9. Computing π – The Indian Method
Articles,  Blog

9. Computing π – The Indian Method


There are certain interesting numbers in Mathematics like Pi, e, etc. The numbers basically started with natural numbers one to infinity of course infinity was not even conceived as a number and so too was zero so in the initial phases and therefore we start with natural numbers and then we move on to whole numbers, then we move on to integers, the negative numbers, that itself is an interesting history on its own, anyway, The point that I wanted to make is this number π(pi) which we refer to, to represent the ratio of circumference to the diameter has a very interesting history. Why were they interested in calculating the value of Pi to high degree? One way to see is, it has to do with the area of a circle and you cannot avoid it, but then its nature was never understood. Infact there are books which describe it as the most mysterious number. And today we have understood that to be a transcendental number. It was first recognized as irrational number, and irrationality itself is not so easy to understand, so people struggled for thousands of years, and to have an exact representation of Pi, so since it is irrational, it may not be possible for you unless you have an infinite series in the right hand side. So any series that you have, which adds up, if it is a terminating series, so it’ll turn into a rational number. So unless you have infinite series, there are various infinite
series representations for Pi. One interesting series which is today ascribed to Gregory – Leibniz, is a series which was codified in the form of verse almost 300 years before
Gregory – Leibniz by a mathematician called Madhava, as we understand today, Madhava was in 14th Century, so these people have been in 17th century. So historically this is something which is not so well known. And if the series were to be given a name which actually honours the founder, then it should be really called Madhava series, instead of Gregory–Leibniz series, this is a historical note about Pi. What was the context in which people wanted to know the value of Pi in the Indian tradition? If we look back, the kind of altars which
people have been using thousands and thousands of years before, for performing various rituals, every household was suppposed to have 3 altars, one in the form of a circle, the other in the form of square and the third in the form of semi circle. So these were called as “Tretaagni”. One is called “Aahavaneeya”, “Dakshina” and “Garhapatya”. The constraint that people imposed, was, that the three altars should have the same area. Now you can very easily see as to why they wanted to know the value of Pi. So it occurs in that context. So if the circle area as we understand today is “pi r squared” and that should be equated to “a squared” and the area should be same. and we have some sort of approximation in the Shulba Sutras. Shulba sutras as I told you is around 800 BC, we do not know exactly, because there is no reference which is found in Shulba Sutras, which gives us a hint to make an estimate on the date of composition as we find for instance in Aryabhatiya or
later works. Aryabhata for instance gives a verse where in he clearly states, Shashtyabdaanam shashtyardaa vyateethaaH trayascha yugapaadaaH TryadhikaavimshatirabdaaH tadehamamajanmanoteetaaH So he very cleary says the year of composition. and after Shulba sutras, we find a nice approximation for Pi which has been given by Aryabhata. Aryabhata’s verse goes like this Chaturadhikam shatamashtaguNam dwashashTistatha sahasraaNaam | Ayutadwayavishkambhasya aasanno vrittapariNaahaH || This is a very interesting verse. Interesting in the sense that it gives you a very clear picture that it has been clearly recognized by Aryabhata that it is only approximate value. So how do we come to know, because he has used the word “aasanna”. “asanna” means that which is close by. aasannaH vrittapariNaahaH – how do they express? So they express the circumference on the left hand side in terms of diameter multiplied by some quantity. This is how it goes. Chaturadhikam shatam ashtaguNam. So he basically gives you a ratio. So circumference should be mentioned in terms of diameter. So here he says, if this were the diameter, then, you have to multiple by this term to get the circumference. ashtaguNa means multipled by 8. Chaturadhikam shatam – 104. So 4 above hundred . dwashashTistatha sahasraaNaam. sahasraaNaam dwashashTiH. dwashashTiH is 62. So Sixty two thousand. Sixty two thousand plus this So it comes to 62,832. Ayutadwayavishkambhasya. Vishkambha is the term used for Diameter. Vistaara, Vishkambha, Vyaasa, these are the terms that they generally use for diameter. Ayutam is 10,000. Ayutadwaya is 20,000. aasanno vrittapariNaahaH. vrittapariNaahaH – circumference of a circle. So circumference of a circle is close to this. What does this boil to? So this boils to getting a value which is 3.1416 as the value of Pi. So this is correct to 4 decimal places. Infact people have been asking, so how did Aryabhata get this value, which is correct to 4 decimal places. So it is not something which one can draw and then keep measuring by something, some scale by which you will be able to get this kind of accuracy, it is almost impossible. So they had a way of doubling methods, Circumference doubling, you divide that into various parts and you can nicely get some formulae, which essentially involves computing square root and square. So if you have a way, technique to find the square root correctly and the square if you have algorithms for that, then you will keep on increasing your accuracy by taking more and more divisions on it. So Polygon Doubling method seems to have been the method which should have been used by Aryabhata to get this value.

13 Comments

  • Klint shayler

    ANY RATIO THAT IS NOT THE RESULT OF A CIRCLE'S CIRCUMFERENCE DIVIDED BY A CIRCLE'S DIAMETER IS NOT PI AND THAT SHOULD BE EASY FOR YOU TO UNDERSTAND.

