© 2012 Frank Chester. All rights reserved.

Squaring the Circle – A new method using the Vesica Pisces

Frank has discovered a new way to square the circle so that the perimeter of the square and the circumference of the circle are almost exactly equal. His method uses only a straight-edge and compass, and is the first to work from the inside out using the Vesica Pisces. The protocol is as follows (see the animated image below for visual reference):

1: Draw a circle (#1).

2: Open your compass to the diameter of circle #1, and draw a second circle (#2) (that has a radius exactly twice the original circle), whose center lies on the perimeter of circle #1.

3: Draw a light line (m) connecting the center of circles #1 and #2, projecting it across most of the page.

4: Use where this line m meets the opposite side of circle #1 to be the center of a new circle, with the same radius as circle #2, this is circle #3.

5: Circles #2 and #3 form the Vesica Pisces, and intersect each other in two new points. Connect these points with a line (n), which is orthogonal to line m.

6: Where line n intersects circle #1 it creates two new points. Use these as the centers of two new circles with the same radius as Circles #2 and #3; let point Q be the center of circle #4 and point R be the center of circle #5. These new circles, #4 and #5, form another Vescia Pisces at 90 degrees to the first. The two Vescia Pisces cradle circle #1 where they overlap.

7: Circles #4 and #5 intersect in two points that lie on line m. Choose one of these points (P), and draw a line (s) orthogonal (at 90 degrees) to line m (and thus parallel to line n) at this point. From point P, extend line s on either side of line m so that its total length is just larger than the diameter of circle #1.

8: Note the center points of circles #4 and #5, Q and R. From each of these center ponts draw a line perpendicular to line n (parallel to line m), extending them to meet line s in two new points (S and T). Let S be the point made by extending from point Q and T be the point made from extending point R.

9: Open your compass to the length made between points S and R, and draw a circle (#6) of that radius with its center on point R.

10: Draw another circle (#7) of the same radius, with its center on point Q.

11: Circles #6 and #7 intersect line n in two new points (U and V) towards the outsides of the construction. Open your compass to the length made between the center of circle #1 and one of these new points, and draw a new circle (#8) with the same center as circle #1.

12: Circle #8 meets lines m and n in four points. Construct a total four lines (one on each point), two parallel to n and two parallel to m, such that the lines form a square in which circle #8 is circumscribed.

13: Open your compass to the original radius of circle #1. With the point of your compass on point U, mark a new point (Y) on line n towards the outside of the construction.

14: Draw a new circle (#9) with a center on point Y, with a radius equal to circle #1. Circle #9 is tangent to circle #8.

15: Open your compass to the distance between the centers of circles #1 and #9, and draw a final circle (#10) with the same center as circle #1. This circle has a perimeter almost exactly equal to the square constructed in step 12.

squaring the circle

 Click the image below for a bigger, static version of the full construction.

The mathematics of this construction are known exactly. Measurements of the circumference of the circle and the perimeter of the square agree to within 99.9+%, with the perimeter of the square being ever so slightly smaller than the circumference of the circle. It was proven in 1882 that it is impossible to perfectly square the circle using a finite number of steps with a straightedge and compass. This is a result of the Lindemann-Weierstrass theorem, which was involved in showing that the number pi is transcendental, and thus a non-algebraic number. Straightedge and compass constructions involving non-algebraic numbers can only be approximations.


  1. Rose

    Geometry is the basis of many of life’s decisions.

  2. Christopher Ricci

    I found your presentation most intriguing. I invite you to review my observations on this subject. http://www.circleissquared.com

  3. I wanted to analyse how accurate this method was for squaring the circle so I wrote a computer program to generate the image an calculate the accuracy. I found myself baffled to discover that the area of the final circle here is nowhere near the area of the square. According to this definition http://en.wikipedia.org/wiki/Squaring_the_circle, this is an altogether different problem than that of “Squaring the Circle”.

    However, analysing your solution to this different problem, the following results were found
    Circumference of circle: 2513.27412287
    Circumference of square: 2400
    Difference: 113.274122872
    Accuracy %: 4.71975511966

  4. Diogo

    Not a very good method… better, terrible method!

    But you gave me an idea…

  5. Rod

    I’m still running in squared circles but bonding well with a certain triangle …

    “Morbus cyclometricus” may eventually translate to “sanitas cyclometricus” if this new perspective of Pi confirms the value of all human effort to “square the circle”: http://www.aitnaru.org/images/Pi_Corral.pdf (file attached to web page)

    This unique scalene triangle, inscribed within a circle, should at least help promote a new generation of ideas for food and festivities for Pi Day … and new ideas for squaring the circle!

    By the next Pi Day celebration, the new slogan might be: “Some Pi are square, some are round, but no Pi is triangular!”

  6. Rod

    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”.

  7. Dear Frank and all.
    Does the postulate and demonstration of Miles Mathis on Pi = 4 (when in vorticial, radial or helical motion)? have any bearing on your twisting or vorticing of shapes transforming into each other?
    Might all ‘geo-metry’ be choreographic or vibrational shorthand step notes for a dance we’ve forgotten that we are? Perhaps BECAUSE of making models and ‘subjecting’ within or under them as ‘ideals, and idols, taking shape in-formation of rules that also rule out.
    I have some intuitive appreciations that are ‘triggered’ or resonate with fields that I have not much studied – ie geometry and math – but as will all innate learning, it rises from within as a result of an abiding curiosity or movement of unfolding focus.

    On a practical note – your video demonstrations so far – often suffer the liability of anyone who is speaking from ‘its easy/obvious once you know’. I feel there is a genuine value in meeting you in your sharing of the process of dis-covery. But I also see merit in crafting educational resources that pause, zoom in or allow replay or slower consideration of some of what you have almost moved on from as soon as you think you have shown your point – when it has not found a full or crystallized reception – though it may still seed that process. Perhaps you are guided to do it just as you do – and trust that those who ‘get it’ will be those who are seed-resonant.
    I do not mean to encourage over slick presentational hyped info-tainment. I am speaking only to the establishing of a resonance between giving and receiving or teaching and learning.

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