**To see previous years of eclectic work/progressive and developing thoughts go to; www.geometry-mass-space-time.com**

**Which has so far; received some 268, 000 + hits from 136 Nations, of the 195 Nations that go into the international make-up of the World.**

**CONTENT**

**Circumferential Length**

**Sumerian Method; Finding The Area of A Circle**

**Archimedes Proposition 1; Finding The Area Of A circle**

**Further Methods - Concurrencies (11)**

**Calculating The Area Of A Ring**

**Symmetry Of The Circle**

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

**12 Steps To The Sphere**

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

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

**Geometry Meets Physics**

**Curved Space**

**The Number Of Latitudinal And Longitudinal Degrees To The Surface Of A Sphere**

**The Area Of 1 Degree, Of The Number Of Degrees To The Surface Area Of A Sphere**

**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**

**Pythagoras Theorem**

**EMAIL:**

**unialphaomega@hotmail.com**

**For the first time in human history; the surface areas and volumes of cylinders, spheres, and ovoids can now be exactly calculated.**

**This being supported by the fact; that despite the numerous challenges that I have put forward on my web pages over the past four four years; not one mathematician or geometer, has ever sought to challenge the mathematics involved, or the axiomatic results obtained and revealed on that web page (Though some [3] have since tried on Ted Talks, and failed.**

**Therefore given the profound silence, that has been emanating from the realms of academia over this time; one is only left to despair at the lack of integrity and of honesty, that clearly pervades through the reams of academia and wonder, if as with Galileo, it will take a century before truth is allowed out. .**

**Quote;**

**Stuart Chase 1888 - 1985:**

**For those who believe no proof is necessary -**

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

**Author**

**Only the Open Mind of the Honest Sceptic, can**

**learn; that**

**the impossible is possible, and the possibilities are endless.**

**Galileo Galilei**

**“In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.”**

**“I have never met a man so ignorant, that I couldn't learn something from him.”**

**"Measure what can be measured, and make measurable what cannot be measured"**

CIRCUMFERENTIAL LENGTHCIRCUMFERENTIAL LENGTH

**Beginning with a Square, with each line measuring 120 centimetres**

**Take one line of 120 centimetres, and consider it to be a Diameter line**

**Multiply the Diameter line by 3, and the circle is 360 centimetres in length**

**Multiply the Diameter line by 4, and the Square of the Diameter is 480 centimetres in length**

__Therefore As;__**1. The circle has 360 degrees to its length**

**Each degree is 1 centimetre in length.**

**2. Each side of the square is 120 centimetres in length**

**3 Diameter lengths is = to**

**3/4 of the 480 centimetre perimeter length.**

__It Follows__**That as the 360 cm circle is 3/4 of the 480 cm perimeter length - the area of the circle, will be 3/4 that of the square.**

__Concurrence 1__

*SUMERIAN METHOD*

*FINDING THE AREA OF A CIRCLE*

*SUMER 4,000 BC***SUMER (Wikipedia)**

Mathematics[edit]

The Sumerians developed a complex system of metrology c. 4000 BCE. This metrology advanced resulting in the creation of arithmetic, geometry, and algebra. From c. 2600 BCE onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[43] The period c. 2700 – 2300 BCE saw the first appearance of the abacus, and a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system.[44] The Sumerians were the first to use a place value numeral system. There is also anecdotal evidence the Sumerians may have used a type of slide rule in astronomical calculations. They were the first to find the area of a triangle and the volume of a cube.[45]

Geometry[edit] Babylonians knew the common rules for measuring volumes and areas. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if

Babylonian mathematics (also known as

Mathematics[edit]

*Main article: Babylonian mathematics*The Sumerians developed a complex system of metrology c. 4000 BCE. This metrology advanced resulting in the creation of arithmetic, geometry, and algebra. From c. 2600 BCE onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[43] The period c. 2700 – 2300 BCE saw the first appearance of the abacus, and a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system.[44] The Sumerians were the first to use a place value numeral system. There is also anecdotal evidence the Sumerians may have used a type of slide rule in astronomical calculations. They were the first to find the area of a triangle and the volume of a cube.[45]