    YOUR VALUE OF PI = 3.141592653589793 IS WRONG BECAUSE IT HAS NOT BEEN DERIVED FROM DIVIDING THE CIRCUMFERENCE OF A CIRCLE BY THE DIAMETER OF A CIRCLE, INSTEAD YOUR VALUE OF PI WAS ORIGINALLY DERIVED FROM ARCHIMEDES’ MULTIPLE POLYGON LIMIT CALCULUS APPROACH THAT INVOLVES CONSTRUCTING CIRCLES AROUND POLYGONS AND ALSO CONSTRUCTING CIRCLES INSIDE OF POLYGONS.

    CONSTRUCTING CIRCLES INSIDE OF POLYGONS AND ALSO CONSTRUCTING CIRCLE'S AROUND POLYGONS IS NOT THE SAME AS CIRCUMFERENCE OF CIRCLE DIVIDED BY DIAMETER OF CIRCLE.

    IT IS IMPOSSIBLE FOR A POLYGON TO BECOME A CIRLCE AND THAT MEANS THAT IT DOES NOT MATTER HOW MANY EDGES THAT A POLYGON HAS THERE WILL FOREVER BE A GAP BETWEEN THE EDGE OF THE POLYGON AND THE CURVATURE OF THE CIRCLE THAT CONTAINS THE POLYGON.
    A CIRCLE DOES NOT HAVE ANY EDGES.

    IT IS IMPOSSIBLE FOR A POLYGON WITH AN INFINITE AMOUNT OF EDGES TO EXIST BECAUSE A POLYGON IS KNOWN AND IDENTIFIED BY THE AMOUNT OF EDGES THAT A POLYGON HAS FOR EXAMPLE A DECAGON IS A POLYGON THAT IS KNOWN TO HAVE 10 EDGES.

    ARCHIMEDES’ MULTIPLE POLYGON CALCULUS LIMIT APPROACH TO FINDING PI CAN ONLY PRODUCE APPROXIMATIONS FOR PI BUT NEVER PRODUCE THE REAL VALUE OF PI.

    USING CALCULUS TO DISCOVER PI IS A WASTE OF TIME AND EFFORT BECAUSE THERE WILL FOREVER BE AN AREA UNDER THE CURVATURE OF A CIRCLE BECAUSE THE CURVATURE OF SPACE IS FRACTAL IN NATURE. THE MORE AREA UNDER A CURVE IS MAGNIFIED THE MORE CREVICES CAN BECOME VISIBLE.

    ACADEMIC MATHEMATICIANS OF TODAY ARE NOW USING COMPUTER SIMULATIONS BASED ON A INFINITE SERIES OF NUMBERS THAT THEY ASSUME WILL JUST MAGICALLY RESULT IN THE CORRECT VALUE OF PI BUT THE PROBLEM WITH INFINITE SERIES IS HOW CAN ANYBODY USE A RANDOM SERIES OF NUMBERS TO CONVERGE TO PI WHEN THEY HAVE NOT DISCOVERED PI DUE TO THE FACT THAT THEY HAVE NEVER DIVIDED THE CIRCUMFERENCE OF A CIRCLE BY THE DIAMETER OF A CIRCLE IN THERE ENTIRE LIFES ?

    INFINITE SERIES IS NOT THE SAME AS CIRCUMFERENCE OF CIRCLE DIVIDED BY DIAMETER OF CIRCLE AND THAT MEANS THAT ANYBODY THAT IS USING INFINITE SERIES TO FIND PI IS EITHER KNOWINGLY OR UNKNOWINGLY AN IDIOT.

    PI MEANS CIRCUMFERENCE OF CIRCLE DIVIDED BY DIAMETER OF CIRCLE.

    I AM HERE TO STOP MATHEMATICIANS FROM COMMITTING FRAUD.

    IF YOU DO NOT UNDERSTAND THAT ANY RATIO THAT IS NOT DERIVED FROM A CIRCLE'S CIRCUMFERENCE DIVIDED BY A CIRCLE'S DIAMETER IS NOT PI THEN YOU ARE CONFUSED.

    YOU ARE COMMITTING FRAUD BY IGNORING THAT FACT AND I CANNOT TOLERATE THAT.

    YOU MUST DIVIDE THE CIRCUMFERENCE OF A CIRCLE BY THE DIAMETER OF A CIRCLE TO DISCOVER PI OR ALTERNATIVELY DIVIDE THE SURFACE AREA OF A CIRCLE BY THE SURFACE AREA OF THE SQUARE THAT IS LOCATED ON THE RADIUS OF THE CIRCLE.

    YOU CAN DIVIDE THE CIRCUMFERENCE OF A CIRCLE INTO ALMOST ANY NUMBER OR RATIO USING COMPASS AND STRAIGHT EDGE AND THAT IS SO EASY TO DO.

    COMPUTE THE MEASURE FOR THE DIAMETER OF THE CIRCLE BY USING THE DIAMETER OF THE CIRCLE AS THE EDGE OF A RIGHT TRIANGLE.

    I MUST REPEAT: ANY RATIO THAT IS NOT THE RESULT OF A CIRCLE'S CIRCUMFERENCE DIVIDED BY A CIRCLE'S DIAMETER IS NOT PI AND IF YOU STILL DO NOT UNDERSTAND THAT THEN YOU ARE AN IDIOT.

    I AM NOT SAYING THAT YOU’RE A TOTAL IDIOT I AM JUST JUDGING YOUR BEHAVIOUR.

