Update README.md

Author: Gmac4247

README.md

@@ -1,11 +1,17 @@
-Providing the best-established and most accurate framework to calculate area and volume using the 3D coordinate system.
+# Core Geometric System 
 
+Providing the best-established and most accurate framework to calculate area and volume using the 3D coordinate system.
 
 <!---
-Gmac4247/Gmac4247 is providing the best-established and most accurate functional frameworkto calculate area and volume using the 3D coordinate system.
+Basic Geometry is providing the best-established and most accurate functional frameworkto calculate area and volume using the 3D coordinate system.
 --->
 
-Geometric Calculations Framework
+Key Points
+
+- Comparative Geometry: Using geometric relationships to derive areas and volumes.
+- Scaling and Proportions: Applying proportional relationships for accurate calculations.
+- Algebraic Manipulation: Simplifying equations to ensure coherence and precision.
+- Exact vs. Approximate Values: Recognizing the use of simpler constants for practicality while maintaining exact values for accuracy.
 
 1. Area of a Circle:
    - Compared to a square, using geometric properties and the Pythagorean theorem.
@@ -23,12 +29,6 @@ Geometric Calculations Framework
    - Compared to an octant sphere and a quarter cylinder.
    - Formula:  V = 3.2r²height/√8 
 
-Key Points
-
-- Comparative Geometry: Using geometric relationships to derive areas and volumes.
-- Scaling and Proportions: Applying proportional relationships for accurate calculations.
-- Algebraic Manipulation: Simplifying equations to ensure coherence and precision.
-- Exact vs. Approximate Values: Recognizing the use of simpler constants for practicality while maintaining exact values for accuracy.
 
 Applications
 
@@ -38,18 +38,46 @@ Applications
 - Optimization Problems: Logistics, resource management.
 - Quantum Computing: Algorithm development, simulation.
 
-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.
+
+How Accurate Are the Conventional Geometry Formulas?
+
+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.
+
+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 unit of measurement of the area is square units and the volume can be expressed as cubic units. 
+
+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. 
+
+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. 
+
+Other shapes are more challenging. The ratios are in the shapes; one just has to write them down algebraically.
+
+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. 
-A Greek mathematician is credited with refining these approximations through the method of in- and circumscribed polygons.
-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 in- and a circumscribed polygon. The polygons can be divided into triangles. The ratio between the triangles' legs and their hypotenuses can be measured linearly.
-That's where the pi/delta=3.14... denotation might originate from.
-This method has several limitations. He tried to increase the accuracy by increasing the number of the sides of the polygons. This approach cannot yield an accurate result. 
+
+A Greek mathematician is credited with refining these approximations through the method of inscribed and circumscribed polygons.
+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. The polygons can be divided into triangles. The ratio between the legs of the triangles and their hypotenuses can be measured linearly.
+That is where the pi/delta=3.14 notation might originate from.
+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. 
+
 The same coefficient was used to calculate the ratio between the squared radius and the area of a circle.
-Despite these early advancements, a precise, universally accepted value for this constant remained elusive for centuries. 
-With its value believed to be an infinite fraction, it seemed necessary to denote it with a sign in equations.
-It wasn’t until the 18th century that the symbol π, popularized by the mathematicians of the time gained widespread acceptance.
-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. 
+
+Despite these early advances, a precise, universally accepted value of this constant remained elusive for centuries. 
+With its value believed to be an infinite fraction, it seemed necessary 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.
+
+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. 
+
+All of the comparison methods mentioned above have one thing in common. They are estimating the perimeters of polygons and do not account for the curved shape of the circle.
+
+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.
+
+Unfortunately, the exact details of the proposed method in the Indiana Pi Bill are somewhat obscure and have been interpreted differently by various accounts. 
+
+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.
 
 
 The area of a circle is defined by comparing it to a square, as that’s the base of area calculation.