Geometry[edit] Babylonians knew the common rules for measuring volumes and areas. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if

*π*is estimated as 3.Babylonian mathematics (also known as

*Assyro-Babylonian mathematics*[1][2][3][4][5][6]) was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited.[7] In respect of time they fall in two distinct groups: one from the Old Babylonian period (1830-1531 BC), the other mainly Seleucid from the last three or four centuries BC. In respect of content there is scarcely any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for nearly two millennia.[7]

__Sumerian Method__**The Circumferential length of the circle was multiplied by itself, and then divided by 12 to give the area to the circle.**

**Given a circle with a circumference length of**

**360 cm**

**Which is 3/4 of the length, to the 480 cm long perimeter length of its surrounding square**

**360 cm x 360 cm = 129, 600 sq cm**

**129, 600 sq cm divided by 12 = 10, 800 sq cm to the circles area**

**Which is 3/4 of the area to its surrounding 14, 400**

**sq cm square.**

__Concurrence 2__

*ARCHIMEDES PROPOSITION 1*

*FINDING THE AREA OF A CIRCLE***ARCHIMEDES 287 - 212 BC**

**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.**

**Archimedes Triangle for finding the Area to a Circle**

**In the above diagram the base right angle of the rectangle has been given the radius length of 60 cm; and the second right angle of the rectangle, has been given a circumferential length of 360 cm.**

**The area of the rectangle therefore has an area of 21, 600 sq cm; and this area divided by 2 gives an area of 10, 800 sq cm to the area of each triangle; the same amount of area as to that of the circle.**

**Which concurs with the results of both the first method and the Sumerian method, and also those gained by the following three methods**

**.**

*FURTHER METHODS*

**Concurrence's 3 - 6**

__1st Method__

*3 x r 2***Diameter 120 Centimetres x Diameter 120 Centimetres - 14, 400 Square Centimetres to the Square**

__2nd Method__**The 60 centimetre radius is squared = 3, 600 square centimetres**

**Then multiplied by 3 =**

__10, 800__square centimetres to the area of the circle**Equating to being three quarters of the area, of**

__14, 400__square centimetre square__3rd Method__

**The 14, 400 square centimetre area of the square, is divided by 4 = 3, 600 square centimetres**

**This is then multiplied by 3 =**

__10, 800__square centimetres to the area of the Circle**Equating to three quarters of the area, to the overall square**

**Diameter 120 Centimetres x Diameter 120 Centimetres - 14, 400 Square Centimetres to the Square**

__4th Method__**The**

**60 cm Radius is squared = 3, 600 sq cm**

**The 3, 600 sq cm is then divided by 4 = 900 sq cm**

**The 900 sq cm is then multiplied by 3 = 2, 700 sq cm**

**Which is three quarter area of the square, of the 60 cm radius**

**The three quarter 2, 700 sq cm area, of the square of the 60 cm radius,**

**Is then multiplied 4**

**Giving an area of**

**10, 800**

**sq cm, to the area of the overall Circle**

**Which is three quarters**

**of the area to the overall square**

**And as the four corners of the square, added together give an area of 3, 600 sq cm**

**When this area of 3, 600 sq cm is added to the 10. 800 sq cm area of the Circle**

**This gives an area of 14, 400 sq cm, to the Overall Square.**

__Absolutes__**The**

__"non-linear" circumferential length__to the area of a circle; is six times that of the circles radius length, and three times that of the circles diameter length.**The**

__"non-linear" circumferential length__to the area of a circle; is three quarters of the length, to the perimeter, of the square of the circles diameter length.**The**

__"non-linear" area__of a circle; is three quarters of the area, to the area of the square of the circles diameter length.

