Squaring the Circle

Since the summer, I have been obsessed with the problem of squaring the circle—that is, finding the square whose area is equal to that of a given circle, with no tools other than a straight edge and compass.

I took interest in the problem when reading the ending of Dante’s Paradiso (in Allen Mandelbaum’s translation):

 

 

As the geometer intently seeks
to square the circle, but he cannot reach,
through thought on thought, the principle he needs,

so I searched that strange sight; I wished to see
the way in which our human effigy
suited the circle and found place in it—

and my own wings were far too weak for that.
But then my mind was struck by light that flashed
and, with this light, received what it had asked.

Here force failed my high fantasy; but my
desire and will were moved already—like
a wheel revolving uniformly—by

the Love that moves the sun and the other stars.

I, too, had difficulty reaching the principle—and it was precisely the principle that I needed to find. The difficulty lies in determining what the problem exactly is.

No one has found a way to square the circle with compass and straight edge alone; in fact, I believe it has been proved that it can’t be done. Yet I kept on trying, thinking that I would learn something from the attempt.

Yesterday morning, something close to a solution came to me—not a solution, mind you, but something that points in the direction of one. It’s probably old hat—or ancient hat—but it’s interesting, all the same.

We will consider the circle with radius r=1, since expanding or shrinking it (and the square) is then a trivial matter.

We know that if the circle has radius 1, its area is π. Thus, for a square to have that area, it must have sides of length √π. But how do you find length √π? There’s the puzzle.

When the square and circle have the same area, the ratio of the circle’s circumference to the square’s perimeter is a constant, 2π divided by 4√π ; that is, π/2 divided by √π; that is, √π/2.

In other words, the ratio of one-quarter of the circle’s circumference t0 one side of the square will always be √π/2 when the square and circle have the same area.

You could consider the ratio in terms of this figure:

|AO| is equal in length to the side of the square  (√π when the circle has radius 1); |OP| is one-quarter of the circle’s circumference (1/4 *  2π =  π/2), which you can measure with a string. (I know you’re only supposed to use a straight edge and compass, but this exception will prove helpful.)

Now, the ratio of |OP| to |AO| is equal to the ratio of|BC| to |AB|, where |BC| = √π = |AO|, and |AB| = 2. We already established this ratio a few paragraphs up.

(Again, the ratio of one-quarter of the circle’s circumference to the square’s side must be √π/2.)

So now the challenge is to tweak the figure until |AO|=|BC| (while keeping points A, P, and C on a single line and A, O, and B on a single line). When we get the two segments to equal length (without changing |OP| or |AB|), we have brought both |AO| and |BC| to √π. Now, you must do this by trial and error, by shifting the OP segment. If |BC| is greater than |AO|, then move OP closer to BC; if it is smaller than |AO|, then move OP closer to A. Your adjustments will be smaller and smaller until you either make |AO| and |BC| equal or come as close as you wish. The next step would be to find a mathematical way of doing this.

Once you have made |AO| and |BC| as close to equal as you wish, you then make AO the side of your square, which you superimpose on a circle of radius 1 (so that both have a common center). You will see that the square is mostly, but not entirely, contained within the circle. It will look like the first of the two figures above.

Although this is not a satisfactory solution, it results in a close approximation and seems to be on the right track.

Update (on a tangent, so to speak): The angle in the figure is the inverse tangent of √π/2, or 41.54822621257918513… Its common logarithm (1.6185524875…) is tantalizingly close to the golden ratio, but not close enough for the trumpets to sound.

Note: This is a reposting of a piece that originally appeared on October 19.

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50 Comments

  1. I believe that the core of the problem is that the way we calculate the area of a circle uses an irrational number, because it is based on the irrational relationship between the diameter and the circumference, which means in real life we are always approximating it. This means all further calculations will also be slightly inaccurate. And it may also be that in any circle, either the radius (or diameter) or the area will always be irrational as well. Squares on the other hand, do not always have the same irrational relationship between their side lengths and their area. Fun stuff.

