**See previous years of eclectic work/thoughts at; www.geometry-mass-space-time.com**

**CONTENT**

**Methods - For Finding The Area of A Circle**

**Archimedes Proposition 1, Method 5; Area of a Circle + Sixth & Seventh Methods**

**Perfect Symmetry Of A Circle**

**What Is The Shortest Distance Between Two Points**

**Calculating The Area Of A Ring**

**From The Cube To It's Sphere**

**12 Steps To The Sphere**

**The Area And Volumes Of Symmetrical Ovals And Ovoids**

**The Number Of Degrees To A Sphere**

**The Area Of 1 Degree, Of A 120 cm Diameter Sphere**

**22/7**

**Irrational 3.14159**

**Non-linear Geometry of A Natural Circle**

**Minus Pi In Black And Yellow**

**Compass And Lines**

**Bi-Radial Arithmetic Of The Circle And Square**

**Context Of A Degree**

**EMAIL:**

**unialphaomega@hotmail.com**

**It is notable/interesting that during the four years that www.Geometry-Mass-Space-Time.com has been on the www, and despite thousands of academics and others having logged on, in 132 nations and in all States of the USA; only two individuals have made contact, and none in regard to this page.**

**One is left to draw ones own conclusions, in regard to human nature.**

**Quote**

*Stuart Chase 1888 - 1985*

*For those who believe no proof is necessary -*

*For those who do not believe, no proof is possible.*

*I would add;*

*For those who have seen it over and over again repetitively; but have failed to both see it and recognize it for what it actually is; due to their obliviousness or unwillingness to be concerned - to think - to concentrate;*

*If and when realization does finally dawn, their response is always "*

*I Knew That!".*

**ARCHIMEDES 287 - 212 BC**

**Beginning with a square, with each line measuring 120 cm**

**Take one line of 120 cm, and consider it to be a diameter line**

**Multiply the diameter line by 3, and the circle is 360 cm in length**

**Multiply the diameter line by 4, and the square of the diameter is 480 cm in length**

**Therefore it follows.**

**1. As every circle has 360 degrees to its length, each degree is 1 cm in length.**

**2. As each side of the square is 120 cm in length; 3 diameters is 3/4 of the 480 cm perimeter length.**

**3. As the 360 cm circle is 3/4 the of the 480 cm perimeter; the area of the circle will be, 3/4 of the area to the square.**

__PROOF's__

**METHODS - FOR FINDING THE AREA OF A CIRCLE**

1st Method:

**14, 400 sq cm divide by 4 = 3, 600 sq cm multiplied by 3 =**

__10, 800__sq cm to the circle = 3/4 of the area to the overall square.**2nd Method:**

**60 cm Radius squared = 3, 600 sq cm, multiplied by 3 =**

__10, 800__sq cm to the circle = 3/4 of the area to the overall sq.**Which concurs with the result of the 1st method**

**3rd**

__Ancient Sumerian Method__(1,500 + years BC):**Circumference length of 360 cm, multiplied by itself = 129, 600 sq cm, divide by 12 =**

__10. 800__sq cm to the area of the circle = 3/4 of the area to the overall square.**Which concurs with the result of the 1st & 2nd method**

4th Method as per Diagram

4th Method as per Diagram

60 cm Radius squared = 3, 600 sq cm, divide by 4 = 900 sq cm; multiply 900 sq cm by 3 = 2, 700 sq cm to 3/4 of the area to the square of the radius.

60 cm Radius squared = 3, 600 sq cm, divide by 4 = 900 sq cm; multiply 900 sq cm by 3 = 2, 700 sq cm to 3/4 of the area to the square of the radius.

**Multiply 2, 700 square cm by 4 =**

__10, 800__**sq cm to the circle = 3/4 of the area to the overall square;**

**And each corner of the square, has an area of 900 sq cm.**

**Which concurs with the results of the 1st, 2nd & 3rd methods**

**ARCHIMEDES PROPOSITION 1 - METHOD 5**

**AREA OF A CIRCLE**

**Proposition 1**.

**The area of any circle is equal to a right-angle triangle in which one of the sides about the triangle is equal to the radius, and the other to the circumference of the circle.**

**5. Fifth Method: Archimedes Triangle for finding the area to a circle**

**In the above diagram using a 12 x 12 square (144 squares) rather than a 120 x 120 cm square.**

**It can be seen that if we give the vertical a circumferential length three times that of the circles 12 square diameter length = 36 squares.**

**And then multiply the 6 square radius of the circle by the 36 square circumferential length we have a rectangle containing 216 squares, which when we then divide by 2, in order to obtain the exact area of the circle it is 108 squares, and exactly three quarters of the 12 x 12 squares' 144 square area.**

**Therefore had we had used a 120 x 120 cm square, rather than a 12 x 12 square; the area of the circle would be 1, 800 sq cm**

**Which concurs with the results of the 1st, 2nd, 3rd, & 4th methods**

**Therefore Archimedes proposition 1, is proven to be correct.**

6. Sixth Method: "

6. Sixth Method: "

__Applicable only to__" the area to a circle, with a 12 square length diameter

12 square/

12 square/

__right angle__length diameter x 3 = a 36 square length circumference

36 square length circumference x 3 = 108 square area

36 square length circumference x 3 = 108 square area

And each one of the 36 square lengths of circumferential length, is equal to 10 degrees of the 360 degrees to the circles circumferential length

And each one of the 36 square lengths of circumferential length, is equal to 10 degrees of the 360 degrees to the circles circumferential length

**Therefore each 1/10 length; of each of the 36 square lengths, is equal to a 1 degree length of the 360 degree length of the circle.**

**And had we used a 120 square diameter length rather than a 12 square diameter length; the circles area would have been 1, 800 sq cm.**

**Which concurs with the results of the 1st, 2nd, 3rd, 4th, and 5th methods.**

**7. Seventh Method: "**

__Applicable only to__" the area to a circle, with a 10 square (right angle/length diameter.

10 square length right angle x 10 square length right angle = 100 square area to the square, of the circle.

