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Thread: squaring the circle with compass/straightedge to better than 5/100,000th's numerical accuracy

  1. #1 squaring the circle with compass/straightedge to better than 5/100,000th's numerical accuracy 
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    squaring the circle with only compass and straight edge... the impossible geometry problem.

    i thought i'd share my extremely close approximation of squaring the circle with compass and straight edge (numerically off by 0.000048132910080579383).

    ...& suppose you have a brand new apple macbook with retina display 2560x1600 pixels and you fill your screen top to bottom with a perfectly pi-area circle and square... the error in pi-area-square-height produced by this method would be much less than a single pixel on the screen... roughly 1/50th of a retina-pixel error by my calculations.

    method:

    step 0: construct a circle with radius equal to 1-- area equals pi exactly

    step 1: construct a regular pentagon side length equals 1
    instructions can be found here: https://en.wikipedia.org/wiki/Pentagon
    *please note in my drawing below i erroneously used an approximated rather than exact pentagon construction method, but i've included instructions for the exact method at the bottom

    step 2: create a line between 2 non-adjacent corners of the pentagon -- this line is exactly length Phi=1.6180339887...
    PHI: The Divine Ratio

    step 3: from the Phi line construct a square area=Phi^2=2.6180339887... -- side length Phi=1.6180339887...
    https://mathbitsnotebook.com/Geometr...ionSquare.html

    step 4: divide the square into 5 equal sections
    https://www.mathopenref.com/printdividesegment.html

    step 5: extend the square and create a rectangle beside it equal to one of the 5 sections of the square

    3.141640786499873817846 = rectangle area = 6/5*Phi^2
    3.141592653589793238463 = pi

    approximation error = 0.000048132910080579383 = 3.141640786499873817846 - 3.141592653589793238463

    step 6: square the rectangle
    Squaring A Rectangle

    square area = 3.141640786499873817846
    circle area = 3.141592653589793238463
    approximation error = 0.000048132910080579383

    phi = (1+sqrt(5))/2 = 1.6180339887
    source: https://en.wikipedia.org/wiki/Phi

    (6/5)*((1+sqrt(5))/2)^2 = 3.141640786499873817846

    ---------------------------------
    Please note: the pentagon construction method i used in the first step of the drawing is approximated but at the very bottom I have included instructions for constructing the pentagon with compass and straightedge exactly.






    please note the exact method of pentagon construction is as follows:

    Source: https://en.wikipedia.org/wiki/Pentagon
    "Pentagon at a given side length
    Draw a segment AB whose length is the given side of the pentagon.
    Extend the segment BA from point A about three quarters of the segment BA.
    Draw an arc of a circle, centre point B, with the radius AB.
    Draw an arc of a circle, centre point A, with the radius AB; there arises the intersection F.
    Construct a perpendicular to the segment AB through the point F; there arises the intersection G.
    Draw a line parallel to the segment FG from the point A to the circular arc about point A; there arises the intersection H.
    Draw an arc of a circle, centre point G with the radius GH to the extension of the segment AB; there arises the intersection J.
    Draw an arc of a circle, centre point B with the radius BJ to the perpendicular at point G; there arises the intersection D on the perpendicular, and the intersection E with the circular arc that was created about the point A.
    Draw an arc of a circle, centre point D, with the radius BA until this circular arc cuts the other circular arc about point B; there arises the intersection C.
    Connect the points BCDEA. This results in the Pentagon"

    Source: https://en.wikipedia.org/wiki/Pentagon


    Image Source: https://en.wikipedia.org/wiki/Pentagon


    Last edited by devin-m; December 1st, 2017 at 11:47 AM.
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  3. #2  
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    https://science.howstuffworks.com/ma...cci-nature.htm

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



    Image Source: http://webecoist.momtastic.com/asset...piral-main.jpg


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  4. #3  
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    Explaining the causes of the quadrature of the circle.

    First regarding the creation of a circle and a square with equal perimeters meaning the circumference of the circle is equal in measure to the perimeter of the square, the width of the square must be equal to 1 quarter of the circle’s circumference resulting in the perimeter of the square being the same measure as the circumference of the circle. For example if the circumference of the circle is 8 then the edge of the square with a perimeter that is equal to a circle with a circumference of 8 must be 2. Also if the square and circle share the same centre and the circumference of the circle is equal to the perimeter of the square then the radius of the circle must be the longer measure of a 1.272019649514069 ratio rectangle while half the central width of the square must be the shorter measure of a 1.272019649514069 ratio rectangle. If a circle and square are created with the perimeter of the square being the same measure as the circumference of the circle and the circle and square do NOT share the same centre then the diameter of the circle CAN be the longer measure of a 1.272019649514069 ratio rectangle while the edge of the square CAN be the shorter measure of a 1.272019649514069 ratio rectangle but this is NOT compulsory.