    Reply
    • Thank you for this insightful comment. Yes, that seems to be the problem. And yet here’s the tantalizing thing: whenever you have a square and circle of equal area, the ratio of the circle’s circumference to the square’s perimeter is a constant. So is the ratio of the circle’s radius to the square’s side (or perimeter).

      In fact, the ratio of the circle’s radius to the square’s side is equal to the ratio of the square’s side to half of the circle’s circumference.

      Reply
      • You’re right–they would be a constant. But I think the issue is that irrational numbers, while real, are sometimes unmeasurable due to their precision. If you really want to find the accurate area of a circle, I think you actually need to use a liquid and find the volume of the cylinder it would make, but even this might not work.

  2. Rod

     /  July 19, 2013

    Consider the geometry of this recent “Pi Corral” design that displays the unique scalene triangle which squares the circle: http://www.aitnaru.org/picorral.html

    Reply
  3. Rod

     /  July 20, 2013

    Click on the “Pi Corral” design to view the attached PDF file (new perspective on “squaring the circle”): http://aitnaru.org/images/Pi_Corral.pdf

    Reply
  4. Rod

     /  July 23, 2013

    Re: http://aitnaru.org/picorral.html (updated design)

    The green diagonal line (half the length of a side of the inscribed square) in the upper right of this design appears to define the right diagonal of the golden scalene triangle. With the left diagonal of the scalene equal to the length of one side of the inscribed square (reflects the square root of 2), the bottom horizontal line (reflects the square root of Pi) completes this circle-squaring scalene triangle.

    Reply
  5. Rod

     /  July 23, 2013

    By the way, the numbers are more revealing when the circle’s diameter = 2 or increments thereof (D = 2,000,000 units in the Pi Corral design).

    Reply
  6. Rod

     /  July 24, 2013

    “Back to the drawing board!”
    The length of the green diagonal line is not equal to half the length of a side of the inscribed square.

    Reply
  7. I should have a chance to take a look at this on Sunday. (Before then, probably not.) I look forward to it.

    Reply
  8. Rod

     /  July 27, 2013

    Re: http://www.aitnaru.org/images/Pi_Corral.pdf (see last design)

    The design displays the circle-squaring magic of this unique scalene triangle; not a solution for “squaring the circle” but further evidence that squaring may indeed be possible … and might be proven geometrically!

    Reply
  9. This is interesting. I see what’s going on with the transformation.

    Finding that scalene is another matter. It seems it could be done with some raising, lowering, and tilting–but again, that’s a trial-and-error method (somewhat like mine).

    Reply
  10. Rod

     /  July 28, 2013

    The intrigue of this perspective is that the perfect scalene triangle does exist somewhere in the transition. When discovered, the bottom horizontal line will reflect the square root of Pi … regardless of the number of decimal digits!

    My recent months of exploration have focused on how to prove geometrically that the circle is squared. Once such proof is discovered, a solution to squaring the circle might be found.

    The still-popular 1882 “impossible” verdict for this Greek challenge may yet be reversed. Now, mathematicians must prove that geometry does not exist to create this unique scalene triangle. Future supercomputers might be able to analyze the millions (trillions?) of possible geometric lines and objects.

    Reply
  11. Rod

     /  July 29, 2013

    Speaking of “new perspective”, here’s an interesting site:
    http://troika.uk.com/squaringthecircle/
    http://troika.uk.com/about/

    “Often merging technology with their projects as a point of departure paired with a particular interest in perception and the spatial experience, their work explores the intersection of rational thought, observation and the changing nature of reality and human experience.

    And while their work is often filled with polarities — solitude and interaction, transition and permanence, the artificial and the natural — their interest lies within man’s resolve to synthesize these opposites.”

