**Differential Geometry**

**Four Methods For Finding The Area To A Circle**

**Archimedes: Measurement of a Circle; Proposition 1 (Fifth & Sixth method for finding the exact area of a Circle)**

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

**Symmetrical Validation**

**Context Of A Degree**

**Finding The Area Of A Ring**

**Finding The Areas Of Concentric Rings**

**From The Cube To The Sphere**

**Twelve steps To The Sphere**

**Nine Spheres Of Creation**

**Nine Spheres Of The Solar System**

**Areas And Volumes Of Symmetrical Ovals And Ovoids**

**Minus Pi In Black And Yellow**

**Archimedes String**

See previous eclectic work at:

**geometry**-

**mass**-

**space**-

**time.com**/

**Beginning with a square measuring 120 cm x 120 cm, the perimeter of the square is 4480 cm and its area 14, 400 square cm.**

**Taking one side of the square measuring 120 cm to use as a diameter length, we multiply this length by 3 in order to obtain a length of 360 cm, to the circles circumferential length.**

**Therefore it follows.**

**1. As every circle has 360 degrees to its circumferential length, so each degree of the circle measures 1 cm in length.**

**2. As each side of the square measures 120 cm in length, so the three diameter length of the circles circumference are equal to, three quarters of the squares 480 cm perimeter length.**

**3. As the 360 cm circumference of the circle, measures three quarters of the 480 cm perimeter length of the square, the area of the circle will be three quarters of the square area to the square, 10, 800 square cm.**

*Four Methods For Finding The Area Of A Circle*

1. 14, 400 sq cm divide by 4 = 3, 600 sq cm multiplied by 3 =

1. 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.**2. 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.**3. Sumerian Method: 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.

4. 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.

4. 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, as per diagram below.**

*ARCHIMEDES: MEASUREMENT OF A CIRCLE***Archimedes: 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 which is 108 squares and exactly three quarters of the 12 x 12 squares' 144 square area.**

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

**However conversely and ironically, relative to his later formulation of the approximation Pi**

**Of**

**22/7 as an improper fraction =**

**3 & 1/7 as whole numbers (Whole diameter/unit lengths) and a 1/7 fraction of a single diameter length**

**22/7 converted to decimal; divide 22 by 7 = 3. 1428591**

**Although as his first proposition shows he was on the right track from the very beginning, the Euclidean and general belief that the exact length of a circle could not be measured, subsequently served to lead him astray.**

**6. Sixth Method "**

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

__right angle__length diameter x 3 = a 36 square length circumference**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**

**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.**

**7. Seventh Method "**

__Applicable only to__" the area to a circle, with a 10 square (right 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 diameter x 3 = a 30 square circumferential length**

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

**In sum**

**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**

**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.**

**12 X 12 square = 144 squares**

**144 divide 4 = 36 squares x 3 = 108 squares to the Circle**

**66 squares + 24 half squares = 72 squares to the Inner Square**

**Therefore as the square area of the Circle is three quarters that of its Square**

**So the square area of the Inner Square of a Circle, is 1/2 of the square area of the overall square, and 2/3 rds of the area of the Circle.**

Bi Radial Arithmetic Of Circle And SquareBi Radial Arithmetic Of Circle And Square

*Context Of A Degree*

*Finding 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 The 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.**

*Twelve 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**

Nine Spheres Of Creation

Nine Spheres Of Creation

*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**

*Minus Pi, In Black And Yellow*

We have two yellow square cards each measuring 120 cm x 120 cm

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

Each 60 x 60 square card holds one quadrant of the yellow square, and one quadrant of the black circle.

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

We have two yellow square cards each measuring 120 cm x 120 cm

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

Each 60 x 60 square card holds one quadrant of the yellow square, and one quadrant of the black circle.

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

**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

C. All Areas

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

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

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**

**In Sum**

Pi represents the physical inequality of the thickness of a length of string, being mathematically inducted into the Archimedes formula.

Pi represents the physical inequality of the thickness of a length of string, being mathematically inducted into the Archimedes formula.

*Archimedes String***During the times of Archimedes and extending up unto the present day, it has been believed that it is impossible to measure or calculate the exact length of a circle, and therefore the closet approximation was/is acceptable.**

**Therefore when Archimedes decided to devise a formula for calculating the length to circle, he was happy to accept the closest approximation he could obtain for his formula, using the most accurate, and only method available to him during those times; a cylinder, a length of string, and a straight rule.**

**Based on his formula Pi, which as an improper fraction is 22/7 or 3 & 1/7 as a whole number and a fraction, and given that we now know that a a circles length is 3 x its diameter length; it is a simple matter, to divide 3 into 22 to obtain 3 equal seven measurement unit diameter lengths, or 21/7 measurement units to the circles exact length, with 1/7th of a diameter length remaining over, due to the inaccuracy of using a length of string.**

**One might wonder why Archimedes did not twig the fact, that given 3 equal diameter lengths within the 22/7 measurement unit length he had obtained, that he did not realise that the thickness of the string was causing an excess.**

**However one has to remember that due to his mind set, that it was impossible to find the exact length of a circle; it was also impossible for him to accept, that a circle could be exactly 3 diameter lengths in total length.**

**When measuring the the cylinders circumference with a length of string, he would have first encircled the cylinder with the length of string, and then having brought one end of the string around the cylinder and into contact with the main length of string, he would have marked on the length of string, the point at which the end of the string met the main length; and then measured the length backwards from the marked point on the main length to the end of the string, up against a straight rule.**

**And this is where the error creeps in, because when the string is stretched around the cylinder, it is the inside of the circle of string that is directly in contact with the cylinders curvature, is constricted, while the opposite outside of the circle of string is stretched; and therefore when removing and then straightening the length of string to measure against a straight rule; it is the longer outer side of the stretched length that is being measured, rather than the constricted inner side, which was directly in contact with the cylinders surface.**

And as to Archimedes having transacted the circle to check or confirm his result, and so proceed to develop his formula; refer back to "Pi In Black And Yellow, because the linear area of every transaction line drawn, only serves to add to the error.

** **

And as to Archimedes having transacted the circle to check or confirm his result, and so proceed to develop his formula; refer back to "Pi In Black And Yellow, because the linear area of every transaction line drawn, only serves to add to the error.

*Compass & 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.

Pythagoras TheoremPythagoras Theorem

*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.*