    In another example the edge of the square is the longer measure of a 1.272019649514069 ratio rectangle while the shorter measure of the 1.272019649514069 ratio rectangle is equal in measure to 1 quarter of the real value of Pi = 3.144605511029693.

    Second regarding the creation of a circle and a square with equal areas the radius of the circle must be the longer measure of a 1.127838485561682 ratio rectangle while half the central width of the square must the shorter measure of a 1.127838485561682 ratio rectangle if the circle and square share the same centre. If the circle and square do NOT share the same centre and the circle and square have the same surface area then the diameter of the circle CAN be the longer measure of 1.127838485561682 ratio rectangle while the edge of the square CAN be the shorter measure of a 1.12783848556 ratio rectangle but this is NOT compulsory.

    The relationship between the circle and the square having the same perimeter or the same area is a result of 2 ratios that are related to the Golden ratio of cosine (36) multiplied by 2 = 1.618033988749895 being used and those 2 ratios again are:


    ˇ The square root of the Golden ratio also called the Golden root = 1.272019649514069. The Golden root 1.272019649514069 is the result of either the diameter of a circle being divided by 1 quarter of a circle’s circumference or the radius of a circle being divided by one 8th of a circle’s circumference. The square root of the Golden ratio = 1.272019649514069 also applies to the perimeter of a square divided by the circumference of a circle with a diameter equal to the width of the square. The square root of the Golden ratio = 1.272019649514069 also applies to the surface area of a square divided by the surface area of a circle with a diameter equal to the width of the square. The square root of the Golden ratio = 1.272019649514069 also applies to the surface area of a square divided by the surface area of a circle with a circumference equal in measure to the perimeter of the square. The second longest edge length of a Kepler right triangle divided by the shortest edge length of a Kepler right triangle is the square root of the Golden ratio also called the Golden root = 1.272019649514069.The square root of the Golden ratio = 1.272019649514069 can also be gained if the surface area of circle is multiplied by 16 and then the result of the surface area of a circle being multiplied by 16 is then divided by the circumference of the circle squared. If the measure for the diameter of a circle is multiplied by 4 and the result of multiplying the measure of a circle’s diameter by 4 is divided by the measure for the circumference of a circle the result is also the square root of the Golden ratio also called the Golden root = 1.272019649514069.



    ˇ The square root of the Golden root = 1.127838485561682. The square root of the Golden root 1.127838485561682 can be gained if the diameter of a circle that has the same surface area as a square is divided by the width of the square that has the same surface area as the circle. The square root of the Golden root 1.127838485561682 can also be gained if the radius of a circle that has the same surface area as a square is divided by half the width of the square that has the same surface area as the circle. The square root of the Golden root 1.127838485561682 can also be gained if a circle and a square with the same surface area are created and the perimeter of the square is divided by the circumference of the circle. The second longest edge length of a Illumien right triangle divided by the shortest edge length of a Illumien right triangle is the ratio The square root of the Golden root = 1.127838485561682.If a circle and a square are created with equal areas of measure and the width of the square of the square is divided by the ratio 1.127838485561682 the result is equal to 1 quarter of the circle’s circumference that has the same surface area as the square and if 1 quarter of the circle’s circumference is multiplied by the ratio 1.272019649514069 the result is the measure for the diameter of the circle . If the width of the square is multiplied by the ratio 1.127838485561682 the result is the measure for the diameter of the circle with the same surface area as the square.
    If a circle and a square are created with equal areas of measure and half the width of the square of the square is divided by the ratio 1.127838485561682 the result is equal to 1 eighth of the circle’s circumference that has the same surface area as the square and if 1 eighth of the circle’s circumference is multiplied by the ratio 1.272019649514069 the result is the measure for the radius of the circle. If half the width of the square is multiplied by the ratio 1.127838485561682 the result is the measure for the radius of the circle with the same surface area as the square.

    If a circle and a square have been created with the circumference of the circle being equal in measure to the perimeter of the square and the desire is to gain the measure for the radius or the diameter of a circle that has the same surface area to the already existing square that already has a circle with a circumference that is equal to the perimeter of the square a solution is to divide the radius or the diameter of the circle that has a circumference equal to the perimeter of the square by the square root of the square root of Phi = 1.127838485561682 resulting in the measure for the radius or diameter of the circle that has the same surface area to the already existing square that has a perimeter that is equal in measure to the circumference of the already existing circle.Alternatively If a circle and a square have been created with the circumference of the circle being equal in measure to the perimeter of the square and the desire is to gain the measure for the edge of a square that has the same surface area as the already existing circle that has a circumference equal in measure to the perimeter of the already existing square a solution is to divide the radius or the diameter of the circle that has a circumference equal to the perimeter of the already existing square by the square root of the square root of Phi = 1.127838485561682 resulting in the measure for half the edge of the square or the edge of the square that has the same surface area as the already existing circle that has a circumference that is equal in measure to the perimeter of the already existing square. 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.
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  5. #4  
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    Squaring the circle with equal areas:

    Squaring the circle involves creating a circle with a circumference equal to the perimeter of a square. Also squaring the circle can involve creating a circle and a square with equal areas or approximate equal areas. Squaring the circle can also include harmonious relationships such as the part of the square that intersects the circle’s circumference can be similar to the radius of the circle or the same as the radius of the circle or equal to half of the square’s edge length. Squaring the circle with the area of the square being equal to the area of the circle usually cannot be achieved with 100% accuracy because traditional Pi 3.141592653589793 has been proven to be Transcendental in addition to being irrational. Traditional Pi 3.141592653589793 is Transcendental because Traditional Pi 3.141592653589793 does not fit any polynomial equations. Squaring the circle becomes possible and easy after traditional Pi 3.141592653589793 has been rejected and replaced with other values of Pi that are NOT transcendental. Golden Pi = 3.144605511029693 is irrational but Golden Pi is NOT transcendental because Golden Pi = 3.144605511029693 is the only value of Pi that fits the following polynomial equations: 8th degree polynomial for Golden Pi: π8 + 16π6 + 163π2 = 164.

    4th dimensional equation/polynomial for Golden Pi = 3.144605511029693 (x4 + 16x2 – 256 = 0).

    A polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1.
    Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.

    Both Golden Pi = 3.144605511029693 and Pi accepted as 22 divided by 7 = 3.142857142857143 can be used to create a circle and a square with equal areas of measure involving 100% accuracy. 2 examples of creating a circle and a square with 100% accuracy:

    Example 1 creating a circle and a square with equal areas involving 100% accuracy with Golden Pi = 3.144605511029693 :

    My question is it possible to create a circle with a surface area of 106 equal units because if we can create a circle with a surface area of 106 equal units of measure then we can also create a square with a surface area of 106 equal units of measure by creating a scalene right triangle with the second longest edge length as 9 equal units of measure taken from the diameter of the circle that has a surface area of 106 equal units of measure, while the shortest length of the scalene right triangle has 5 equal units of measure. The hypotenuse of a scalene right triangle with the second longest length as 9 equal units of measure and the shortest length of the scalene right triangle as 5 equal units of measure is equal in measure to the width of a square that has a surface area of 106 equal units of measure. We can use the theorem of Pythagoras: to prove that the square with a width equal to the longest length of the scalene right triangle also called the hypotenuse also has 106 equal units of measure.

    Area of circle = 106.
    Rational measure for the diameter of circle = 11.62.
    Irrational measure for the diameter of the circle = 11.61180790611399 according to Golden Pi = 3.144605511029693.

    Irrational measure for the diameter of the circle = 11.61180790611399 divided by the width of the square the square root of 106 = the square root of the Golden root = 1.127838485561683.
    The Golden root = 1.272019649514069.The Golden root = 1.272019649514069 is the square root of Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.

    Irrational measure for the circumference of the circle = 36.514555134584213 according to Golden Pi = 3.144605511029693.

    Square root of Golden Pi = 3.144605511029693 = 1.773303558624324


    9 squared = 81.
    5 squared = 25.
    81 + 25 = 106.

    Most values of Pi can confirm that if a circle has a rational measure for the diameter as 11.62 equal units of measure then the surface area of the circle with a rational measure for the diameter of 11.62 equal units of measure is 106 equal units of measure.
    “The ancient Egyptian square root of Pi rectangle and the ancient Egyptian square root for the Golden root rectangle and also the Pythagorean theorem”:

    Example 2 creating a circle and a square with equal areas involving 100% accuracy with 22 divided by 7 = 3.142857142857143 as Pi.

    Is it possible to create a circle with a surface area of 154 equal units because if we can create a circle with a surface area of 154 equal units of measure then we can also create a square with a surface area of 154 equal units of measure by creating a scalene triangle with the second longest edge length as 12 equal units of measure taken from the diameter of the circle that has a surface area of 154 equal units of measure, while the shortest length of the scalene triangle has 3 plus 1 equal units of measure. The hypotenuse of a scalene right triangle with the second longest length as 12 equal units of measure and the shortest length of the scalene triangle as 3 plus 1 equal units of measure is equal in measure to the width of a square that has a surface area of 154 equal units of measure. We can use the theorem of Pythagoras to prove that the square with a width equal to the longest length of the scalene right triangle also called the hypotenuse also has a surface area of 154 equal units of measure.


    Area of circle = 154.

    Diameter of circle = 14.

    Circumference of circle = 44.

    Ancient Egyptian Pi = 22 divided by 7 = 3.142857142857143.

    12 squared = 144.
    3 squared = 9.
    1 squared = 1

    144 + 9 + 1 = 154.
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