    COMMON SENSE SHOULD TELL YOU THAT TO GET PI YOU MUST DIVIDE THE CIRCUMFERENCE OF A CIRCLE BY THE DIAMETER OF A CIRCLE BECAUSE THE DICTIONARY SAYS THAT PI IS THE RATIO OF A CIRCLE'S CIRCUMFERENCE DIVIDED BY A CIRCLE'S DIAMETER.

    THE DICTIONARY SAYS THAT PI IS THE RATIO OF A CIRCLE'S CIRCUMFERENCE DIVIDED BY A CIRCLE'S DIAMETER.
    THERE ARE ONLY 2 WAYS TO DISCOVER PI AND THAT IS TO DIVIDE THE CIRCUMFERENCE OF A CIRCLE BY THE DIAMETER OF A CIRCLE OR ALTERNATIVELY DIVIDE THE SURFACE AREA OF A CIRLCE BY THE SURFACE AREA OF THE SQUARE THAT IS LOCATED ON THE RADIUS OF THE CIRCLE. ANYTHING OTHER THAN CIRCUMFERENCE OF CIRCLE DIVIDED BY DIAMETER OF CIRCLE OR SURFACE AREA OF CIRCLE DIVIDED BY THE SURFACE AREA OF THE SQUARE THAT IS LOCATED ON THE RADIUS OF THE CIRCLE IS NOT PI.

    IF YOU ARE NOT DIVIDING THE CIRCUMFERENCE OF A CIRCLE BY THE DIAMETER OF A CIRCLE OR DIVIDING THE SURFACE AREA OF A CIRLCE BY THE SURFACE AREA OF THE SQUARE THAT IS LOCATED ON THE RADIUS OF THE CIRCLE THEN YOU CANNOT HONESTLY DISCOVER PI AND IF YOU CLAIM TO HAVE DISCOVERED PI WHILE REFUSING TO DIVIDE THE CIRCUMFERENCE OF A CIRCLE BY THE DIAMETER OF A CIRCLE OR DIVIDE THE SURFACE AREA OF A CIRLCE BY THE SURFACE AREA OF THE SQUARE THAT IS LOCATED ON THE RADIUS OF THE CIRCLE THEN YOU ARE A FRAUD BECAUSE THE ONLY 2 WAYS TO DISCOVER PI IS TO DIVIDE THE CIRCUMFERENCE OF A CIRCLE BY THE DIAMETER OF A CIRCLE OR DIVIDE THE SURFACE AREA OF A CIRLCE BY THE SURFACE AREA OF THE SQUARE THAT IS LOCATED ON THE RADIUS OF THE CIRCLE.

  • Klint shayler

    Proof that the shortest edge length of a Kepler right triangle is equal in measure to the circumference of a circle with a diameter that is equal in measure to 1 quarter of the second longest edge length of the Kepler right triangle:

    The shortest edge length of the Kepler right triangle is 12.

    If the hypotenuse of a Kepler right triangle is divided by the measure for the shortest edge length of the Kepler right triangle the result is the Golden ratio of cosine (36) multiplied by 2 = Phi = 1.618033988749895.

    The shortest edge length of the Kepler right triangle is 12.

    12 multiplied by the Golden ratio of cosine (36) multiplied by 2 = Phi = 1.618033988749895 = 19.41640786499874.

    The hypotenuse of a Kepler right triangle that has its shortest edge length equal to 12 is equal to 19.41640786499874.

    Apply the Pythagorean theorem to the hypotenuse of the Kepler right triangle and the shortest edge length of the Kepler right triangle to get the measure for the second longest edge length of the Kepler right triangle.

    19.41640786499874 squared = 376.996894379984929.

    12 squared = 144.

    376.996894379984929 subtract 144 = 232.996894379984929.

    The square root of 232.996894379984929 = 15.26423579416883.

    The second longest edge length of the Kepler right triangle that has its shortest edge length equal to 12 is equal to 15.26423579416883.

    15.264235794168832 divided by 4 = 3.816058948542208 the diameter of the circle.

    15.264235794168832 divided by the square root of the Golden ratio = 1.272019649514069 = 12.

    12 is the measure for both the circumference of the circle and the shortest edge length of the Kepler right triangle that has its hypotenuse equal to 19.41640786499874 while the second longest edge length of the Kepler right triangle is equal to 15.264235794168832.

    Circumference of circle and the shortest edge length of the mentioned Kepler right triangle are both equal to 12.

    1 quarter of the second longest edge length of the Kepler right triangle and the diameter of the circle are both equal to 3.816058948542208.

    12 divided by 3.816058948542208 = Pi = 3.144605511029692.

    Remember to multiply the shortest edge length of the Kepler right triangle that is 12 by Cosine (36) multiplied by 2 = The Golden ratio = Phi = 1.6180339887499 to get the measure for the hypotenuse of the Kepler right triangle = 19.41640786499874 then apply the Pythagorean theorem to the shortest edge length of the Kepler right triangle and the hypotenuse of the Kepler right triangle to get the measure for the second longest edge length of the Kepler right triangle.

    The diameter of the circle is equal to 1 quarter of the second longest edge length of the Kepler right triangle.

    The shortest edge length of the Kepler right triangle is equal in measure to the circumference of the circle that has a diameter that is equal in measure to 1 quarter of the second longest edge length of the Kepler right triangle.