__Concurrence 7__

*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.**

__Concurrence 8__

SYMMETRY OF THE CIRCLESYMMETRY OF THE CIRCLE

**A circular protractor has**

180 Degrees to its Diameter

360 Degrees to its Circumference

Therefore by giving 180 Centimetres 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 Centimetres 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

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

**360⁰/cm**

**÷ 5 = 72 ÷ 5 = 14.4 x 10 = 144 square centimetres ÷ 4 = 36 x 3 = 108 square centimetres to the circle**

And 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.

And 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.

**And any part of any circle multiplied by 360, will give 360 degrees to the circle with each degree exactly identical in curvature of length.**

__Concurrence 9__

FROM THE CUBE TO ITS SPHEREFROM THE CUBE TO ITS 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.**

**Concurrence 10**

TWELVE STEPS TO THE SPHERE

TWELVE STEPS TO THE SPHERE

**CUBE TO CYLINDER**

**Calculating the surface area and volume of a 6 centimetre diameter sphere, obtained from a 6 centimetre cube.**

**Note we use 6 centimetres rather than 120 cm, in order to make the numbers easier to follow.**

**1. Measure cube height to obtain a/its Diameter line = 6 cm**

**2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of perimeter to the square face = Length 24 cm, Square area 36 sq cm.**

**3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm. 4. Divide the cubic capacity by**

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

**5. Multiply the one quarter cubic capacity by 3. to obtain the 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 of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm. CYLINDER TO SPHERE**

**9. Divide the Cylinders cubic capacity by 4, to obtain one quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm.**

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

**11. Divide the Cylinders surface are by 4, to obtain one quarter of the surface area of the Cylinder = 40 & a half square cm.**

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

__Concurrence 11__

*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**

WHAT IS THE SHORTEST DISTANCEWHAT IS THE SHORTEST DISTANCE

*BETWEEN TWO POINT*S**Geometry Meets Physics And Curved Space**

*Please note: This chapter may (more than likely) be subject to further revision, as the subject matter relative to my thoughts; is more difficult to convey via way of the written, rather than spoken word.*

**In the previous chapters we have used five methods, inclusive of the Sumerian method and Archimedes first proposition, in order to obtain five identical results that serve to prove; that the 360 curved degrees to a circles circumferential length, possess the same length, as that of 360 straight linear degrees to a circles circumferential length.**

**Therefore one must question how this can be possible?**

**When it is apparent; that given the space that exists between an A to B straight line, and a parallel line of curvature, that is accompanying it, between the same two A to B points; that the line of curvature, has the same length as the straight line.**

**On the one hand, the mathematical results are indisputably correct.**

**On the other hand as the diagram of the hexagram above serves to empirically prove.**

**A.**

**The straight linear radius transects the circles curved circumferential length six times.**

**B.**

**Each transect-ion, serves to create space between it, and the accompanying parallel arc of curvature.**

**Therefore regardless of eleven concurring mathematical results, our eyes and our common sense, tell us that the curved line has to be longer than the straight line.**

**Which raises the question: Given two correct results that are in conflict with each other, how do we set about resolving this quandary?**

**The first step, is to accept that both results are correct; and therefore it is not a fact that they are in conflict, but rather that we do not know how or why they apparently seem to be so.**

**The next step, is put aside our straight linear Euclidean based preconceptions, and examine the nature of the circle for what it is; rather than with the straight linear based preconceptions, that we have “**

__imposed__” upon it.

**Essentially a circle is a cross section of a sphere, and no**

*“*incorporates any form of straight line or linear measurements.__natural circle or sphere of geophysics__”**Which is why the geometry of spheres circles and curvatures is called differential/different geometry, as opposed to the straight linear geometry we are familiar with, and is generally used.**

**Therefore given that two measurements of a straight line and a curvature are the same, and yet the curved line is apparently longer than the other; one is only left to conclude that given;**

**1. A curved**

**linear dimension of distance, extending between two A to B spatial points is**

**longer**

*(and multi - dimensionally, its areas and volumes are spatially larger)*than; a**straight linear dimension of distance, extending between the same two A to B spatial points.**