10 square length right angle x 10 square length right angle = 100 square area to the square, of the circle.

10 square length diameter x 3 = a 30 square circumferential length.

10 square length diameter x 3 = a 30 square circumferential length.

30 square length circumference x 2.5 (Not 3) = 75 (%) square area to the circle

30 square length circumference x 2.5 (Not 3) = 75 (%) square area to the circle

In sum

In sum

3 x the length of any straight line or right angle is a circumference.

3 x the length of any straight line or right angle is a circumference.

4 x the length of any straight line or right angle is a

4 x the length of any straight line or right angle is a

**perimeter.**

And given any length of straight line that is used as a diameter; the resultant circumferential length of the circle; will always be three quarters that of the straight lines perimeter length.

And given any length of straight line that is used as a diameter; the resultant circumferential length of the circle; will always be three quarters that of the straight lines perimeter length.

Which concurs with the results of the 1st, 2nd, 3rd, 4th, 5th, & 6th methods.

Which concurs with the results of the 1st, 2nd, 3rd, 4th, 5th, & 6th methods.

**Unlike Pi which is not a measurement nor is it approximate to anything, because unlike scales of measurement which are founded in reality; Pi is an improper fraction of an imperial/natural number, that has been decimalised/decimated into a permanent state of infinite irrationality.**

**And the decimal (10) based number/unit system of weights, measures, and money of itself; is an abstract concoction of "valueless/Abstract numbers" developed in the realms of dishonesty, cunning, and Machiavellian fraudulences**

*(Modern day Propaganda - Marketing - Advertising).*

That were inherent to and universally practised 2000 years ago, within the Greco

That were inherent to and universally practised 2000 years ago, within the Greco

*(Merchant)*- Roman*(Fascist-Military)*Empire*(Defined at the Temple of Jerusalem)*;**Which have subsequently evolved over the past two millennia, into becoming the universally wide spread and virulently more destructive; Machiavellian scheming of "Banks - Retailers - Corporations - Political Parties";**

**W**

**ho have gradually - insidiously - surreptitiously, conned the gullible masses into believing over the millennia;**

**That Roman Fascist Capitalism**

*(Privatisation of the common wealth/property/work/money/resources/public purse/taxes)*, equates with Democracy; when in truth it has always been "since the sacking, and the genocide of Carthage";**Constantine's Roman Fascistic Wolf - Universal Capitalistic Empire, hiding in Shepherd's Clothing.**

**PERFECT SYMMETRY OF A CIRCLE**

**A circular protractor has**

180 Degrees to its Diameter

360 Degrees to its Circumference

Therefore by giving 180 Centimeters to the length of Diameter

The Area to the "Square" of the Diameter is

32, 400 "Square Centimeters"

Rotate the 90 degree radius 360 Degrees; and the Area to the "Circle" of the Diameter is;

32, 400 "Circumnavigation Degrees"

Proving that the Circumferential length to a Circle has an "Exact Ratio" to its Radius (radii) length, and hence its Diameter length.

Therefore the "

Which is reinforced by the following; which serve to confirm that a Circles circumference length far from being irrational;

180 Degrees to its Diameter

360 Degrees to its Circumference

Therefore by giving 180 Centimeters to the length of Diameter

The Area to the "Square" of the Diameter is

32, 400 "Square Centimeters"

Rotate the 90 degree radius 360 Degrees; and the Area to the "Circle" of the Diameter is;

32, 400 "Circumnavigation Degrees"

Proving that the Circumferential length to a Circle has an "Exact Ratio" to its Radius (radii) length, and hence its Diameter length.

Therefore the "

*irrational/non-ratio*" formula of Pi cannot be considered or said to be; the ratio of a Circles circumference length, to Diameter length.Which is reinforced by the following; which serve to confirm that a Circles circumference length far from being irrational;

Is perfectly symmetrical

90⁰/cm radius x 360⁰ rotation = 32, 400 circumnavigation degrees

60⁰/cm radius x 360⁰ rotation = 21, 600 circumnavigation degrees

30⁰/cm radius x 360⁰ rotation = 10. 800 circumnavigation degrees

10⁰/cm radius x 360⁰ rotation = 3, 600 circumnavigation degrees

This is why a circles circumference length, can be equally subdivided by any "whole number", into that number of exactly equal, and identical parts of the circle.

Is perfectly symmetrical

90⁰/cm radius x 360⁰ rotation = 32, 400 circumnavigation degrees

60⁰/cm radius x 360⁰ rotation = 21, 600 circumnavigation degrees

30⁰/cm radius x 360⁰ rotation = 10. 800 circumnavigation degrees

10⁰/cm radius x 360⁰ rotation = 3, 600 circumnavigation degrees

This is why a circles circumference length, can be equally subdivided by any "whole number", into that number of exactly equal, and identical parts of the circle.

