You Cannot Square A Circle.

You Cannot Square A Circle.

You Cannot Square a Circle.

“…so, they spend all this money, but they’re not doing it right,” my friend Mitch said. “I know there’s a big business there. All we have to do is square the circle.”

“Huh? I interrupted. “What does square the circle mean?” You can’t turn a circle into a square!”

David was waving his hand like that obnoxious kid in your sixth-grade class who always had the answer. “I know, I know,” David said, barely able to control himself.

“Okay, David, tell us… but please spare us the long mathematical explanation” I begged.

Even though David’s a math savant, he doesn’t get to show off his knowledge very often. He could barely control himself.

You Cannot Square A Circle

“That a circle cannot be squared was not known to the ancients because the Greeks didn’t grasp the concept of irrational numbers,” he started. “Indeed, one of the Pythagoreans was killed for attempting to explain that the square root of two is indeed an irrational number. But before considering irrational numbers like the square root of two or pi, let’s consider rational numbers. Rational numbers are the RATIO of two integers.”

I already regretted asking David to explain. But it was too late.

“Rational numbers come in two flavors, those that repeat and those that terminate. 3/4 is the ratio of the integers three and four. Its decimal equivalent is .75. 3/4 terminates. 2/3 is a rational number that repeats. 2/3 equals .66666… The sixes repeat forever. 2/3 does not terminate.

But irrational numbers neither terminate nor repeat. The square root of two is equal to precisely 1.414213562.”

“I’ll bet you didn’t have to look that up,” I smirked.

David nodded and continued: “Pi is also an irrational number. The infinite decimal expansion of pi neither terminates nor repeats. The first few digits of pi are 3.14159265358979323846264338327950288. I could go on.”

“Please don’t” we said in unison.

“But you get the point. Unlike 3/4 which terminates, unlike 2/3 which repeats, irrational numbers neither terminate nor repeat.”

We didn’t get the point but he was on a roll.

“Now that we have distinguished between rational and irrational numbers, we have to distinguish between two kinds of irrational numbers. The square root of two is an irrational number. Pi is an irrational number that is also transcendental. Irrational numbers, like the square root of two, can be the answer to quadratic equations. X squared -2 equals zero has an answer. The answer is the square root of two. But there are no quadratic equations that have pi as an answer. Pi is not only irrational but transcendental.

The area of a circle is pi times the radius squared. A circle with a radius five has an area of 25 pi.

The area of a square is the length of a side times itself. The area of a square with a side of eight is equal to eight squared or 64.

Squaring the circle means finding a circle whose area is exactly equal to the area of a square using only a finite number of steps. Since the area of the circle will always be a transcendental number and the area of a square has to be an integer, this can never happen in a finite number of steps. Therefore, you cannot square a circle. It’s a metaphor for that which cannot be done.”

“You mean it’s impossible,” said Mitch. “Why didn’t you just say that in the first place?”

David was knowledgeable. Mitch was erudite.

Which are you? Are you presenting your brand, your business, and yourself in simple terms that your customers can understand? Or are you wrapping yourself in acres of tiresome talk? Are you using big words when small ones will do? Are you using long rationalizations when simple examples offer more clarity? Are you writing an SAT essay when you should be tweeting a competitive advantage?

If you are, you are squaring the circle.

By | 2017-11-28T12:23:48+00:00 November 28th, 2017|8 Comments

8 Comments

  1. Len Kaufman November 29, 2017 at 11:15 am - Reply

    All of which goes to explain why 45’s tweets (unfortunately) appeal to his base. Even when they are not true, they’re easy to understand.

  2. John Calia November 29, 2017 at 11:26 am - Reply

    This might be my favorite post yet. #mathgeek

  3. Hank Yunes November 29, 2017 at 12:45 pm - Reply

    Very interesting. I am reading the Da Vinci book and he spent a great amount of time throughout his life trying to solve the most famous of the ancient math puzzles: squaring the circle.

  4. john demarchi November 29, 2017 at 12:56 pm - Reply

    This email was several orders of magnitude more impressive than other emails I received today.

    Proving you can have your Pi, and eat it, too.

    • Bruce Turkel November 30, 2017 at 2:09 pm - Reply

      …proving you can bore some of the people some of the time, John?

  5. David J. Hawes MAS+ November 29, 2017 at 5:36 pm - Reply

    Bruce,

    Good message. I tend to be a knowledge flaunter. It’s a highly effective way to be percieived as condescending which is typically the beginnng of the ending.

    It’s the perfect way NOT to make people feel good about themselves.

    Thanks for the reminder!

  6. Anonymous November 30, 2017 at 10:47 am - Reply

    Bruce! Great message and to build on Hank’s comment Da Vinci actually continued to try to solve this math problem until his dying breath. I heard an interview with Walter Isaacson where he said that in the last page of Da Vinci’s notebook he had doodles trying to square the circle. You can also see it on his vitruvian man studies.

  7. Liddz February 23, 2018 at 5:47 pm - Reply

    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 + 16×2 – 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”: https://en.wikipedia.org/wiki/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.

