortable to denote it by a sign in the equations.
+It was not until the 18th century that the symbol π, popularized by the mathematicians of the time, gained widespread acceptance. 
+</p>
+</div>
 <br>
 <br>
-It was not until the 18th century that the symbol π, popularized by the mathematicians of the time, gained widespread acceptance.
+<div>
+<p style="margin:12px;"><b>∫ Calculus: Summary, Not Source</b>
 <br>
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 Several complex formulas were introduced by different mathematicians, aimed at more accurately estimating this ratio, based on a theoretical polygon with an infinite number of sides. 
 <br>
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-All of the above mentioned approximation methods have one thing in common. They are estimating the perimeters of polygons and do not account for the curved shape of the circle.
+All of the above mentioned approximation methods have two things in common: 
 <br>
 <br>
-Historical records suggest that a legislative process took place in 1897, Indiana, USA, known as House Bill 246, or Indiana Pi Act, aiming to replace the numeric value 3.14 by 3.2.
+- They assume that the circle maximizes the area with a given perimeter, and
 <br>
 <br>
-Unfortunately, the exact details of the proposed method in the Indiana Pi Bill are somewhat obscure and have been interpreted differently by various accounts. 
+- they estimate the perimeters of polygons and do not account for the curved shape of the circle.
 <br>
 <br>
-The π is a fundamental constant in the geometry of idealized circles and plays a crucial role in many mathematical theories. 
+Modern calculus summarizes these approximations with elegant notation, such as:
+</p>
+<br>
+<math style="margin:12px;" xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+  <mrow>
+    <mi>C</mi>
+    <mo>=</mo>
+    <msubsup>
+      <mo>&#x222B;</mo> <!-- Integral symbol -->
+      <mn>0</mn>
+      <mrow>
+        <mn>2</mn>
+        <mi>&#x03C0;</mi> <!-- pi -->
+      </mrow>
+    </msubsup>
+    <mi>r</mi>
+    <mo>&#x2062;</mo> <!-- Invisible times -->
+    <mi>d</mi>
+    <mi>&#x03B8;</mi> <!-- theta -->
+  </mrow>
+</math>
+<br>
+<br>
+<p style="margin:12px;">
+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. 
+<br>
+<br>
+Calculus can be a useful mathematical tool, but it calling it exact is a bold statement. 
+<br>
+<br>
+It can be exact with exact limits and basic operations, but if those are given then they can be calculated directly without calculus.
 </p>
 </div>
 <br>
+<br>
 <div>
-<p style="margin:12px;">My work, however, suggests 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. 
+<p style="margin:12px;"><b>The golden ratio</b>
+<br>
+<br>
+Some relate the numeric value of 3.14… to the so-called “golden ratio” of ( √5 + 1 ) / 2.
+<br>
+<br>
+4 / √( ( √5 + 1 ) / 2) ~ 3.1446
 <br>
 <br>
-By focusing on area relationships and direct comparisons between shapes, my method emphasizes a more intuitive and potentially more fundamental understanding of geometric concepts.
+That has no logical ties to the area nor the circumference of a circle.
 </p>
+</div>
 <br>
-<strong style="font-size:160%;margin:7px;">THE AREA OF A CIRCLE is defined by comparing it to a square since that is the base of area calculation.
-</strong>
+<br>
+<div>
+<p style="margin:12px;">
+<b>A Rational Alternative: 3.2</b>
+<br>
+<br>
+Historical records suggest that a legislative process took place in 1897, Indiana, USA, known as House Bill 246, or Indiana Pi Act, aiming to replace the numeric value 3.14 by 3.2.
+<br>
+<br>
+Unfortunately, the exact details of the proposed method in the Indiana Pi Bill are somewhat obscure and have been interpreted differently by various accounts. 
+<br>
+<br>
+The π 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>
+By focusing on area relationships and direct comparisons between shapes, the following method emphasizes a more intuitive and potentially more fundamental understanding of geometric concepts.
+<br>
+<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.
+</p>
+</div>
+<br>
+<br>
+<div>
+<strong style="font-size:160%;margin:7px;">THE AREA OF A CIRCLE is defined by comparing it to a square since that is the base of area calculation.</strong>
 <br>
 <div>
 <iframe title="Geometry puzzle" width="420" height="315" src="https://youtube.com/embed/-gvXLaLFXuM">
@@ -777,8 +876,7 @@ <h1 style="font-size:160%;margin:7px;">How Accurate Are The Conventional Geometr
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-<strong style="font-size:160%;margin:7px;">THE CIRCUMFERENCE OF A CIRCLE can be derived from the area algebraically.
-</strong>
+<strong style="font-size:160%;margin:7px;">THE CIRCUMFERENCE OF A CIRCLE can be derived from the area algebraically.</strong>
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     <img class="center-fit" src="circumference.jpg" alt="figure-circumference=6.4r" id="circumference">
@@ -1060,6 +1158,7 @@ <h1 style="font-size:160%;margin:7px;">How Accurate Are The Conventional Geometr
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+<div>
 <p style="margin:12px;">
 Irrational or not, with an infinitesimally small thickness the circumference practically equals 6.4 × radius.
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@@ -1087,6 +1186,7 @@ <h1 style="font-size:160%;margin:7px;">How Accurate Are The Conventional Geometr
 These are two aspects of that.
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+</div>
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@@ -1388,6 +1488,8 @@ <h2 style="margin:6px;">SURFACE AREA OF A SPHERE</h2>
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+</div>
+</div>
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