Simply:

Simply:

**360⁰ ÷ by the whole number of 360 = 360 equal and identical parts to the Circles integrity.**

**WHAT IS THE SHORTEST DISTANCE BETWEEN TWO POINTS**

**In the first chapter we used a three to one ratio of circumferential length, to that of diameter length, in order to find the exact area of a circle.**

**All seven methods used inclusive of the ancient Sumerian method, and Archimedes first proposition; served to prove by each giving exactly the same mathematical result, that;**

**1. A circles circumferential length is indeed, three times that of the circles diameter length.**

**2. The length of circumference to a circle, is indeed three quarters that of the square of its diameter length.**

**3. The area of a circle, is indeed three quarters that of the square of its diameter length.**

**90⁰/cm radius x 360⁰ rotation = 32, 400 circumnavigation degrees**

**60⁰/cm radius x 360⁰ rotation = 21, 600 circumnavigation degrees**

30⁰/cm radius x 360⁰ rotation = 10. 800 circumnavigation degrees

10⁰/cm radius x 360⁰ rotation = 3, 600 circumnavigation degrees

30⁰/cm radius x 360⁰ rotation = 10. 800 circumnavigation degrees

10⁰/cm radius x 360⁰ rotation = 3, 600 circumnavigation degrees

**And in the previous chapter by rotating the 90 degree radius 360 degrees; we proved that the circumferential length and the area of a circle, is not irrational relative to its diameter length; but rather, that both the circumference length and the circumnavigation area of a circle, is perfectly symmetrical in nature.**

**However given these concurring mathematical results, they serve to leave us in somewhat of quandary, as to the nature of differential geometry/mathematics, and their relativity, in regard to our perceptions of what we observe to be, and accept as being our physical reality.**

**There is no doubt given the fact that all seven of these "**

*mathematical*" results concur, that a straight linear 1 centimeter degree of a circle, is the same length as a curved 1 centimeter degree of a circle; but how can this be? When it is "*visually apparent*" that the straight linear distance between the two A to B spatial points "*that they both share*", is shorter than that of the linear length to the degree of curvature.**Therefore we are left with the question; how on Earth can it be, that our linear based mathematical results tell us one story, and yet our five senses and our empirical experiences, tell a different story.**

**So the question remains "What is the shortest distance between two points"?**

**And the only answer I have to this quandary; is to put it to the back of my mind, and later the subconscious may allow a greater deal of insight.**

**For now at least the one thing I do know; is that linear based mathematics alone, cannot serve to explain the differential geometry/mathematics/mysteries; of the quantum of the Alpha Omega Continuum**

*(cycles)*relative to its centre*(of all universal particulates)*.

**CALCULATING THE AREA OF A RING**

**Beginning with a square measuring 60 cm x 60 cm, the area of the square is 3, 600 square cm.**

**Therefore as the area of the circle is three quarters of the squares area, the area of the circle is 2. 700 square cm.**

**Therefore we only need to find the area of the central circle and deduct this from the 3, 600 square cm area of the larger circle, in order to give the area of the ring.**

**The diameter of the central circle is 30 cm, therefore the area of the square of the 30 cm diameter is 30 cm x 30 x cm, 900 square cm. And as the square area of the circle is three quarters that of its square, so the area of the 900 square circle is 775 square cm.**

**2, 700 square cm to the area of the larger circle minus 775 square cm; gives 1, 925 square cm to the area of the ring.**

**FROM THE CUBE TO IT'S SPHERE**

**From the Cube to its Cylinder**

**Diagrams 1 - 5 serve to depict the potential cylinder within a cube, diagrams 5 - 6 serve to show that when a three dimensional cylinder is rotated, so that its lateral length is face on to us, if we imagine it rather as being as a flat square, with the circle of the potential sphere within the cylinder being visibly apparent; we can see that when the four corners of the cylinder are carved away, the shape of the sphere will be released.**

**The diagram above depicts the three quarter area of the circle of the square relative to the potential cylinder lying within a 16 cm x 16 cm wooden cube. and in order to form the cylinder from the wooden cube; we place it on a wood lathe, and then rotate the cube and shave away the four lateral corners, which are equal one quarter of the cubes mass, leaving the three quarter mass of the cylinder remaining.**

**The two diagrams above serve to demonstrate both visually, and via our minds eye, that when the four-dimensional (lateral-Vertical-Angled- Curvature) cylinder is given an angled frontal aspect of view, and then turned towards us to a full face on view, and while at the same time mentally discarding its dimension of the curvature; the front facial view of the cylinder, then in visual effect, becomes a flat square.**

**Therefore if we use our minds eye, to imagine a circle of the same height and width, relative to the area of the square;**

**It then becomes apparent, that when we shave off the four corners of the square, which are in reality the corner ends to the cylinder, we will be in reality shaving away one quarter, of the mass of the cylinder.**

**The first diagram above serves to depict the cube fixed on a wood lathe, prior to carving away the four corners to form the cylinder; and the second diagram depicts the cylinder placed lengthwise and laterally away from us, prior to shaving away the corners of the circular face, to form the sphere.**

**This visual perspective then allows us to imagine, that as we use our chisel in a left and right circular motion, acting between and towards each of the two central spindles, we are shaving away the circular dark flat aspect, of the front of the cylinder.**