    Reply
  12. Rod

     /  July 29, 2013

    This Mormon site well explains the benefit of resurrecting the quest to “square the circle”: http://mormonmatters.org/2010/01/29/squaring-the-circle-balance-and-ideals/

    … since discovery of a solution is not as important as believing that a solution can be discovered.

    Reply
  13. Rod

     /  July 29, 2013

    Re: http://www.aitnaru.org/images/Pi_Corral.pdf (another design added)

    The “Pivotal Confirmation?” design suggests that the circle must be squared if the two Pythagorean triangles align on their common hypotenuse (a side of the circle’s inscribed square).

    Oh, the power of those right triangles!

    Reply
  14. Rod

     /  July 31, 2013

    Re: http://www.aitnaru.org/images/Pi_Corral.pdf
    Discussion about new formulas added at the bottom of this file.

    Geometers will easily comprehend that this new concept of Pi simply substitutes one ratio (ASR) for another (Pi) … and both ratios include the mysterious and stimulating essence of irrationality! Such is the nature of squared circles!

    However, the ASR perspective presents intuitive geometry for conceptualizing solutions to “squaring the circle”.

    Reply
  15. Rod

     /  August 2, 2013

    Re: http://www.aitnaru.org/images/Pi_Corral.pdf (Pop Quiz #2)

    Pi are both square and rational … according to Pythagorean geometry.

    Reply
  16. Rod

     /  August 2, 2013

    Thanks for your responses – I had included a few more notes for closure since the research has achieved a good plateau.

    Reply
  17. Rod

     /  September 23, 2013

    A colorful drawing and concise review of this research plateau:
    http://www.aitnaru.org/images/Pi_Corral.pdf (see design: Impossible Balance)

    “Quick Guide for square Pi” – Simple display of geometric balance in a squared circle.

    The circle (D = 2,000,000 units) is drawn next to last, confirming the starting object:
    a green perpendicular line set, representing the square root of Pi (1,772,453.850.. units)
    and half the square root of Pi (886,226.925..).

    The circle is drawn via a perpendicular line set (magenta, perpendicular line not shown).
    The mid-point distance of the two magenta lines is known when the first object is drawn;
    these lines will have length equal to the side of a square inscribed in the golden circle.

    Of course, “stuff” happens along the way but can be guessed from the starting geometry. The red lines, drawn last, confirm all of the good stuff that happened.

    Why “Impossible”? This is not a solution but contemplation of proof …
    and supportive of the whimsical “To square the circle, one must circle the square.”

    Reply
  18. Rod

     /  September 27, 2013

    Regarding the Impossible Balance design …
    “simple display” had to yield to more instructive display.

    Learning from forest fire firefighters that one can control fire with fire (re: backfiring), I
    now understand that one can control the “impossible” Pi with more Pi: most of the T-shaped, perpendicular line sets are constructed of lengths equal to the square root of Pi and half the square root of Pi.

    Pi may be impossible but reveals its own constraints in symmetrical patterns which permit easier management of this geometric “impossible”.

    “Can’t see the forest because of all the trees” can now be inverted:
    “Can see the squared circle because of all the Pi”.

    Reply
  19. Rod

     /  October 19, 2013

    Squaring the circle with a protractor and compass (I-Square method):

    Draw a 45-degree angle with the sides at 135 and 180 degrees.
    Draw a center line at 152.403 degrees* from the vertex of the angle.
    Mark the length of the circle’s radius along the center line from the vertex.
    Draw the circle with the compass point at the left end of the radius.

    The two sides of the 45-degree angle and the vertex identify 3 points of the circle’s square and creates a chord that has length equal to the side length of the circle’s square.

    This is not a solution to the Greek challenge of “squaring the circle” but reveals the scalene triangle that might help prove that the circle is squared (the shorter diagonal chord has length equal to the side length of a square inscribed in the circle).

    * Draw center line at 152.40288736430939554826779524767 degrees for best precision. Precision can be increased beyond this, complementing half the square root of Pi digit for digit (value related to the cosine of a unique right triangle).