    Divide the measure of the shortest edge length of the Kepler right triangle by the measure for 1 quarter of the second longest edge length of the Kepler right triangle to get Pi = 3.144605511029.

    Pi can also be calculated from the diagram of a circle contained inside of a square if the width of the square is the same measure as the diameter of the circle because if the perimeter of a square is divided by the square root of the Golden ratio = 1.272019649514069 then the result is the measure for the circumference of a the circle that has a diameter that is the same measure as the width of the square.

    Example:

    The width of the square = 3.816058948542208.

    Diameter of the circle that is contained inside of the square = 3.816058948542208.

    3.816058948542208 multiplied by 4 = 15.264235794168832.

    Perimeter of square that contains the circle that has a diameter that is the same measure as the width of the square = 15.264235794168832.

    15.264235794168832 divided by the square root of the Golden ratio = 1.272019649514069 = 12.

    12 is the measure for both the circumference of the circle that has a diameter that is the same measure as the width of the square.

    12 divided by 3.816058948542208 = Pi = 3.144605511029692.

    (Please click on to the following links or copy and them into your web browser):

    PLEASE DOWNLOAD THE GOOLE DRIVE LINKS,

    https://drive.google.com/file/d/1XIL8p7aqN3yQ0i7VCzAenfVLAu_gc0RJ/view?usp=sharing

    https://drive.google.com/file/d/1_d0-qWx1fc4rOZgvUvlXyYW15_udaPoM/view?usp=sharing

    https://drive.google.com/file/d/1XeYiSO8ls2vAnGOkxCdf82KnxXUv507y/view?usp=sharing

  • Klint shayler

    Pi can also be calculated from a square and a circle with the same surface area because if the edge of a square is multiplied by √√φ = 1.127838485561682 then the result is the diameter of a circle with the same surface area as the square and if the perimeter of a square is divided by √√φ = 1.127838485561682 the result is the circumference of a circle with the same surface area as the square. Circumference of circle divided by diameter of circle = π = 3.144605511029693144.

    Also you do NOT need Pi to create a circle and a square with the same surface area instead you MUST use √√φ = 1.127838485561682: http://www.wolframalpha.com/input/?i=√√φ

    The creation of a circle and a square with the same surface area can also be used to find Pi:

    If only the surface area for a circle is known and the desire is to know both the measure of the circumference and the diameter of the circle a solution for finding the measure for the diameter of the circle is to multiply the square root for the surface area by the ratio 1.127838485561682. To find the measure for the circumference of a circle when only the surface area of a circle is known a solution is to multiply the square root of the circle's surface area by 4 then divide the result of multiplying the square root of the circle's surface area by 4 by the ratio 1.127838485561682.Please remember that the ratio 1.127838485561682 is the square root of the ratio 1.272019649514069 and the ratio 1.272019649514069 is the square root of the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.

    If only the surface area for a circle is known and the desire is to know both the measure of the circumference and the diameter of the circle a solution for finding the measure for the diameter of the circle and the circumference of the circle is to divide the square root for the surface area of the circle by the ratio 1.127838485561682 resulting in quarter of the circle’s circumference. If 1 quarter of a circle’s circumference is multiplied by the square root of the Golden ratio = 1.272019649514069 the result is the diameter for the circle. Please remember that the ratio 1.127838485561682 is the square root of the ratio 1.272019649514069 and the ratio 1.272019649514069 is the square root of the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.

    For example.

    The edge of square is equal to 3.383515456685047.

    3.383515456685047 divided by 3 = √√φ = 1.127838485561682.

    3.383515456685047 multiplied by √√φ = 1.127838485561682 =

    3.816058948542207 the diameter of a circle with the same surface area as the square with a width equal to 3.383515456685047.

    3.383515456685047 multiplied by 4 = 13.534061826740188 the perimeter of the square.

    13.534061826740188 divided by √√φ = 1.127838485561682 = 12 the circumference of a circle with the same surface area as the square with a width equal to 3.383515456685047.

    12 divided by 3.816058948542207 = Pi = 3.144605511029693.

    Quadrature of the circle constants again:

    The Golden ratio Phi = Cosine (36) multiplied by 2 = 1.618033988749895.

    The square root of Phi = 1.272019649514069.

    1.272019649514069 squared = 1.618033988749895.

    The square root of the square root of Phi = 1.127838485561682.

    1.127838485561682 squared = 1.272019649514069.

    The true value of Pi = 4 divided by 1.272019649514069 = 3.144605511029693144.

    The square root of Pi = 2 divided by 1.127838485561682 = 1.773303558624324.

    2 divided by the square root of Golden Pi = 1.773303558624324 = 1.127838485561682.

    Please remember that the ratio 1.127838485561682 is the square root of the ratio
    1.272019649514069 and the ratio 1.272019649514069 is the square root of the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.

    1.773303558624324 squared = 3.144605511029693144.

    The ratio for the diameter of a circle divided by 1 quarter of the circle's circumference is the square root of the Golden ratio = 1.2720196495141 because if the circumference of a circle is divided into 12 equal parts and 3 chords are placed from the circle's circumference that has been divided into 12 at any of the 2 poles for the circle's diameter on a horizontal straight line perpendicular to the vertical diameter of the circle a Kepler right triangle can be formed with the diameter of the circle as the second longest edge length of the Kepler right triangle while the shortest edge length of the Kepler right triangle is equal in measure to 1 quarter of the perimeter of a regular polygon with 8 or more edges contained inside of the circle.