**And as we do so the round and darker frontal facial aspect of the cylinders length, will gradually move upwards and forwards away from us, and grow smaller, as the final curvature of the sphere, takes its full form.**

**In sum in regard to the mass of wood removed from the cube**

**With our first cut we removed one quarter of the mass of wood from the cube**

**With our second cut we removed one quarter of the mass of wood from the cube**

**Therefore we can say**

**A**

**circle is three quarters of the area of its square**

**A a cylinder is three quarters of its cube**

**A sphere is three quarters of its cylinder.**

**Confirmation by Mass - Weights**

**Given that the cube weighed 160 grams prior to being converted into a sphere**

**The cylinder would weigh 120 grams**

**The wood shavings would weigh 40 grams**

**Given that the cylinder weighed 120 grams**

**The wood shavings would weigh 30 grams**

**Confirming a cylinder is three quarters of its cube, a sphere is three quarters of its cylinder.**

**12 Steps To The Sphere**

__Cube To Its Cylinder__

1. Measure cube height to obtain its diameter = 6 cm

1. Measure cube height to obtain its diameter = 6 cm

**2. Multiply 6 cm x 6 cm, to obtain the square area, and the length of the perimeter to one face of the cube; length = 24 cm, area = 36 square cm**

**3. Multiply the square area, by the length of diameter, to obtain the cubic capacity = 216 cubic cm**

**4. Divide the cubic capacity by 4, to obtain one quarter of the cubic capacity = 54 cubic cm**

**5. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the cylinder = 162 cubic cm**

**6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm**

**7. Divide the cubes surface area by 4, to obtain one quarter of the cubes surface area = 54 square cm**

**8. Multiply the one quarter surface area by 3, to obtain the three quarter surface area of the cylinder = 162 square cm**

**Cylinder To Its Sphere****9. Divide the cylinders cubic capacity by 4, to obtain the three quarter cubic capacity of the sphere = 40 and a half cubic cm**

**10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter capacity of the sphere = 121 and a half cubic cm**

**11. Divide the cylinders surface area by 4, to obtain one quarter of the surface area = 40 and a half square cm**

**12. Multiply the one quarter surface area by 3, to obtain the three quarter surface area of the sphere = 121 and a half square cubic cm**

**AREAS AND VOLUMES OF SYMMETRICAL OVALS AND OVOIDS**

**Space, volume, and area, are relative interchangeable commodities, (see space - volume - Area - Mass after the next chapter) therefore and as such;**

**A symmetrical oval is a squashed circle**

**A symmetrical ovoid is a squashed sphere**

**When a straight flat downward pressure is applied to the centre of an inflated sphere/ball, resting on a flat solid surface, the air within the ball is displaced equally in all directions away from the the centre of gravity of the ball; as the passive resistance of the atomic tensile strength, of the solid surface below, causes the air pressure already within the ball, plus the pressure being applied to the ball, to equate in all directions away from the centre of gravity of the ball.**

**Resulting in an equal expansion of the balls envelope, in all directions away from the centre of gravity of the ball.**

**Therefore this same effect can be relatively be applied by the use of the minds eye, to the amount of space that exists within a circle, existent upon a flat surface.**

**Whereby by given the symmetrical confining stricture, to the line of the circles circumference, the simulated downward pressure, can only expand symmetrically, laterally, and geometrically along the increasingly oval length of the former symmetrical circular shape., as per the diagram below.**

**Therefore given that the diameter of the ball in the diagram above is 6 cm, the area of the square of the diameter will be 36 square cm; and the three quarter square area of the circle will be 27 square metres.**

**Therefore if the diameter length is compressed down to 3 cm**

**The area of half of the oval will be 13 and a half square cm, and the total area of the oval will be 27 square cm; which will be three quarters of the area to the ovals rectangle.**

**As the rectangles area would have consisted of 3 rows of one cm squares; totalling 36 square cm of area to the rectangle.**

**Therefore we can say**

**As a symmetrical circle is three quarters of its square, and its sphere is three quarters of its cylinder**

**So a symmetrical oval is three quarters of its rectangle, and its ovoid is three quarters of its cylinder**

**THE NUMBER OF DEGREES TO A SPHERE**

Square Degree

From Wikipedia, the free encyclopedia

A

or

For example, observed from the surface of the Earth, the sun or the full moon has a diameter of

Assuming the Earth to be a sphere with a surface area of 510 000 000 km2, the area of Northern Ireland, 13 600 km2 represents a solid angle of 1.10 deg2.[2] Similarly, the area of Connecticut, 14 356 km2represents a solid angle of 1.16 deg2.[3]

From Wikipedia, the free encyclopedia

A

**square degree**is a non-SI unit measure of solid angle. It is denoted in various ways, including**deg2**,**sq.deg.**, (°)², and ☐°. Just as degrees are used to measure parts of a circle, square degrees are used to measure parts of a sphere. Analogous to one degree being equal to*π*/180 radians, a square degree is equal to (*π*/180)2, or**about**1/3283 or 3.0462×10−4 steradian (0.30462 msr). The number of square degrees in a whole sphere isor

**approximately**41 253 deg2. This is the total area of the 88 constellations in the list of constellations by area. The largest constellation, Hydra covers a solid angle of 1303 deg2, whereas the smallest, Crux covers 68 deg2.[1]For example, observed from the surface of the Earth, the sun or the full moon has a diameter of

