Update about.html

Author: Gmac4247

about.html

@@ -69,7 +69,7 @@
 
     </style>
 <meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0">
-<meta name="description" content="History and disapproval of the mathematical constant π; derivation of the properties of shapes from the area of a circle.">
+<meta name="description" content="History and detailed disapproval of the mathematical constant π; derivation of the properties of shapes from the area of a circle.">
 <meta name="author" content="type=PostalAddress,
   addressLocality=Szentendre,
   addressRegion=Hungary,
@@ -173,7 +173,7 @@
 },
 "dateCreated" :"2024-08-31",
 "datePublished":"2024-08-31",
-"dateModified" :"2025-01-28",
+"dateModified" :"2025-01-30",
 "description" :"History and disapproval of the mathematical constant π; derivation of the properties of shapes from the area of a circle.",
 "disambiguatingDescription": "Exact formulas. No pi.",	
 	"image":[
@@ -191,12 +191,41 @@
 "administrativeArea",
 "name" :"Szentendre, Hungary"},
 "mainEntity" : 
- {
- "@type": "LearningResource",
- "name" : "About-Basic Geometry",
+"@type": ["MathSolver", "LearningResource"],
+ "name" : "Exact Geometry",
  "inLanguage":"en-us",
-"url": "https://basic-geometry.github.io/about", 
-"usageInfo":"Learn about the history of the mathematical constant π, why it is wrong, and how the area and the circumference of a circle can be calculated."},
+"url": "https://basic-geometry.github.io/", 
+"usageInfo":"Learn about historical and exact geometric concepts.",
+ "potentialAction": [
+     "@type": "SolveMathAction",
+     "target": "/",
+     "mathExpression-input": "required name=radius default=1 name=angleOfRotation default=360",
+     "mathExpression-output": 
+     "angleOfRotation / 360 * 3.2 * Math.pow(radius, 2)", 
+     "image": "areaOfACircle.jpg",
+     "eduQuestionType": "equationSolving",
+     "name": "Area of a circle"
+   },
+{
+     "@type": "SolveMathAction",
+     "target": "/",
+     "mathExpression-input": "required name=radius default=1 name=angleOfRotation default=360",
+     "mathExpression-output": "angleOfRotation / 360 * 6.4 * radius", 
+     "image": "circumference.jpg",
+     "eduQuestionType": "equationSolving",
+     "name": "Circumference of a circle"
+   },
+{
+     "@type": "SolveMathAction",
+     "target": "/",
+     "mathExpression-input": "required name=radius default=1 name=angleOfRotation default=360",
+     "mathExpression-output": "angleOfRotation/360*(3.2^(1/2)*radius)^3", 
+     "image": "sphereAndCubeMarkup.jpeg",
+     "eduQuestionType": "equationSolving",
+     "name": "Volume of a sphere"
+   }
+]
+},
 "offers": {
       "@type": "Offer",
  "name": "Surface area of a sphere formula",
@@ -225,6 +254,7 @@
 },
 "sameAs" : "basic-geometry.pages.dev/about",
 "subjectOf": 
+[
  {
 "@type": "VideoObject",
 "description": "Geometry puzzle",
@@ -234,6 +264,17 @@
 "uploadDate":"2022-12-15",
 "contentUrl":"https://youtu.be/-gvXLaLFXuM"
        },
+ {
+"@type": "VideoObject",
+"description": "Determining the volume of a sphere",
+"duration": "PT1M26S",
+"name": "The sphere experiment",
+"thumbnailUrl":"https://i.ytimg.com/vi/7uxyLfh2B38/mqdefault.jpg",
+"uploadDate":"2024-04-19",
+"contentUrl":"https://youtu.be/7uxyLfh2B38"
+       }
+	]
+	},
 "thumbnail" :"android-chrome-256x256",
 "typicalAgeRange" :"12-102",
   "url": "https://basic-geometry.github.io/about",
@@ -252,10 +293,22 @@
 		</div>
 </div>
 <div>
-  <h1 style="margin:6px;">How Accurate Are The Conventional Geometry Formulas?</h1>
+  <h1 style="font-size:160%;margin:7px;">How Accurate Are The Conventional Geometry Formulas?</h1>
+<br>
+<br>
+ <p style="margin:12px;">Historically, Euclidean geometry has provided a framework for understanding and describing the physical world. It is based on axioms and postulates, leading to well-defined formulas for the calculation of areas and volumes of shapes such as circles and spheres.
 <br>
 <br>
- <p style="margin:12px;">The ratios are in the shapes; one just has to write them down algebraically.
+The concept of setting the square and the cube as the basis of the area and the volume calculation is well established and straightforward. Regardless of the shape of the measured object, the result is in square or cubic units, depending on if it is a plane or a solid shape. 
+<br>
+<br>
+In the case of the area of a triangle, it is an easy task because multiplying the base by the height gives a rectangle with an area exactly the double of the triangle. The square root of half of the area of the rectangle is the side length of the theoretical square that has the same area as the triangle. 