    The real value of Pi = 4/√φ = 3.144605511029693144 can be confirmed by always remembering the ratio of the diameter of a circle divided by 1 quarter of the circle's circumference = the square root of the Golden ratio Phi = 1.272019649514069.

    (Please click on to the following links or copy and them into your web browser):

    PLEASE DOWNLOAD THE GOOLE DRIVE LINKS,

    https://drive.google.com/file/d/1Gm6zTLz-dniKqiwEbtbCfl4mIJB4KIg0/view?usp=sharing

    https://drive.google.com/file/d/1C-UEOMRY0MOsMyPJp1jmT-PZnz4VjeDS/view?usp=sharing

    The true value of Pi = 3.144605511029 is NOT Transcendental:

    Pi = 4/√φ = 4 divided by 1.2720196495141 = 3.144605511029.
    π = 4/√φ = 3.144605511029693144.

    THE REAL VALUE OF Pi IS NOT TRANSCENDENTAL BECAUSE THE REAL VALUE OF PI = 4/√φ = 3.144605511029693144 IS THE ONLY VALUE OF PI THAT CAN FIT THE FOLLOWING POLYNOMIAL EQUATION:

    4th dimensional equation/polynomial for Golden Pi = 3.144605511029693

    Minimal Polynomial:

    x4 + 16×2 – 256 = 0.

    https://www.tiger-algebra.com/drill/x~4-16x~2-256=0/

    THE REAL VALUE OF PI = 4/√φ = 3.144605511029693144:

    Please copy and paste the following link into your web browser if you cannot click onto the following link:

    https://www.wolframalpha.com/input/?i=4+divided+by+the+square+root+of+the+golden+ratio

    PLEASE CLICK ON THE RED DOTS IN THE FOLLOWING LINK TO CONFIRM THAT THE REAL VALUE OF PI = 4/√φ = 3.1446 IS NOT TRANSCENDENTAL.

    THE REAL VALUE OF PI = 4/√φ = 3.144605511029693144.

    Minimal polynomial:

    x4 + 16×2 – 256 = 0

    https://www.wolframalpha.com/input/?i=x4+%2B+16×2+%E2%80%93+256+%3D+0

    3D plot of a graph proving that the real value of Pi is NOT transcendental:

    (Please click on to the following links or copy and them into your web browser):

    PLEASE DOWNLOAD THE GOOLE DRIVE LINK

    https://drive.google.com/file/d/1nT0xGIYHHim49pwjo9oo80MmyWaqOWj1/view?usp=sharing

    • Panagiotis Stefanides fourth order equation:

    http://www.stefanides.gr/Html/piquad.html

    • Panagiotis Stefanides: Quadrature of circle, theoretical definition:

    http://www.stefanides.gr/Html/QuadCirc.html

    • 2/Sqrt[Sqrt[GoldenRatio]]

    2/√√φ = the square root of 3.144605511029693144.

    (Square root of Pi = 2 divided by 1.127838485561682 = 1.773303558624324)

    http://www.wolframalpha.com/input/?i=2%2F%E2%88%9A%E2%88%9A%CF%86

    (-256 + 16 x^4 + x^8)

    (x8 + 16×4 – 256)

    http://www.wolframalpha.com/input/?i=-256+%2B+16+x%5E4+%2B+x%5E8&lk=1&assumption=%22ClashPrefs%22+-%3E+%7B%22Math%22%7DShow+less

    The Non Transcendental, Exact Value of π and the Squaring of the Circle 1:

    https://www.youtube.com/watch?v=ccxVW2MIbxA&lc=z232inlhiwiowtokuacdp4345l4off3xqdsfga0vawtw03c010c.1512761876258142

    The Non Transcendental, Exact Value of π and the Squaring of the Circle 3:

    https://www.youtube.com/watch?v=-QCtnZjZIsw

    Pi by phi quadrature: https://www.youtube.com/watch?v=CRkIKSkVzPA

    Pi by Phi saved archive: http://archive.is/b02DL

    Pi by Phi quadrature: http://quadrature-code.blogspot.co.uk/

    Quadrature blogspot conclusions http://quadrature-code.blogspot.co.uk/p/some-conclusions.html

    Quadrature blogspot Holistic: http://quadrature-code.blogspot.co.uk/p/holistic-view.html

    √√φ = 1.127838485561682 is the key to creating a circle and a square with the same surface area.

    The following Wolfram alpha site gives us information about the ratio √√φ = 1.127838485561682 =

    http://www.wolframalpha.com/input/?i=√√φ

    MEASURING PI SQUARING PHI: www.measuringpisquaringphi.com

    The square root of Phi = 1.272019649514069:

    (-1 – x^2 + x^4) http://www.wolframalpha.com/input/?i=%E2%88%9A%CF%86

    The square root of the square root of Phi = 1.127838485561682 .