**approximately**0.5°, so it covers a solid angle of**approximately**0.2 deg2 (≈*π*(0.5/2)2 deg2), which is 4.8 × 10−6 of the total sky sphere.Assuming the Earth to be a sphere with a surface area of 510 000 000 km2, the area of Northern Ireland, 13 600 km2 represents a solid angle of 1.10 deg2.[2] Similarly, the area of Connecticut, 14 356 km2represents a solid angle of 1.16 deg2.[3]

__Geometric Sphere__**300 BC 2015 AD**

__360 Degree Sphere__**180 DEGREES TO VERTICAL AXIS 180 DEGREES TO VERTICAL AXIS**

**The 360 degree**

**protractor represents a vertical cross section of the sphere**

**There are 180 degrees to the height of the protractor, and the sphere**

**Therefore there are 180 lateral rings/circles of 360 degrees, to the height of the sphere**

**180 x 360 degrees = 64, 800 degrees to the whole of the sphere**

**OR**

**The left side edge of the protractor represents 180**

__vertical__degrees and zero__lateral__degrees**Rotating the protractor 360 degrees; will carry the 180 vertical degrees, through 360 degrees of lateral rotation**

**180 x 360 degrees = 64, 800 degrees of rotation**

**The answer is very simple when you finally get there, but it takes one hell of a journey to actually get there; with everyone obtusely doing their damnedest to stop you, while telling you that you are wrong; and then when you do finally break through their obtuseness, they all tell you "I knew that!".**

**Well the 2,300 year straight linear geometric history of Euclid - Archimedes - Einstein; says no, you most certainly did not!.**

**And if Incorporated Mankind - Political Parties - Gangs - Vandals; continue to Attack - Poison - Spoil - Kill - Destroy, all of the Creators natural wonders of**

__"This Living Planet";__it*(this knowledge)*and anything else concerned of, or with human kind; will not matter a damn anyway.

**THE AREA OF 1⁰ OF A 120 CENTIMETER DIAMETER SPHERE**

**120 cm Cube 120 cm Diameter Sphere**

**One face of the six faces of a 120 cm x 120 x 120 cm cube = 14, 400 square**

**centimeters**

**Six faces x 14, 400 square centimeters = 86, 400 square**

**centimeters**

**86, 400 sq cm ÷ by 4 = 21, 600 square centimeters to one quarter of the surface area of the cube**

**21, 600 sq cm x 3 = 64, 800 square centimeters to the surface area of the ¾ Cylinder of the Cube**

**64, 800 sq cm ÷ by 4 = 16, 200 sq cm to one quarter of the cylinders surface area**

**16, 200 sq cm x 3 = 48, 600 square centimeters to the surface area of the ¾ Sphere of the Cylinder**

**48, 600 sq cm ÷ by 64, 800 degrees to the sphere = 0∙75 or ¾ "**

__" to one square centimeter, to the area of 1 degree of the 64, 800 degree Sphere.__*of the area***IRRATIONAL 3.14159**

*Oxford English Dictionary**Pi symbol of the*

__"ratio"__of the circumference of a circle to its diameter, approximately 3.14159

FACTS

FACTS

1: Pi cannot be and is not; the symbol of the

1: Pi cannot be and is not; the symbol of the

__"ratio"__of the circumferential length of a circle to its diameter length; because it is an__"irrational number"__.

2: Pi is the symbol of the

2: Pi is the symbol of the

__"decimal ir-rational"__of the circumferential length of a circle to its diameter length; gained by dividing a decimal 7, into the 22 whole measurement units of the__approximate__circumferential length__given to the circle by Archimedes.__**3: Pi is simply the**

__"symbol of the result"__of the decimalization of the__(approximate)__circumference length of 22 measurement units, given by Archimedes to the length of the circle; relative to being divided by the 7 measurement units to diameter length.

Which equates to three whole units of diameter length, with each whole unit of diameter length measuring seven measurement units = 21/7. and one measurement unit of diameter length 1/7, remaining.

Which equates to three whole units of diameter length, with each whole unit of diameter length measuring seven measurement units = 21/7. and one measurement unit of diameter length 1/7, remaining.

**Over time similar improper fractions were used by others e.g. Ptolemy (150 AD) who used 377/120, which equates to 360/120 with 17 parts remaining, or 3 17/120. 3 whole units of diameter length, and 17 parts of the 120 diameter length.**

**Until finally, the original fractional value of 22/7 Pi; was**

*(decimated)*after the decimal system was adopted by France in 1790, during the French Revolution.**As decimals**

**22 measurement units divided by 7 = 3.14185714285 diameter lengths**

*(Note not 3.14259)*

**22 measurement units divided**

**3 = 7.3333333 diameter lengths**

**Neither of which unlike the imperial/empirical fractional system, serves to provide an exact ratio of measurement, between the diameter and circumferential length of a circle.**

**And contrary to the claim that the decimal form of 22/7, provides the closest approximation of the ratio of a circles circumferential length, to that of its diameter length; this is totally disproved, by the fact that if we divide 22 measurement units of circumferential length by 7.33 measurement units instead of 7, we obtain the result of;**