+<br>
+<br>
+In the case of the volume of a cuboid, it is a simple multiplication of the edges. The cubic root of the product of the edges is the edge length of the theoretical cube that has the same volume as the cuboid. 
+<br>
+<br>
+Other shapes are more challenging.
 <br>
 <br>
 The constant relationship between a circle's circumference and its diameter has captivated mathematicians for millennia. 
@@ -279,13 +332,17 @@ <h1 style="margin:6px;">How Accurate Are The Conventional Geometry Formulas?</h1
 <br>
 Several complex formulas were introduced by different mathematicians, aiming to more accurately estimate this ratio, based on a theoretical polygon with an infinite number of sides. 
 All the above mentioned comparison methods have one thing in common. They are estimating the perimeters of polygons and do not account for the curved shape of the circle.
- </p>
+<br>
+<br>
+The ratios are in the shapes; one just has to write them down algebraically.
+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>
+</p>
 </div>
 <br>
 <br>
 <br>
 <div>
-<p style="margin:12px;">The area of a circle is defined by comparing it to a square since that is the base of area calculation.
+<h2 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.</h2>
 <br>
 <br>
 The circle is cut to four quadrants, each placed with their origin on the vertices of a square.
@@ -692,20 +749,19 @@ <h1 style="margin:6px;">How Accurate Are The Conventional Geometry Formulas?</h1
 </p>
 <br>
 <br>
-</div>
 <br>
 <br>
 <div>
 <iframe width="420" height="315" src="https://youtube.com/embed/-gvXLaLFXuM">
 </iframe>
+</div>
+</div>
 <br>
 <br>
 <br>
 <br>
-</div>
 <div>
-<p style="margin:12px;">
-The circumference of a circle is derived algebraically from its area by subtracting a theoretical circle with a ray shorter than the ray of the actual circle by the theoretical width of the circumference.
+<h3 style="font-size:160%;margin:7px;">The circumference of a circle is derived algebraically from its area by subtracting a theoretical circle with a ray shorter than the ray of the actual circle by the theoretical width of the circumference.</h3>
 <br>
 <br>
 The x represents the width of the circumference, which is just theoretical, hence a very small number.
@@ -958,46 +1014,51 @@ <h1 style="margin:6px;">How Accurate Are The Conventional Geometry Formulas?</h1
 <br>
 <br>
 <div>
-<p style="margin:12px;">The volume of a sphere can also be derived from the area of a circle as <math xmlns="http://www.w3.org/1998/Math/MathML" >
+<h4 style="font-size:160%;margin:7px;">The volume of a sphere is defined by comparing it to a cube, since that is the base of volume calculation.
+</h4>
+<br>
+<div>
+ <iframe width="420" height="315" src="https://youtube.com/embed/7uxyLfh2B38">
+ </iframe>
+</div>
+<br>
+<p style="margin:12px;">
+Just as the volume of a cube equals the square root of its cross section cubed - <math xmlns="http://www.w3.org/1998/Math/MathML" >
 <mrow>
- <mi>V</mi>
-      <mo>=</mo>
-      <msup>
+<mi>V</mi>
+<mo>=</mo>
+<msup>
 <mrow>
 <mo>(</mo>
 <msqrt>
-<mn>3.2</mn>
+<mi>Area</mi>
 </msqrt>
-<mo>×</mo>
-<mi>r</mi>
 <mo>)</mo>
 </mrow>
 <mn>3</mn>
 </msup>
 </mrow>
-</math>
-</p>
-<br>
-<br>
-<p style="margin:12px;">
-The volume of a cone can be derived from the volume of a sphere as <math xmlns="http://www.w3.org/1998/Math/MathML" >
+</math> -,
+so is the volume of a sphere equal to the area of its cross section cubed.
+<br><br>
+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>
+<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <mrow>
-<mi>V</mi>
+ <mi>V</mi>
 <mo>=</mo>
-<mfrac>
+<msup>
 <mrow>
+<mo>(</mo>
+<msqrt>
 <mn>3.2</mn>
-<msup>
-<mi>r</mi>
-<mn>2</mn>
-</msup>
+</msqrt>
 <mo>×</mo>
-<mi>H</mi>
+<mi>r</mi>
+<mo>)</mo>
 </mrow>
-<msqrt>
-<mn>8</mn>
-</msqrt>
-</mfrac>
+<mn>3</mn>
+</msup>
 </mrow>
 </math>
 </p>
@@ -1007,8 +1068,8 @@ <h1 style="margin:6px;">How Accurate Are The Conventional Geometry Formulas?</h1
 <br>
 <br>
 <div>
-<p style="margin:12px;">
-Using the same model, in which we were able to find a direct relationship between the radius of the circle and the side length of the square by ensuring that the overlaps equal the unfilled space, 
+<h5 style="font-size:160%;margin:7px;">Disapproval of the mathematical constant π</h5>
+<p style="margin:12px;">Using the same model, in which we were able to find a direct relationship between the radius of the circle and the side length of the square by ensuring that the overlaps equal the unfilled space, 
 and the radius of the circle equals <math xmlns="http://www.w3.org/1998/Math/MathML" >
 <mrow>
 <msqrt>