    (-1 – x^4 + x^8) http://www.wolframalpha.com/input/?i=√√φ

  • Klint shayler

    THE REAL VALUE OF Pi IS NOT TRANSCENDENTAL BECAUSE THE REAL VALUE OF PI = 4/√φ = 3.144605511029693144 IS THE ONLY VALUE OF PI THAT CAN FIT THE FOLLOWING POLYNOMIAL EQUATION:

    4th dimensional equation/polynomial for Golden Pi = 3.144605511029693

    Minimal Polynomial:

    x4 + 16×2 – 256 = 0.

    https://www.tiger-algebra.com/drill/x~4-16x~2-256=0/

    THE REAL VALUE OF PI = 4/√φ = 3.144605511029693144:

    Please copy and paste the following link into your web browser if you cannot click onto the following link:

    https://www.wolframalpha.com/input/?i=4+divided+by+the+square+root+of+the+golden+ratio

    PLEASE CLICK ON THE RED DOTS IN THE FOLLOWING LINK TO CONFIRM THAT THE REAL VALUE OF PI = 4/√φ = 3.1446 IS NOT TRANSCENDENTAL.

    THE REAL VALUE OF PI = 4/√φ = 3.144605511029693144.

    Minimal polynomial:

    x4 + 16×2 – 256 = 0

    https://www.wolframalpha.com/input/?i=x4+%2B+16×2+%E2%80%93+256+%3D+0

    3D plot of a graph proving that the real value of Pi is NOT transcendental:

    (Please click on to the following links or copy and them into your web browser):

    PLEASE DOWNLOAD THE GOOLE DRIVE LINK

    https://drive.google.com/file/d/1nT0xGIYHHim49pwjo9oo80MmyWaqOWj1/view?usp=sharing

    • Panagiotis Stefanides fourth order equation:

    http://www.stefanides.gr/Html/piquad.html

    • Panagiotis Stefanides: Quadrature of circle, theoretical definition:

    http://www.stefanides.gr/Html/QuadCirc.html

    • 2/Sqrt[Sqrt[GoldenRatio]]

    2/√√φ = the square root of 3.144605511029693144.

    (Square root of Pi = 2 divided by 1.127838485561682 = 1.773303558624324)

    http://www.wolframalpha.com/input/?i=2%2F%E2%88%9A%E2%88%9A%CF%86

    (-256 + 16 x^4 + x^8)

    (x8 + 16×4 – 256)

    http://www.wolframalpha.com/input/?i=-256+%2B+16+x%5E4+%2B+x%5E8&lk=1&assumption=%22ClashPrefs%22+-%3E+%7B%22Math%22%7DShow+less

  • Klint shayler

    To get the correct measure for a circle’s diameter and to prove that Golden Pi = 4/√φ = 3.144605511029693144 is the true value of Pi by applying the Pythagorean theorem to all the edges of a Kepler right triangle when using the second longest edge length of a Kepler right triangle as the diameter of a circle then the shortest edge length of a Kepler right triangle is equal in measure to 1 quarter of a circle’s circumference. Also if the radius of a circle is used as the second longest edge length of a Kepler right triangle then the shortest edge length of a Kepler right triangle is equal to one 8th of a circle’s circumference:

    Example 1:

    The circumference of the circle is 12 but the measure for the diameter of the circle is not yet known. To discover the measure for the diameter of the circle apply the Pythagorean theorem to both 1 quarter of the circle’s circumference and also the result of multiplying 1 quarter of the circle’s circumference by the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.

    Divide the diameter of the circle by the square root of the Golden ratio = 1.272019649514069 to confirm that the edge of the square that has a perimeter that is equal to the numerical value for the circumference of the circle is equal to 1 quarter of the circle’s circumference.

    Multiply the edge of the square by 4 to also confirm that the perimeter of the square has the same numerical value as the circumference of the circle.

    Divide the measure for the circumference of the circle by the measure for the diameter of the circle to discover the true value of Pi.

    Multiply Pi by the diameter of the circle to also confirm that the circumference of the circle has the same numerical value as the perimeter of the square.

    The second longest edge length of a Kepler right triangle is used as the diameter of a circle in this example. 12 divided by 4 is 3 so the shortest edge length of the Kepler right triangle is 3.

    The hypotenuse of a Kepler right triangle divided by the shortest edge length of a Kepler right triangle produces the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.

    According to the Pythagorean theorem the hypotenuse of any right triangle contains the sum of both the squares on the 2 other edges of the right triangle.

    The shortest edge length of the Kepler right triangle is 3 and since the ratio gained from dividing the hypotenuse of a Kepler right triangle by the measure for the shortest edge of the Kepler right triangle is the Golden ratio of Cosine (36) multiplied by 2 = 1.61803398874989 then the measure for the hypotenuse of a Kepler right triangle that has its shortest edge length as 3 is 4.854101966249685.

    3 times the Golden ratio = 4.854101966249685.

    4.854101966249685 divided by 3 is the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.

    The square root of the Golden ratio = 1.272019649514069

    4.854101966249685 squared is 23.562305898749058.

    3 squared is 9.

    23.562305898749058 subtract 9 = 14.562305898749058

    The square root of 14.562305898749058 is 3.816058948542208.

    Remember that the second longest edge length of the Kepler right triangle is used as the diameter of a circle.

    The measure for both the second longest edge length of this Kepler right triangle and the diameter of the circle is 3.816058948542208.

    3 times the square root of the Golden ratio = 3.816058948542208.

    Remember that the shortest edge length of this Kepler right triangle is 3 and is equal to 1 quarter of a circle’s circumference that has a measure of 12 equal units.

    Circumference of circle is 12

    Diameter of circle is 3.816058948542208.

    Diameter of circle is 3.816058948542208 divided by the square root of the Golden ratio = 1.272019649514069 = 3 the edge of the square.