**2. A straight linear "measurement" of distance between two A to B spatial points is shorter than; a curved linear dimension of distance, extending between the same two A to B spatial points.**

**That straight linear "measurements" of distance extending between two A to B spatial points, are not applicable to curved linear dimensions of distance, extending between the same two spatial points;**

**Because**

**Unlike straight linear dimensions of distance, which extend between two A to B spatial points; curved linear dimensions of distance, do not directly extend between two A to B spatial points;**

**But rather they extend from spatial point A, and then through multiple divergent spatial points, before reaching spatial point B.**

__Further Reasoning__**Despite our awareness, that we and all other bodies are comprised of infinitesimally small microcosmic particles of matter; due to the vastness of the surrounding Universe, and the huge sizes of its celestial bodies, we tend to think of the Universe as being like ourselves, macro-cosmic in nature.**

**However this is far from being case, because our solid macro-cosmic bodies, are essentially made up of concentrated microcosmic particles; of the infinite and eternal Microcosmic Universe, that surrounds and pervades through every space and particle of our seemingly solid beings.**

**When we observe any solid solid body, we are simply observing the external solid linear aspects/dimensions of its being. However if we consider the three straight linear dimensions of vertical, lateral, and depth relative to the internal structure of a solid body in the form of a sphere.**

**What we can discern, is that the fundamental source of all forms of "microcosmic and macro-cosmic" dimension; begins at and lies at the centre of every microcosmic particle (sphere of radiation/influence), of the surrounding infinite - eternal - boundless microcosmic Universe.**

**Essentially we and all forms of living and none living solid matter, are just as with the fish swimming in our watery oceans; also swimming in the boundless and "voluminous" vastness of the Microcosmic Universe that surrounds, and pervades through us.**

**And given this aspect of micro and macro reasoning; realise that that although the straight linear measurements of the number of 360 degrees to a circle, apparently give the same mathematical result.**

**The curved arcs to the degrees of a circle are in fact longer than the straight linear, because the dimensions of curvature, belong to the "expansive voluminous" nature of microcosmic space; while the straight linear aspects to the number of curved degrees, belongs to the "inclusive concentrated" nature of macro-cosmic solid matter.**

**And relative to the surrounding and boundless voluminous microcosmic ocean; the transition from "voluminous microcosmic matter" to "concentrated solid macro-cosmic matter" is an ongoing process that extends from that of microcosmic gravitation, whereby particles are attracted together and coalesce; up to macro-cosmic gravitation, whereby the stars and other huge celestial bodies and complexes, serve to create the gravitational fields, currents, and vortices of interstellar space.**

**In sum**

**The size/length/ratio of the dimension of curvature relative to straight linear dimension; equates "visually and physically" to 180 degrees of circular length, relative to the straight linear length, of the two lateral radius's combined.**

__Natural Kingdom Geometry__**Unlike the educated minds of we humans, the**

**“**

*” do not possess any conscious mathematical knowledge, relative to measuring distances or objects in their surrounding environments.*__natural minds of other living creatures__

**which raises the questions**

**e.g.**

**How does a Spider measure, in order to be able to build its geometric web?**

**How does a Bee measure, in order to be able to build hexagonal honey cells, all of the same size?**

**How does an Eagle equate its own height distance and velocity, relative to the height distance and the flying speed of its prey below?**

**How does a Bat in total darkness equate its height distance and velocity, relative to surrounding objects, and the height distance and flying speed of its prey?**

**And the answer is, that they are all "**

__naturally able__" to take extremely precise mathematical measurements.

**Which is what we were all formerly able to do; before we became almost totally reliant, on straight linear mathematics.**

**So what is the common natural denominator, in regard the taking of these measurements?**

**It is the direction of and the “focal lengths” of vision, sound, radar, or electrical impulses etc. that are specific to a type of a species.**

**Many of us who work in such fields as construction, landscaping, sculpting, and painting etc. make use of this natural ability to assess the lengths of, distance, depth, perspective etc.**