**3.00136425648**

**Which is a closer approximation of circumferential length, to diameter length; than either 3.1429 or 3.14285714285.**

****

**NON-LINEAR GEOMETRY OF A NATURAL CIRCLE**

****

*E**nergy cannot be separated from energy, because no lines of separation exist, between the energy of any particle of a body of matter; and the adjacent particles of energy, of its surrounding environment.*

*Hence:*

*As any body of matter, may be converted back into the original elemental particles of its matter; and any particulate of matter, may converted back into the quanta of its smaller particulates, and so on.*

*So it follows;*

*That Energy may be converted and so deployed, but Energy cannot be created or destroyed.*

*......*

**Imagine you are viewing a steel sphere, from a direct frontal aspect/viewpoint.**

**Does the body of the steel sphere possess a circular outline to its shape?**

**It would certainly seem that it does, however this is a false impression that is gained by the fact, that our vision is limited to the level of our macro-cosmic reality.**

**In reality, and if we were able to alter the power of our vision down to the microcosmic level; what we would see as we observed the supposed outline of the steel sphere; is that the atoms of steel of the outline of the sphere, are directly in contact with, merging into, and interacting with the gaseous atoms of the surrounding atmosphere**

*(e.g. oxidization)*. There is no linear separation existent, between the atoms of steel of the sphere, and the atoms of the atmosphere.**This applies equally when we consider the nature of a drawn (linear) circle, if e.g. we use a compass to draw a circle of a sheet of paper; the area of the circle merges with the inner side of the area of the line, that is surrounding the inner area of the actual circle; and the outer side of the area of the line that is surrounding the inner area of the circle, merges with the area of paper that is surrounding the area of the drawn circle.**

**Therefore a natural circle only consists of its own inner area, and therefore it is impossible to exactly calculate the inner area of a circle, by the use of any physical means or drawn line; the only way this can be achieved, is via the means of the circles mathematical ratio of radial length, to diameter and circumferential length.**

**Which is; R x 2 = D**

**D x 3 = C or R x 6 = D**

**D x 4 = Sq or R x 8 = Sq**

**Euclid: Book 1 of The Elements; Definition 17**

Quote: A diameter of the circle is "any straight line" drawn through the centre, and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

Quote: A diameter of the circle is "any straight line" drawn through the centre, and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

Facts

Facts

1. A drawn line cannot be drawn through the centre of a circle, as its width/area of thickness merely occludes and covers over the centre of the circumnavigation of the circle; with a variance of thickness/area to each side of the centre of the circle.

1. A drawn line cannot be drawn through the centre of a circle, as its width/area of thickness merely occludes and covers over the centre of the circumnavigation of the circle; with a variance of thickness/area to each side of the centre of the circle.

2. The proposition/concept that a diameter is/can be "any straight line" is totally in error. What is proposed in linear terms as being a straight diameter length; is two single lines of radial length, that have been added together to form one length.

2. The proposition/concept that a diameter is/can be "any straight line" is totally in error. What is proposed in linear terms as being a straight diameter length; is two single lines of radial length, that have been added together to form one length.

Whereas in reality; two radial lengths of a circle cannot be added together, as there is a gap in the form of a hole at the centre of the radiated circle; which serves to separate each of the two central ends of the radial lines, from the other.

Whereas in reality; two radial lengths of a circle cannot be added together, as there is a gap in the form of a hole at the centre of the radiated circle; which serves to separate each of the two central ends of the radial lines, from the other.

3. Given the linear concept that two radial lines of a circle do exist, and are in direct opposition to each other; and therefore in effect they are serving as a single line of diameter length bisecting the circle into two halves, this makes sense; because 1 unit

3. Given the linear concept that two radial lines of a circle do exist, and are in direct opposition to each other; and therefore in effect they are serving as a single line of diameter length bisecting the circle into two halves, this makes sense; because 1 unit

*(e.g. circle)*divided by "two ends/terminal degrees" of a single line = two halves of the unit.

And therefore every time a single "natural number of length" is used to divide a circles "natural circular length"; the circle is divided into that natural number of lengths.

And therefore every time a single "natural number of length" is used to divide a circles "natural circular length"; the circle is divided into that natural number of lengths.

****

4. A circle is "perfectly symmetrical in nature", which is why its circumferential length can be sub divided by any number, into "identical equal lengths" of that number

4. A circle is "perfectly symmetrical in nature", which is why its circumferential length can be sub divided by any number, into "identical equal lengths" of that number

*(Proof; hours, minutes, seconds, of a circular clock-face)*.

Therefore given that a circle has been s

Therefore given that a circle has been s

**ub divided by the number 3; into "3 identical equal lengths" of the number of 3.**

And each identical equal length, of the number of 3

And each identical equal length, of the number of 3

Has 7 identical and equal parts to it length

Has 7 identical and equal parts to it length

It follows that the circle will have 21/7 identical equal parts to its length not 22/7, because 22 cannot be subdivided into equal whole units of length, by either the number of 7 or the number of 3.

It follows that the circle will have 21/7 identical equal parts to its length not 22/7, because 22 cannot be subdivided into equal whole units of length, by either the number of 7 or the number of 3.

In Sum

In Sum

The nature of a circle is defined by the extent of area, that is radiated from its centre; not by the thickness/area of a line, that is used to circumvent and enclose the extent of area radiated from the centre of the circle

The nature of a circle is defined by the extent of area, that is radiated from its centre; not by the thickness/area of a line, that is used to circumvent and enclose the extent of area radiated from the centre of the circle

*(Refer to Minus Pi in Black & Yellow)*.