    3 multiplied by 4 = 12.

    The perimeter of the square = 12.

    12 divided by 3.816058948542208 = Golden Pi = 3.144605511029693144.

    12 divided by 3 times the square root of the golden ratio = Pi = 3.144605511029693144.

    4/√φ = Pi = 3.144605511029693144 multiplied by the diameter of the circle = 3.816058948542208 = 12.

    The circumference of the circle is the same measure as the perimeter of the square.

    4/√φ = 3.144605511029693144 is the true value of Pi.

    (Please click on to the following links or copy and them into your web browser):

    PLEASE DOWNLOAD THE GOOLE DRIVE LINKS,

    Squaring the circle geometry of 2 Kepler right triangle Golden Pi proof(main proof):

    https://drive.google.com/file/d/1_cIt1pUbu2TvPzb17TnI9AQeoSOEvNrD/view?usp=sharing

    Scan of 2 Kepler right triangle Golden Pi proof ( Main diagram)

    https://drive.google.com/file/d/1zjAUOo_iK2pBmuDTqdHZqa3XwmubW61D/view?usp=sharing

    Kepler right triangle diagram with squares upon the edges of the Kepler right triangle:

    https://drive.google.com/file/d/1iBtXYy06yv9UWtGP5mwMXyt80vaysFvR/view?usp=sharing

    Kepler right triangle construction method:

    https://drive.google.com/file/d/15DNXB_xNP2f2jCroC0FyNUyBYGhVAA2J/view?usp=sharing

    PYTHAGOREAN THEOREM:

    https://en.wikipedia.org/wiki/Pythagorean_theorem

    Golden ratio: https://en.wikipedia.org/wiki/Golden_ratio

    THE REAL VALUE OF Pi IS NOT TRANSCENDENTAL BECAUSE THE REAL VALUE OF PI = 4/√φ = 3.144605511029693144 IS THE ONLY VALUE OF PI THAT CAN FIT THE FOLLOWING POLYNOMIAL EQUATION:

    4th dimensional equation/polynomial for Golden Pi = 3.144605511029693

    Minimal Polynomial:

    x4 + 16×2 – 256 = 0.

    https://www.tiger-algebra.com/drill/x~4-16x~2-256=0/

    THE REAL VALUE OF PI = 4/√φ = 3.144605511029693144:

    Please copy and paste the following link into your web browser if you cannot click onto the following link:

    https://www.wolframalpha.com/input/?i=4+divided+by+the+square+root+of+the+golden+ratio

    PLEASE CLICK ON THE RED DOTS IN THE FOLLOWING LINK TO CONFIRM THAT THE REAL VALUE OF PI = 4/√φ = 3.1446 IS NOT TRANSCENDENTAL.

    THE REAL VALUE OF PI = 4/√φ = 3.144605511029693144.

    Minimal polynomial:

    x4 + 16×2 – 256 = 0

    https://www.wolframalpha.com/input/?i=x4+%2B+16×2+%E2%80%93+256+%3D+0

    3D plot of a graph proving that the real value of Pi is NOT transcendental:

    (Please click on to the following links or copy and them into your web browser):

    PLEASE DOWNLOAD THE GOOLE DRIVE LINK

    https://drive.google.com/file/d/1nT0xGIYHHim49pwjo9oo80MmyWaqOWj1/view?usp=sharing

    • Panagiotis Stefanides fourth order equation:

    http://www.stefanides.gr/Html/piquad.html

    • Panagiotis Stefanides: Quadrature of circle, theoretical definition:

    http://www.stefanides.gr/Html/QuadCirc.html

  • Klint shayler

    PI MEASUREMENT CONFIRMED TO BE 3.1446:

    THE FOLLOWING VIDEOS HAVE BEEN SEEN MANY TIMES WHERE THE CURVATURE OF A CYLINDER WITH A DIAMETER OF 1 METER IS MEASURED AND THE PI CIRCUMFERENCE IS REVEALED TO BE 3.1446 AND NOT 3.1415 OR 3.1416

    Pythagorean theorem: https://en.wikipedia.org/wiki/Pythagorean_theorem

    YouTube downloader, download videos from YouTube and watch them later and keep the videos: YouTube downloader for Windows:

    https://ytd-video-downloader-free.en.softonic.com/download

    YouTube downloader for Mac:

    https://www.macxdvd.com/free-youtube-video-downloader-mac/free-update.htm?run.myoutubedl

    Felder group: FORMAT-4® – profit H08 – CNC machining center:

    https://www.youtube.com/watch?v=EkOYPmYUAKU

    Videos for the true value of Pi: 3.144605511029: Introduction to finding the real value of Pi:

    https://www.youtube.com/watch?v=1DawMQ1LV6Q

    Physical measurement for the real value of Pi part 1:

    https://www.youtube.com/watch?v=iVNrhqLN110

    Physical measurement for the real value of Pi part 2:

    https://www.youtube.com/watch?v=Ck_pe81qZ9Q

    Physical measurement for the real value of Pi part 3:

    https://www.youtube.com/watch?v=fATlIzht7VI

    Physical measurement for the real value of Pi part 4:

    https://www.youtube.com/watch?v=DSAK6XcdVuI

    Physical measurement for the real value of Pi part 5:

    https://www.youtube.com/watch?v=KFGww0z6HRw

    Physical measurement for the real value of Pi part 6:

    https://www.youtube.com/watch?v=KTJsCzHLsEo

    Physical measurement for the real value of Pi part 7:

    https://www.youtube.com/watch?v=ehjANdRlktw

    Pi Math Proof: http://measuringpisquaringphi.com/pi-math-proof/

    Proof 7 Part 2 Pi Math Proof:

    https://www.youtube.com/watch?v=ohFXjnPOOnw&t=4s

    Kepler right triangle math proof for Pi;

    https://www.youtube.com/watch?v=nvja8rGCbzY&t=123s

    Fixing Correcting the problems caused by using traditional Pi:

    https://www.youtube.com/watch?v=X1ZAtn6g-Vg

    Geo Proof 1 Brand: https://www.youtube.com/watch?v=FBKl62-13SQ&t=52s

    Geo Proof 2 Brand: https://www.youtube.com/watch?v=E1KUSEfr46k&t=12s

    Geo Proof 4 Brand: https://www.youtube.com/watch?v=d22iTENHGwU&t=69s

    Geo Proof 6 Brand: https://www.youtube.com/watch?v=VVwJ4J4pUFQ

    5 more squared circle mathematical constants:

    https://www.youtube.com/watch?v=BYlEvYflOps&t=6s

    The Non Transcendental, Exact Value of π and the Squaring of the Circle 1:

    https://www.youtube.com/watch?v=ccxVW2MIbxA&lc=z232inlhiwiowtokuacdp4345l4off3xqdsfga0vawtw03c010c.1512761876258142

    Pi by phi quadrature: https://www.youtube.com/watch?v=CRkIKSkVzPA

    Measuring Pi squaring phi: www.measuringpisquaringphi.com

    More information about Pi, Archimedes boundary limit claim is a DOGMA. :

    https://translate.google.com/translate?hl=en&sl=ro&u=http://www.gazetamatematica.net/%3Fq%3Dnode/702&prev=search

    I have found the EXACT VALUE OF PI = 4/√φ = 3.1446.

    3.1446 IS THE EXACT 100% REAL VALUE OF PI.

    PURCHASE A CIRCLE CUTTER WITH A 200-CENTIMETER DIAMETER FOR £65:

    https://www.wholesaleglasscompany.co.uk/acatalog/Silberschnitt-Circle-Cutter-with-6-Steel-Wheels.html?gclid=EAIaIQobChMInd_Uz8GO4AIVyZ3tCh2L3whtEAQYBCABEgIEPPD_BwE#SID=163

    BRISTOL BOARD THAT IS LARGER THAN AO. MEASURMENT FOR BRISTOL BOARD = 3050MM X 1220MM:

    https://www.amazon.co.uk/White-Foamex-Sheet-CHOOSE-1189mm/dp/B00QQD6RIE/ref=pd_rhf_se_p_img_2?_encoding=UTF8&refRID=W6GPAP6ZDGE6ZVX8QZ5N&th=1

    BEAM COMPASS WITH A RADIUS THAT IS LARGER THAN 500MM:

    https://www.londongraphics.co.uk/ecobra-beam-complete-compass

    Pi Tape Easy read: http://www.pitape.co.uk/products.asp#Easy_Read

    Pi tape standard: http://www.pitape.co.uk/products.asp

    Pi tape pricing: http://www.pitape.co.uk/pdf/PiTapeEuroPriceList.pdf

    SOFT 5 METER PLUS ENGINEERING TAPE 1:

    https://www.zoro.co.uk/shop/hand-tools/tape-measures/0-34-297-30m-closed-fibreglass-tape-measure/p/ZT1003127X

    SOFT 5 METER PLUS ENGINEERING TAPE 2:

    https://www.zoro.co.uk/shop/hand-tools/long-tapes/2-34-791-100'-30m-open-fibreglass-tape/p/ZT1003152X?utm_campaign=pla-Hand+Tools+-+&utm_source=google&utm_medium=shopping-pla&utm_keyword=ZT1003152X&istCompanyId=6aa6787b-063e-4414-802d-129f235df603&istItemId=wtqitwppwr&istBid=tztt&gclid=EAIaIQobChMIl6jk3cmO4AIVCp3tCh29wwqdEAQYBSABEgKkvfD_BwE

    SOFT 5 METER PLUS ENGINEERING TAPE 3:

    https://www.zoro.co.uk/shop/hand-tools/tape-measures/surveyor's-fibreglass-tape-measure/f/34064

    5 PLUS METER HARD ENGINEERING TAPES:

    ELECTRONIC 5-METER HARD ENGINEERING TAPE 1:

    https://www.amazon.co.uk/Tacklife-Screwdriver-Magnetic-Self-Calibration-Measurements/dp/B07GS423BG/ref=sr_1_17?ie=UTF8&qid=1548612333&sr=8-17&keywords=Tape+Measures

    ELECTRONIC 5-METER HARD ENGINEERING TAPE 2:

    https://www.amazon.co.uk/dp/B07GNL7KKB/ref=psdc_1939126031_t3_B07GS423BG

    VARIOUS 5-METER HARD TAPE MEASURES WITH METAL BLADES:

    https://www.thetapestore.co.uk/tapes-rules/tape-measures/tape-length/5m-16ft-tape-measures

Leave a Reply

Your email address will not be published. Required fields are marked *