    Reply
  20. Rod

     /  October 29, 2013

    (from the I-Square research)

    Conjecture:

    A transcendental number becomes algebraic when corraled by a triangle, inscribed within a circle.

    Consider the scalene triangle:

    Circle’s diameter = 2 units; 45-degree angle includes Pi side; one non-Pi side has length equal to a side of the circle’s inscribed square (effectively, square root of 2, irrational but not transcendental).

    The length of the other non-Pi side must adjust as the Pi side becomes more precise (has more decimal digits). Does the remaining side balance, absorb, or dissipate Pi’s transcendentalism and the other side’s irrationality?

    Reply
  21. Rod

     /  October 31, 2013

    New conjecture about the scalene triangle in a squared circle:

    The only reason that Pi is transcendental is that its popular early model (a polygon with an increasing number of sides) was by its nature “transcendental” (there will never be a polygon with an infinite number of sides). Thereafter, developments in math perpetuated the transcendental model, converting it from geometry to algebra and beyond. However, the transcendental model may be the only model we currently envisage.

    At least, this might explain why the scalene triangle geometry hints that Pi (relationship of a circle’s circumference to its diameter) is not really transcendental.

    Reply
  22. Rod

     /  November 2, 2013

    An observation …

    Since Pi is an ever increasing (but diminishing) value in its popular calculations, another calculation should balance (limit) this value by working backward from a maximum value. How the increasing and decreasing values are perfectly resolved mathematically is the ripe fruit waiting for picking by a new-era mathematician.

    This non-mathematician observes only that a certain scalene triangle, inscribed within a circle, hints that the real Pi has a finite limit.

    Reply
  23. Go to YOUTUBE and enter HOW TO SQUARE A CIRCLE WITH A COMPASS AND STRAIGHTEDGE by Al Amin , There is also a new value for Pi as well as the square root of PI if you look closely .
    eg 1.732050808 x 1,732050808 = 3.000000001 the N `th degree

    Reply
  24. Rod

     /  November 21, 2013

    A nice approximation by Al Amin, but …

    Using Amin’s 40 mm / 69.2820323 reference, a radius in the upper right quadrant would connect to the circle (the point where it meets the top right side of the square) at 60 degrees – not 62.40288736430939554826779524.. (the necessary reference angle for a circle to be squared … and the reason Pi has been so accurate and formidable).

    I still believe that geometry does exist to prove that a circle is squared, but the necessary reference angle suggests that it may be impossible to construct the square (with absolute precision) using only compass and straightedge.

    Rod

    Reply
    • in researching this problem , i have never heard of any reference angles that are required . What i can say is that this problem was recognised over 500 years ago and therefore they were using a less complex form of numeracy . To try and use current thinking will only allow it to be a problem that can never be solved .Of course the fact that this problem was recognised means the answer was known as well . A bit like a crossword , that is the answer comes before the question ,. You must also remember , i am not being rude here , that what we have been taught , allows us to be products of someone elses` creation , and that of course allows our intellect to be corrupted by a system of design , to which we have been born into
      CHEERS !!!!

      Reply
    • Food for thought Rod . To scrutinise the squaring of the circle , it seems that you have used a programme based on the decimal or metric system . It has to be done with an programme based on imperial measurement , i think that if you follow the same path as you have using imperial measurement , you will hit the proverbial nail on the head

      Cheers !!!!!!

      Reply
      • steve

         /  March 12, 2014

        The answer you need Rod . The programme that you used to find the angle of 60 degrees is in decimal notation and has a predetermined value of Pi being 3.14159 . The squaring of the circle is based upon the old testament , and therefore has the value of 3 , imperial . If you swap the value of Pi from 3.14159 to 3 to create the radius for the reference angle then the answer will appear for 60 degrees to being accurate .Remember the square root of three is 1.732050808 which takes it to the N`th degree

  25. Rod

     /  March 26, 2014

    Thanks for the feedback, Steve, but the Imperial perspective seems to rely on less precise representation of Pi. Here is another concept, nicknamed Onurb em Pi and complementing the long-honored precision of Pi.