**However in the natural kingdom, this natural ability of focal length measurement, and calculation is far more precisely attuned/refined, as the very life of the predator is entirely dependent upon it, in order to feed itself and ensure its survival, and the same can be said of its prey, relative to it eluding the predator.**

**Therefore every living creature, must have a vast library of comparative focal length measurements, relative to all forms of shapes, and structure that exist within its surrounding environment. Which brings us back to the question “What is the shortest distance between two points?”.**

**And the answer is, that the shortest, fastest, and most accurate distance between two spatial points, relative to the natural physical world, is not a straight line; but rather the forward line of sight focal length, that lies between e.g. a bats radar receptors, or an eyes retina, relative to an object.**

**The light of which in regard to the latter, is comprised of reflected electromagnetic photon/particle waves**

*(spirals; refer to Gravity Magnetism and Inertia)*travelling at a speed of 300, thousand kilometres per second*(within a gravitational field)*from their source.**However as the curvature of all lens's, are ground using the ratio of a**

**3.14159 circumferential length to diameter length rather than 3 to 1; it follows, that all subsequent calculations carried out relative to the speed of light, and the sizes of astronomical bodies; are in error due to that additional amount of**

**.14159 of curvature, relative to the diameter of lens.**

*GEOMETRY MEETS PHYSICS*

**A linear drawn radius and circle are essentially a solid/static/inert linear mimicry of a sphere in cross section, with the radius being representative of a single radiation; and the arc of a circle between the termini of two radii, being representative of the “curved spatial distance” between two radii termini.**

**FOUR DIMENSIONS OF A SPHERE OF INFLUENCE/RADIANCE**

**FOUR DIMENSIONS OF A SOLID SPHERE**

**Therefore when we use a solid/static/inert measured straight drawn line, to transect the solid/static/inert circumferential line of a drawn circle; as such the straight line mimics and represents the length of an A - B transect-ion of spatial curvature; the solid drawn line and its physical straight linear measurement, does not equate to, nor does it or can it, represent the actual length of spatial distance, that exists between two A – B termini of a spatial arc of curvature.**

**In the realm of particle physics, the A - B distance between the "**

*termini of two electromagnetic radiations*" is subject to;**A) The amount of power emitted from the central source/energy of the two radiations "**

*in the form of the two radiations*".

**B) The distance of the two termini of the radiations from their central power/energy source.**

**C) The amount of gravitational influence being exerted upon/along the two radiation wave lengths.**

**D) The amount of gravitational influence acting in between the two wave lengths, and towards their central power/energy source.**

**Which can be equated to the angle of focal length, that extends from the two radiations termini, and then in towards the central power/focus/focal point, of their two emissions.**

**Refer to page titled "Gravity - Inertia - Magnetism".**

****

CURVED SPACECURVED SPACE

**Substance of Energy + Motion = Matter: Matter + Rotation/Spin/Charge = Gravitation & Centrifugal Force**

**Gravitation & Centrifugal Force = Electromagnetic Attraction & Repulsion**

**Ratios of Electromagnetic Attraction & Repulsion**

**=**

**Particulate Geometry of; Quantum Physics to Macro Geophysics**

Every-thing/form of matter within both the microcosmic and the macro-cosmic Universe travels in the form of the curvatures of spirals, orbits, rotations/spin “through space”; as we can see from the picture of the galaxy of Andromeda above, which is spiraling within inter-galaxy space; that the inter galaxy space of itself, which Andromeda is spiraling within, is not of itself curved in nature.

Every-thing/form of matter within both the microcosmic and the macro-cosmic Universe travels in the form of the curvatures of spirals, orbits, rotations/spin “through space”; as we can see from the picture of the galaxy of Andromeda above, which is spiraling within inter-galaxy space; that the inter galaxy space of itself, which Andromeda is spiraling within, is not of itself curved in nature.