    “The ancient Egyptian square root of Pi rectangle and the ancient Egyptian square root for the Golden root rectangle and also the Pythagorean theorem”:
    A square with a surface area of 154 equal units of measure can be created if the second longest edge length of a scalene right triangle has 12 equal units of measure while the shortest edge length of the scalene right triangle has 3 plus 1 equal units of measure. According to the Pythagorean theorem if a square has a width that is equal to the hypotenuse of a scalene right triangle that has its second longest edge length as 12 equal units of measure while the shortest edge length for the scalene right triangle has 3 plus 1 equal units of measure then the surface area of the square that has a width equal to the measure of the hypotenuse for the scalene right triangle that has its second longest edge length as 12 while its shortest edge length is 3 plus 1 equal units of measure is 154 equal units of measure. If the width of the square that has a surface area of 154 equal units of measure is then used as the longer length of a square root of ancient Egyptian Pi = 1.772810520855837 rectangle then a circle can be created with the shorter edge length of the square root of ancient Egyptian Pi = 1.772810520855837 rectangle being equal in measure to the radius of the circle with a surface area equal to the surface area of the square that has a surface are of 154 equal units of measure. According to ancient Egyptian Pi = 3.142857142857143 if the radius of a circle has 7 equal units of measure then the surface are of the circle is 154 equal units of measure. The measuring angles for a square root of ancient Egyptian Pi = 1.772810520855837 rectangle are 60.57369496075449 degrees and 29.42630503924551 degrees. 60.57369496075449 degrees can be gained when the square root of ancient Egyptian Pi = 1.772810520855837 is applied to the inverse of Tangent in Trigonometry. 29.4263050392455 degrees can be gained when the ratio 0.564076074817766 is applied to the inverse of Tangent in Trigonometry. If a circle with a diameter of 14 equal units of measure has already been created so that the surface area of the circle can have 154 equal units of measure according to ancient Egyptian Pi = 3.142857142857143 and the desire is to have a square that also has a surface area equal to the circle’s surface area of 154 equal units of measure then a solution is to add 1 quarter of the circle’s circumference that is 11 to the diameter of the circle with 14 equal units of measure and at the division point where 14 is subtracted from the diameter line of 25 equal units of measure draw right angles that can touch the circumference of a circle or a semi-circle if the diameter of 25 equal units of measure is divided into 2 halves. A rectangle with its longest length as 14 while its second longest length is the square root of 154 has the ratio for the square root of the Golden root = 1.127838485561682 approximated to 1.128152149635533. 1.128152149635533 is the square root of 1.272727272727273. 4 divided by 1.272727272727273 is ancient Egyptian Pi = 3.142857142857143. So the longer length of the ancient Egyptian square root for the Golden root = 1.128152149635533 rectangle is 14, the diameter of the circle with a surface area of 154, while the shorter length of the ancient Egyptian square root for the Golden root = 1.128152149635533 rectangle is the square root of 154 = 12.40967364599086, the width of the square.
    1.128152149635532 squared is 1.272727272727272 and 1.272727272727272 squared is the Golden ratio approximated to 1.619834710743799. The ancient Egyptian square root for the Golden root = 1.128152149635532 is important.

    Area of circle = 154.

    Diameter of circle = 14.

    Circumference of circle = 44.

    Pi as 22 divided by 7 = 3.142857142857143.

    3 methods for calculating the surface area of a circle

    Method 1 = Radius of circle = 7 .7 squared = 49. 49 multiplied by Pi as 22 divided by 7 = 3.142857142857143 = 154.

    The surface area of a circle that has 154 equal units of measure can also be calculated if the diameter of the circle is divided by the Square root of Phi = 1.272019649514069 approximated to 14 divided by 11 = 1.272727272727273 resulting in 1 quarter of the circle’s circumference 11 and then half the circumference of the circle 22 is then multiplied by 14 the measure for the diameter of the circle and then the result of multiplying half the circumference of the circle 22 by the measure for the diameter of the circle 14 is divided in 2 resulting in the measure for the surface area of the circle = 154.
    A square with the same surface area as a circle can be created if the diameter of the circle is divided by the square root of the square root of the Golden ratio = 1.127838485561682. Please remember that the irrational ratio 1.127838485561682 is the square root of 1.272019649514068 and the irrational ratio 1.272019649514069 is the square root of the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895. Because the value of Pi that is being used in this example to create a circle and a square and also an isosceles triangle with the same surface area is 22 divided by 7 = 3.142857142857143 then the square root of the square root of the Golden ratio = 1.127838485561682 is approximated to 1.128152149635532. Remember that 1.128152149635532 squared is 1.272727272727272 and 1.272727272727272 squared is the Golden ratio approximated to 1.619834710743799.

    A square with a surface area of 154 equal units of measure can be created according to the Pythagorean theorem through the following formula:

    12 squared = 144.
    3 squared = 9.
    1 squared = 1.
    144 + 9 + 1 = 154.

    The edge of the square with a edge length of 12 is placed on the same angle and line as the extended hypotenuse of the right triangle that has its second longest edge length as 3 while the shortest edge of the right triangle is 1. According to the Pythagorean theorem a right triangle that has its second longest edge length as 3 while the shortest edge of the right triangle is 1 has the hypotenuse equal to the square root of 10.

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