    Visualization to explore the finite limit of all possible* squares of a given circle:

    – Draw a square having sides equal to the square root of Pi.
    – Draw diagonal lines between the opposite corners.

    Imagine that the left vertice (45 degree angle) of the smallest right triangle (hypotenuse equals bottom side of the given square) pivots counter-clockwise, with the triangle increasing in size. Before this triangle becomes as large as the largest right triangle (hypotenuse equals diagonal of given square from bottom left corner to top right) and in the same position, a right triangle is created whose length of hypotenuse equals the diameter of the only circle that is squared by Pi.

    Since the hypotenuses of the right triangles in this finite range represent the diameters of all possible circles squared by Pi, geometry may exist that proves that a circle is squared.

    * All possible refers to the geometric reality that this square may be inscribed in a circle or a circle may be inscribed in this square, effectively defining a finite range of circle-squaring possibilities (of which only one circle is truly squared).

    Reply
  26. Rod

     /  May 29, 2014

    Re: pSymmetry page of Aitnaru site.

    This is the precise geometry of a squared circle. It does not show “how to get there” but what “there” looks like geometrically.

    Each corner (diagonal of a small square) of the circle’s square hosts six 17.4028873643093955482677952.. degree angles, giving 24 triangles arranged symmetrically in the geometry.

    When all 24 triangles have identical 17.402.. degree vertices, the circle is squared regardless of the number of decimal digits (which may number as many as in Pi).

    Rod

    Reply
  27. Rod

     /  June 22, 2014

    Re: Three Points page of Aitnaru site
    (click on page to view Scalene, VP design in PDF)

    From another online discussion about squared circles:
    “It is impossible to find a square of equal area to the area of a circle. You will be dealing with complex numbers and it then becomes impossible.”

    After experiencing a certain scalene triangle, I would say:
    “Even complex math cannot yet describe squared circles
    … and the derivation of Pi may be the reason.”

    Reply
  28. Rod

     /  June 27, 2014

    Regarding the iTrapezoid design in the same PDF …

    This iTrapezoid geometry, a CSC continuum* (four circles), highlights the four integrated isosceles trapezoids (two overlapping scalene triangles per trapezoid; one inscribed trapezoid per circle).

    Noteworthy consistencies of the four trapezoids:
    0. largest circle’s diameter = 2; side of circle’s square = square root of Pi.
    1. many sets of angles in these nested objects are similar – even visually redundant!
    2. longest line of trapezoid always has length equal to the side of that circle’s square.
    3. mid-point to mid-point line of non-parallel lines is also diagonal of next smaller circle’s trapezoid.

    Regarding the following Son-iT design** (subset of iTrapezoid geometry) …
    This geometry appears to be the quintessential geometry of squared circles!

    After realizing that I was pronouncing “Son-iT” like “sonnet”,
    I found this: http://en.wikipedia.org/wiki/Sonnet

    … leading to this caution:
    Aficionados of Son-iT geometry might be labeled “sonneteers”,
    with such labeling reminiscent of “morbus cyclometricus”.

    * Circle inscribed within Square inscribed within Circle …
    ** Not related to Son-aB, truly impossible geometry.

    Reply
  29. L.W.S.

     /  September 14, 2014

    Close, but no cigar.

    https://imageshack.com/i/iqqpnZ3dj

    Reply
  30. Rod

     /  October 30, 2014

    New concept:

    A Pythagorean triangle that defines both a circle and its square (center of circle is located at point where perpendicular to hypotenuse connects with long side; diameter = 4):
    Long side = Pi = 3.14159265358979323846264..
    Hypotenuse = (square root of Pi) x 2
    = 3.54490770181103205459633..
    = side length of circle’s square.