Because a natural circle as with all forms of natural geometry, is a non-linear geophysical and biophysical phenomenon. With the extent and limits of its natural designs and constructs, being subject only to the universal mathematical ratios; that exist between all of the combinations of its "

Because a natural circle as with all forms of natural geometry, is a non-linear geophysical and biophysical phenomenon. With the extent and limits of its natural designs and constructs, being subject only to the universal mathematical ratios; that exist between all of the combinations of its "

*of positive neutral and negative energy/matter; none of which possess a decimal linear based devaluation, into the non-mathematical - non-empirical realm of irrationality/non-ratio.*__3 non-linear forms"__

In conclusion for this chapter

In conclusion for this chapter

We no longer live in the limited age and perspective of Euclid's linear geometry, we live in the age of our developing knowledge of the atom, and its microcosmic universe.

Therefore we know that the centre of a circle cannot be found, as the microcosmic spatial beginnings of it radii disappear into the realm of microcosmic infinity; and equally the same applies to the microcosmic spatial endings of the radii at the circles periphery; which also disappear into the realm of microcosmic infinity.

We no longer live in the limited age and perspective of Euclid's linear geometry, we live in the age of our developing knowledge of the atom, and its microcosmic universe.

Therefore we know that the centre of a circle cannot be found, as the microcosmic spatial beginnings of it radii disappear into the realm of microcosmic infinity; and equally the same applies to the microcosmic spatial endings of the radii at the circles periphery; which also disappear into the realm of microcosmic infinity.

Therefore we can state that although the radius length of a circle, can never be exactly measured;

Therefore we can state that although the radius length of a circle, can never be exactly measured;

The ratio of the circles bi-radial diameter length, to that of its circumferential length;

The ratio of the circles bi-radial diameter length, to that of its circumferential length;

Is dictated by the radial distance that lies between the spatial centre of a circle; relative to the two spatial central points (Degrees), that lie at each end of the bi-radial diameter length.

Is dictated by the radial distance that lies between the spatial centre of a circle; relative to the two spatial central points (Degrees), that lie at each end of the bi-radial diameter length.

MINUS Pi IN BLACK AND YELLOW

MINUS Pi IN BLACK AND YELLOW

****

Hypothetically we have two yellow square cards each measuring 120 cm x 120 cm

Card 1. Has a black circle measuring 120 cm diameter high x 120 cm diameter wide, and the square card has been cut into 4 equal squares, each measuring 60 cm high x 60 cm wide.

Therefore each 60 x 60 square card is one quadrant of the 120 cm yellow square, and holds one quadrant of the 120 cm black circle.

Card 2. Has also been cut into 4 equal 60 x 60 cm square pieces, and each square has a 60 cm diameter black circle.

Hypothetically we have two yellow square cards each measuring 120 cm x 120 cm

Card 1. Has a black circle measuring 120 cm diameter high x 120 cm diameter wide, and the square card has been cut into 4 equal squares, each measuring 60 cm high x 60 cm wide.

Therefore each 60 x 60 square card is one quadrant of the 120 cm yellow square, and holds one quadrant of the 120 cm black circle.

Card 2. Has also been cut into 4 equal 60 x 60 cm square pieces, and each square has a 60 cm diameter black circle.

__A. Black Areas__**1.**

**All black areas have an equal area to each other**

**2. Any number or type or black area combined will give an equal area**

**3.**

**All black areas combined will give an equal area**

__B. Yellow Areas__**1. All yellow areas have an equal area to each other**

2. Any number or type of yellow area combined will give an equal area

3. All yellow areas combined will give an equal area

1. Have an equal area

2. Any number and any combination of black and yellow areas, will give an equal area

3. All black and yellow areas combined, will give an equal area

4. All areas of the two cards combined, will give an equal area (288 squares)

2. Any number or type of yellow area combined will give an equal area

3. All yellow areas combined will give an equal area

__C. All Areas Of The Two Cards__1. Have an equal area

2. Any number and any combination of black and yellow areas, will give an equal area

3. All black and yellow areas combined, will give an equal area

4. All areas of the two cards combined, will give an equal area (288 squares)

__Logic__**1.**

**All of black and yellow areas of circle and square contain an equality of area**

**2. When an equal amount of area is deducted from an equal amount of area, it leaves an equal amount of area remaining**

**3.**

**Pi has a greater sum of inequality to its area, than that of the lesser sum of equality to the area of the circle**

**4.**

**Pi is a mathematical infringement into the area surrounding of a circle**

Pi represents the physical inequality of the thickness/area of a length of line or lines, having been mathematically inducted into the formula.

__In Sum__Pi represents the physical inequality of the thickness/area of a length of line or lines, having been mathematically inducted into the formula.

**COMPASS AND LINES**

**Unlike Archimedes, today we use various types of drawing compass to draw our circles and though they are more complex in nature they are far simpler to use and more accurate than any piece of string used by our ancient mathematicians. However though they are more accurate they are still unable to divide/partition the circumference of the circle into exactly equal lengths. The reason for this being that both the drawing compass and the method used for drawing a circle contain inherent physical limitations in regard to obtaining a high degree of accuracy.**

A compass has two arms with a pencil tip at the end of one arm and a steel tip at the end of the other arm, and prior to drawing a circle the two tips of the compass are separated to a distance of measurement that is to equate to the desired radius of the diameter of the circle that is to be drawn. However it is during this process of physically measuring the distance between the two tips of the compass that the first of many inaccuracies in this procedure begin to manifest.