And when we consider the internal workings of an atom, they are also in the form of the curved orbits around the nucleus of the atom; whereas when we consider these same dimensions from the aspect of Euclidean straight linear geometry, relative to a drawn circle or a sphere; we can see that the straight linear dimensions within the sphere extending from its center are; lateral - vertical – diagonal as with the three external straight dimensions of a cube.

Therefore if we equate the mass of a sphere, with that of the mass of the sphere of the Earth; and then consider the origin of the dimension of curvature relative to the Earths spherical mass; we realize that the origin of the Earth's spherical shape, and hence the dimension of curvature to its circumferential length, is gravity.

And when we consider the internal workings of an atom, they are also in the form of the curved orbits around the nucleus of the atom; whereas when we consider these same dimensions from the aspect of Euclidean straight linear geometry, relative to a drawn circle or a sphere; we can see that the straight linear dimensions within the sphere extending from its center are; lateral - vertical – diagonal as with the three external straight dimensions of a cube.

Therefore if we equate the mass of a sphere, with that of the mass of the sphere of the Earth; and then consider the origin of the dimension of curvature relative to the Earths spherical mass; we realize that the origin of the Earth's spherical shape, and hence the dimension of curvature to its circumferential length, is gravity.

And subsequently we realize that the 4th dimension of curvature, is not a spatial dimension of outer surrounding space;

But rather that it is a dimension

And subsequently we realize that the 4th dimension of curvature, is not a spatial dimension of outer surrounding space;

But rather that it is a dimension

*(Sphere of gravitation)*created "*" non-dimensional inter galactic outer space;*__of and within__

**And also within the gravitational field of single mass, by the traction-al power of the gravitational mass lying at its center**

*(Center of gravity),*subject to the rotational and outwardly generated/radiant centrifugal force, that is being exerted at, and from its centre.

**And relative to a rotating or spiraling galaxy, it is created by;**

**The the collective of gravitational/**

**traction-al forces, and rotational centrifugal forces; being exerted by the rotating masses**

*(e.g. the celestial bodies of our Solar System)*lying within the volume, of the periphery of the galaxy's rotation.

**More simply; any single unit of rotating gravitational mass, or any rotating system/unit of gravitational masses (e.g. as with the particulate structure of an atom) will possess its own surrounding sphere/field of gravitation, and its own surrounding sphere of opposing centrifugal force.**

**And hence as with the refraction of light within the medium of the atmosphere, of the gravitational field of the Earth; so light refracts (Lens effect) within the gravitational medium (gravitons), of the gravitational field of a galaxy.**

**THE NUMBER OF DEGREES OF LATITUDE AND LONGITUDE**

**TO THE SURFACE OF A SPHERE**

**Square Degree**

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

or 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 (≈

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 Number of Lateral and Longitudinal Degrees To A Sphere**

A Circular protractor has 180 Degrees to its "Vertical Diameter"

Equating to 180 "

180 x 360 "

..................

Given 180 cm's to the Diameter of a protractor

The Area of the Square of the Diameter is 32, 400 "Square Centimeters"

Rotate the vertical of the 90 degree radius 360 Degrees clockwise; and the Area to the Circle is;

32, 400 "Circumnavigation Degrees"

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

Therefore the "Irratio-nal" formula of Pi; cannot be said to have, or considered to have a "Ratio" of Circumference to Diameter length.

A Circular protractor has 180 Degrees to its "Vertical Diameter"

Equating to 180 "

__Lateral Circles__of 360 degrees" to the HEIGHT of the Sphere180 x 360 "

__Lateral Circles__of 360 degrees" = 64, 800 shared lateral and longitudinal degrees to the sphere..................

__Oxford English Dictionary__: Pi The numerical value of the ratio of the circumference of a circle to its diameter.Given 180 cm's to the Diameter of a protractor

The Area of the Square of the Diameter is 32, 400 "Square Centimeters"

*(Half of the number of lateral and longitudinal degrees to a Sphere)*Rotate the vertical of the 90 degree radius 360 Degrees clockwise; and the Area to the Circle is;

32, 400 "Circumnavigation Degrees"

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

Therefore the "Irratio-nal" formula of Pi; cannot be said to have, or considered to have a "Ratio" of Circumference to Diameter length.