    Reply
  31. Rod

     /  November 2, 2014

    From the related Plane Askew geometry design:
    “To many, squared circle geometry is plane askew
    … but unites the square roots of Pi and 2.”

    Reply
  32. Rod

     /  November 9, 2014

    If a circle’s diameter = 2(sqrt of Pi)
    and the side of its square = Pi,
    the circle is squared … precisely,
    including all decimal digits of Pi.

    Who knew? Pi are square!

    Reply
  33. Rod

     /  November 9, 2014

    So, what’s the point?

    What is the last decimal digit of Pi?
    The one that is right.
    What is the first decimal digit of Pi?
    The one that is right.
    What digit of Pi is left?
    The whole one.
    So, what’s the point?
    The one that is right of the whole
    and left of the one that is right.

    Reply
  34. Rod

     /  November 20, 2014

    Sanitas Cyclometricus
    (about the new Pinnacle of Pythagoras design) …

    Highlighting the circle-squaring Pythagorean triangle
    (vertex of 27.597112635690604451732204752339.. degrees
    between two sides having lengths of Pi/2 and sqrt of Pi):

    The lower yellow line (length = sqrt of Pi) forms an obtuse triangle
    where the lengths of the two shorter sides = 1 (radii of the inner
    largest circle). This triangle perfectly correlates the two smaller
    squared circles where the length of one diameter = 1 and the
    side length of the other circle’s square = 1.

    All four circles are squared by similar right triangles.

    Is it possible to get there from here?
    Of “there” now familiar to Pythagoras,
    he would say: “Come and see!”

    Reply
  35. steve

     /  January 30, 2015

    I thought the squaring of a circle is a metaphorical explanation of the crucifixion of the carpenter…known as…CHRIST your LORD

    Reply
  36. Rod

     /  July 21, 2015

    Entertainment for Pi Approximation Day, July 22 …

    Re: http://aitnaru.org/images/Three_Point_One_Four.pdf
    (see contrasting proportions in PR 1128 geometry)

    Where is “irrational” and “transcendental” in threse squared circles?
    Are these previously unknown qualities of the number 2?

    Every geometer knows this:

    Pi/2 = 1.5707963267948966192313216916398..
    sqrt(Pi) = 1.7724538509055160272981674833411..
    sqrt(Pi)/2 = 0.88622692545275801364908374167057..
    2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..

    I. Proportions between the two-circle sets:

    (long_side:hypotenuse of right triangles)
    sqrt(Pi) : 2 ~ 1 : 2(sqrt(1/Pi))

    Means/Extremes math property:

    sqrt(Pi) x 2(sqrt(1/Pi)) = 2
    (1.7724538509055160272981674833411..
    x 1.1283791670955125738961589031215..) = 2

    II. Proportions within each two-circle set:

    Pi/2 : sqrt(Pi) ~ sqrt(Pi) : 2
    (2 x 1.5707963267948966192313216916398..)
    = (1.7724538509055160272981674833411..
    x 1.7724538509055160272981674833411..)
    Pi = Pi

    sqrt(Pi)/2 : 1 ~ 1 : 2(sqrt(1/Pi))
    (0.88622692545275801364908374167057..
    x 1.1283791670955125738961589031215..)
    = 1

    Reply
  37. I think my model squares the circle and use alone the allowed premises.
    The idea is the use of a ruler with the same dimension that the circumference exterior to the circle.
    That is, the circumference and the same are the same thing.
    http://fermancebo.com/squaring_the_circle.html

    Reply
    • Thank you! I enjoyed your explanations, graphs, and photos. When I worked through it, it made sense. I only wonder whether it’s permissible to use a flexible ruler in this manner (that is, to use the ruler to take the circumference of the circle). This question applies to my own (semi)-solution too, since I took the circumference with a string.

      Reply
  38. Thank Diana
    I have treated of following the required premises.
    Any way is other form of an exact quadrature.
    Regard.
    Fernando

    Reply
  1. Prelude to a Preliminary Review | Diana Senechal

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