In first place in regard to the inaccuracies inherent to the process of drawing a circle with a compass is the accuracy of the measurements that are marked upon the ruler that we use in order to measure between the two tips of the compass. If there is even the vaguest shade of a degree of an inaccuracy in the ruler or the measurement taken from it between the two tips of the compass, then this degree of inaccuracy will be multiplied 360 times over by the time that we have completed the circle.

A compass has two arms with a pencil tip at the end of one arm and a steel tip at the end of the other arm, and prior to drawing a circle the two tips of the compass are separated to a distance of measurement that is to equate to the desired radius of the diameter of the circle that is to be drawn. However it is during this process of physically measuring the distance between the two tips of the compass that the first of many inaccuracies in this procedure begin to manifest.

In first place in regard to the inaccuracies inherent to the process of drawing a circle with a compass is the accuracy of the measurements that are marked upon the ruler that we use in order to measure between the two tips of the compass. If there is even the vaguest shade of a degree of an inaccuracy in the ruler or the measurement taken from it between the two tips of the compass, then this degree of inaccuracy will be multiplied 360 times over by the time that we have completed the circle.

**We also have to consider the sharpness (thicknesses) of both the steel and the graphite tips of the arms of the compass that we are using, because the accuracy of the circle is not merely dependent on the accuracy of the measurement of distance between the two tips of the compass, it is also dependent upon**

*that distance/measurement throughout the process of drawing the circle.*__maintaining__

A compass works on the principle of one arm of the compass providing a central fixed point around which the second arm is rotated in order to draw or transcribe a circle.

A compass works on the principle of one arm of the compass providing a central fixed point around which the second arm is rotated in order to draw or transcribe a circle.

**In order to provide a fixed central point (B) the steel tip of the first arm is forced**

*and*__into__*the surface that is to be drawn upon. However as can be seen in the first diagram above at point B wherein the steel tip is forced into the surface to fixate the arm of the compass, the length of the surface radius becomes shorter according to the depth/length of penetration of the steel tip into the central point from which the circle will radiate from.*__below__

After the steel tip has been forced into its central point to fixate its central position, the degree of pressure that has been used and placed on the fulcrum of the compass in order to achieve penetration is lessened. The lessening of the downward pressure on the fulcrum of the compass then allows the pencil tip to be rotated around the central point in order to transcribe the circle.

After the steel tip has been forced into its central point to fixate its central position, the degree of pressure that has been used and placed on the fulcrum of the compass in order to achieve penetration is lessened. The lessening of the downward pressure on the fulcrum of the compass then allows the pencil tip to be rotated around the central point in order to transcribe the circle.

**However if the downward force on the fulcrum is too great, it can cause the arms of the compass to spread apart and so increase the length of the drawn radius.**

There are also many other factors of error that can and do creep into this procedure regardless of, and no matter how careful we are in trying to avoid them, and still further errors creep in if and when we then use the compass to sub divide the circumference of the circle into equal lengths. For example, when using the circles radius to divide the circumference of the circle into six (Supposedly) equal arcs in order to draw a six sided hexagon; the amount of error that was present in the original radius of the circle, is repeated six times on the circumference of the circle; however not exactly, because each time we subdivide the circumference, the steel tip of the compass has to be forced down into the surface; and each time the pressure exerted on the fulcrum, is variant.

There are also many other factors of error that can and do creep into this procedure regardless of, and no matter how careful we are in trying to avoid them, and still further errors creep in if and when we then use the compass to sub divide the circumference of the circle into equal lengths. For example, when using the circles radius to divide the circumference of the circle into six (Supposedly) equal arcs in order to draw a six sided hexagon; the amount of error that was present in the original radius of the circle, is repeated six times on the circumference of the circle; however not exactly, because each time we subdivide the circumference, the steel tip of the compass has to be forced down into the surface; and each time the pressure exerted on the fulcrum, is variant.

Regardless of whether we use a piece of string or whether we use a compass, in order to draw a circle; it is a physical/empirical fact, that each of the minute errors that do occur during the subdivision process, are then amplified over the three hundred and sixty degrees of the circle.

Regardless of whether we use a piece of string or whether we use a compass, in order to draw a circle; it is a physical/empirical fact, that each of the minute errors that do occur during the subdivision process, are then amplified over the three hundred and sixty degrees of the circle.

**Therefore in sum: It is a totally impossible physically reality to be able to draw a perfect circle.**

However this is not the case in regard to our minds eye intelligence, logic, and rationality; which do not suffer from any such physical restraints; and it is therefore only within the environs of our minds, that it is possible to mathematically define/rationalise and draw, the perfectly symmetrical non-linear circle.

However this is not the case in regard to our minds eye intelligence, logic, and rationality; which do not suffer from any such physical restraints; and it is therefore only within the environs of our minds, that it is possible to mathematically define/rationalise and draw, the perfectly symmetrical non-linear circle.

**BI-RADIAL ARITHMETIC OF CIRCLE AND SQUARE**

*(Pasted document)*

CONTEXT OF A DEGREE

CONTEXT OF A DEGREE

*The square of the length of the hypotenuse of a right triangle, is equal to the sum of the squares of the other two sides.*