This being confirmed by the following; which determine that a Circles Circumference and Area, far from being irrational;

Are perfectly rational - symmetrical

90⁰/cm radius x 360⁰ rotation = 32, 400 circumnavigation degrees (8, 100 sq cm to the square of diameter)

60⁰/cm radius x 360⁰ rotation = 21, 600 circumnavigation degrees (3, 600 sq cm to the square of diameter)

30⁰/cm radius x 360⁰ rotation = 10. 800 circumnavigation degrees (900 sq cm to the square of diameter)

10⁰/cm radius x 360⁰ rotation = 3, 600 circumnavigation degrees (100 sq cm to the square of the diameter)

__Reiteration__This being confirmed by the following; which determine that a Circles Circumference and Area, far from being irrational;

Are perfectly rational - symmetrical

90⁰/cm radius x 360⁰ rotation = 32, 400 circumnavigation degrees (8, 100 sq cm to the square of diameter)

60⁰/cm radius x 360⁰ rotation = 21, 600 circumnavigation degrees (3, 600 sq cm to the square of diameter)

30⁰/cm radius x 360⁰ rotation = 10. 800 circumnavigation degrees (900 sq cm to the square of diameter)

10⁰/cm radius x 360⁰ rotation = 3, 600 circumnavigation degrees (100 sq cm to the square of the diameter)

**360⁰/cm**

**÷ 5 = 72 ÷ 5 = 14.4 x 10 = 144 square centimetres ÷ 4 = 36 x 3 = 108 square centimetres to the circle**

**Which is why any circumferential length, can be equally subdivided by any "whole number"; into that number of identical parts of the circle.**

*AREA OF 1 DEGREE*

*OF*

*64, 800 DEGREES***a. Circle of 360**

**⁰**

**b. 6 Diameters 6 Square faces c. Shared Degrees d. 1 Square Face of Cube e. 8 Corners of Cube Removed**

**In the previous chapter in order to gain the number of degrees to the surface of a sphere; we rotated the 180**

**⁰ of**

**height of a protractor by 360**

**⁰, and in so doing defined that there are 180 lateral circles of 360⁰ to the height of a sphere; and therefore there are 64, 800⁰ to the surface area of a sphere.**

**However although this lateral rotational method of finding the number of degrees to the surface of a sphere, does indeed serve to define the number of 64, 000 degrees to the surface of a sphere; it does not serve to give each of those degrees an equal amount of surface area.**

**The reason being, that if we were to view such a hypothetical sphere from an overhead aspect of view, extending upwards from the spheres equator; each subsequent circle of 360⁰ to that of the circle of the equator, would appear to be smaller than the one below it.**

**Therefore in order to equate all of the degrees to each other, and equally over the surface of the sphere, we have to divide the number of 64,800 degrees into the overall surface area of the sphere.**

**And in order to achieve this via straight linear geometry, we begin by observing that the number of radii/degrees to a circle is the same number for its surrounding square; and relative to the solidarity of a sphere and its surrounding solid cube, this principle does not change.**

**To grasp this rather than think in terms of linear radii; think in terms of radiations extending directly outwards from the centre of a sphere, and continuing outwards through the surrounding body of the cube.**

**Given this aspect of observation it becomes clear, that if we convert the number of degrees to a circle into the same number of centimeters to find the surface area of the sphere; and then divide this surface area by 64, 800 degrees, this will give the area to each one of those degrees.**

**120 cm Height of Cube = 120 cm Diameter Sphere**

One face of the Six faces of a 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 surface of the Sphere = 0∙75 or ¾ of one square centimeter to 1 degree of the surface, of the Sphere.

__Refer Back To 12 Steps To The Sphere__One face of the Six faces of a 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 surface of the Sphere = 0∙75 or ¾ of one square centimeter to 1 degree of the surface, of the Sphere.

