m³'Pi and the Cube, with the Cylinder, Sphere, and Cone are a Universal Unit and the Foundation of Physical Geometry
by Joseph Schuman Mathematical Analyzer
Since the beginning of time Pi has been regarded as the ratio between the diameter and the circumference; which mathematically is correct. Since then, academia and individuals have spend an enormous amount of time trying to outdo themselves in working out how many numerical digits Pi has, or formulating mathematical series to arrive at Pi's numerical value: its limit here will be 3.14159. But no further consideration was given to it, as to its essence and myraid of meanings in geometry.
I will proceed by first illustrating the four geometry equations as they appear in the textbooks, and Pi's interaction with them. All series begin from zero to one; 1"inch per side, radius .5"inch, height 1"inch.
Equation 1 Equation 2 Equation 3 Equation 4
Cube Cylinder sphere Cone
C³ 1 inch³ Π r² h 4 Π r³/3 Π r²h/3
= 1 cubic inch = .7853981634 = .523598776 = .2617993878
cubic inch cubic inch cubic inch
Another method to derive the answers to the above geometries, and understand the mathematical sequence that binds them, is by first changing the Cube into a cylinder by eliminating the four (4) sides of the cube by dividing the four (4) by Pi(Π), and deriving the universal constant 1.273239545, which is than divided into the one inch cube.
Equation 5
1 inch³/(4 sides/pi(Π) = 1.273239545
= .7853981634 cubic inch cylinder
dividing the cylinder by 1.5 to derive the sphere.
= .523598776 cubic inch sphere
dividing the sphere by 2 to derive the cone.
= .2617993878 cubic inch cone
The Pi in the cylinder and cone equations 2&4 are not ratios. A ratio is beteen two numbers not three. The Pi in the sphere equation 3, is also not a ratio. The divisor 3 in the sphere equation factored, is actually 2 times 1.5. The 2 is divided into the 4, making the equation 2 Π r³ which is the volume of a cylinder, and the 1.5 is the volume difference between the cylinder and sphere of the same height, width, and depth; a sphere within a cylinder.
This was correctly calculated twenty-two hundred years ago by the Greek Archimedes, who regarded the sphere within the cylinder mathematics as his greatest mathematical achievement, and after he was slain by a Roman soldier the people of Syracuse embossed on his burial tablet the spear within th cylinder, to immortalize this achievement.
But in the seventeenth century, the mathematician Bonaventura Cavalier, mistakenly doubled the volume of the cylinder in the sphere equation.
V = 2(vol.of cylinder - vol.of cube)
= 2 (Π r³ - Π r³/3)
= 4 Π r³/3
The above equation is in An Introduction to the History of Mathematics, by Howard Eves, Rinehart and Company, 1956. Its apparent that Cavalier did not understand what binds the four geometries and Pi as a universal unit, otherwise he would not have added the three (3) in equations 3&4 (sphere and cone). These equations shoud read,
Cylinder = Π/4 x 4 x r² x h Volume of cylinder
Cube of cylinder = 4 x r² x h Dividing the Cube of the cylinder by the universal
constant 1.273239545 the volume of the cylinder are
obtained. This also holds for the sequence of the sphere
and cone.
Sphere = Π/6 x 8 x r³ Volume of sphere
Cube of sphere = 8 x r³
Cone = Π/12 x 4 x r² x h Volume of cone
Cube of cone = 4 x r² x h
The essence of Pi is that the physical configurations of the cylinder, sphere, and cone are derived from the universal volume of Pi. That being, 1/4 of the volume of Pi is a cylinder. 1/6 of the volume of Pi is a sphere. And, 1/12 of the volume of Pi is a cone. This the foundation. All three geometries, the cylinder, sphere, and cone are emitted from within the body of the volume of Pi, and not from anywhere else.
Totaling the numerical volume sum that Pi has assigned to these geometries, that being,
1/4 the volume of Pi = .7853981634 cylinder
1/6 the volume of Pi = .523598775 sphere
1/12 the volume of Pi = .261799387 cone
The total volume sum of the above geometries is 1.570796327, which is exactly 1/2 of the volume of Pi. It should be apparent that Pi has a myraid of meanings, physical as well as a ratio.
Viewing Pi and the Cube in an anthropocentric manner, as humans always tend to do, the Cube is the Patriarch and Pi the matriarch of geometry, who controls and brings forth the universal geometric configurations.