**Height 180 degrees x 3 1/3rd Rotation's of 120 degrees (360) = 64' 800 degrees****Height 180 centimetres x 3 1/3rd Rotation's of 120 centimetres (360) = 64' 800 square****centimetres****Which serves to prove once again; that a circles length is 3 x its diameter length & its area is 3/4 that of its surrounding square.****And given that; 48, 600 sq cm ÷ by 64, 800 degrees to the surface of the Sphere =**__¾ of one "Square Centimetre"__to the area of 1 degree of the surface, of the Sphere.**This also serves to inform us;**

That the three-dimensional, 3/4 square centimetre shape of one degree of sphere surface; is in the shape of a circle.

That the three-dimensional, 3/4 square centimetre shape of one degree of sphere surface; is in the shape of a circle.

Which equates to; the cross section of oneWhich equates to; the cross section of one

__radiation__extending from the__centre__of the 120 cm sphere, being circular along its entire radiant length.**Which equates to; the cross section of one**

__radiation__extending from the__centre__of the 120 cm sphere, being circular along its entire radiant length.**Because the straight lines we use to draw a radius or the radii of a circle, are a simply a linear mimicry of the radiations that go into the make up, of every circle and sphere. For although we use a length of string or a compass to draw/circumnavigate a circle around a central focal point, the string or compass in effect represent a single radius, circumnavigating the central focus point.**

**And relative to a solid circle e.g. a wagon wheel, the radii/spokes of the wheel serve as levers; levering/rotating around a central spatial point occupied by the axle.**

**And if we use our minds eye imagination, while thinking in these terms; and then applying them to the action, of rolling a large sphere across the ground, we realise; that the forward pressure we are applying is to the rear upper aspect of the sphere, is causing a leverage effect; that is acting through the centre of gravity of the sphere, and against the resistant traction of the ground below.**

**This same leverage through a spatial centre of gravity principle, also applies in regard to the motion of any object.**

**For example imagine that we are moving a large wardrobe on our own. To move the wardrobe, we first have to push against its upper aspect, in order to lever/tilt the upper aspect of the wardrobe forward; and in so doing shift - rotate its centre of gravitation**

*(central balance of it weight)*forwards, over a smaller focus of area and traction.**We are then able to rotate the height of the body of the wardrobe, around this smaller focus of area and traction; and then continue moving the wardrobe, using alternating ends its length, to provide leverage around the shifted centre of gravity.**

**In sum**

**Just as the centre of gravity and rotation of a wheel, is also its centre of balance; in effect it is also the centre of gravity of the wheel and axle, that serves as its central fulcrum point of leverage; and therefore relative to it counterpart a sphere, the same principle applies; but as a sphere does not possess an axle, it is the body of the sphere, that rotates around its own centre of gravity.**

**As such the centre of gravity of any sphere (or sphere/particle/body of radiance) is its central spatial**

*(64, 800 degree)*fulcrum point of rotation/spin.

*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 = C 360**

**D x 4 = Sq or R x 8 = Sq 480**

**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/geometric concept that a diameter is a line passing through a circle, is in error as a diameter consists of two single lines of radial length; however mathematically these two radial lines can be added together and consider to be a single line; and as such mathematically, any length of straight line e.g. the right angle of a square, can be used as a diameter length.

2. The proposition/geometric concept that a diameter is a line passing through a circle, is in error as a diameter consists of two single lines of radial length; however mathematically these two radial lines can be added together and consider to be a single line; and as such mathematically, any length of straight line e.g. the right angle of a square, can be used as a diameter length.

Geometrically two radial lengths of a circle are not a single length; as there is a gap in the form of a hole at the centre of any radiated circle; which serves to separate the central ends of radiated lines, from each other.

Geometrically two radial lengths of a circle are not a single length; as there is a gap in the form of a hole at the centre of any radiated circle; which serves to separate the central ends of radiated lines, from each 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*****

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 LINE***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*

*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.*