Squaring the Circle

August 21st, 2007

An exercise in contemplative geometry from Robert Lawlor’s Sacred Geometry, 1982.

“There are a number of diagrams in the literature of Sacred Geometry all related to the single idea known as ‘Squaring of the Circle’. This is a practice which seeks, with only the usual compass and straight-edge, to construct a square which is virtually equal in perimeter to the circumference of a given circle, or which is virtually equal in area to the area of a given circle. Because the circle is an incommensurable figure based on π, it is impossible to draw a square more than approximately equal to it. Nevertheless the Squaring of the Circle is of great importance to the geometer-cosmologist because for him the circle represents pure, unmanifest spirit-space, while the square represents the manifest and comprehensible world. When a near equality is drawn between the circle and the square, the infinite is able to express its dimensions or qualities through the finite” (p74).

Here I have drawn the former arrangement (perimeter equal to circumference):

Squaring the Circle

The derivation begins with an initial circle (within the square) of radius unity. Along its horizontal diameter are drawn two tangent circles, each with radius one half. Observe that the total circumference of the smaller circles equals the circumference of the initial circle, but the total area of the smaller circles is one half that of the initial circle: “One has become Two” (p73), an image of the primary duality, of yin-yang.

Next are drawn two arcs from the ends of the initial circle’s vertical diameter with radius tangent to the far sides of the smaller circles. This radius is φ, the golden ratio, dividing the vertical radius of the initial circle into the golden section of lengths 1/φ and 1/φ2. The two arcs meet to create a vesica that encloses the primary duality — the mouth of Ra, the Word, the vibrating string.

Around the initial circle is drawn a tangent square, with side 2, perimeter 8; and, finally, a large circle is drawn with diameter equal to the width of the vesica, 2√φ, giving a circumference of 2π√φ = 7.993, approximately equal to 8. The circle is squared.

14 Responses to “Squaring the Circle”

  1. Mirco Says:

    A nice & instructive construction. A few comments are in order:

    1) the passage from the ONE to the TWO always happens by MITOSIS, as it is witnessed innumerable times in the biological world: see


    2) the WORD (logos), aka the number 3, is the MEDIATOR, i.e. the RATIO (it is not by chance that the word reason comes from the latin root ratio). This ratio is well represented by the number phi, the RATIO AUREA, or the eternal harmony inside duality.

    3) out of the 3 comes the 4, the FOUR VISIBLE ELEMENTS, the SQUARE of directions (the fifth element being the original circle itself, itself an expansion of the central dot). The 4 externalized is the VISIBLE WORLD, represented by the outer circle. The fact that the square and the outer circle are indeed the same in perimeter, points to their fundamental identity.

    A good Mandala for endless meditations….

  2. Greg Pass Says:

    Mirco — 1, 2, 3, 4 (+1)…

    From Lawlor’s original diagram I excluded 5 — a pentagon inscribed within the initial circle — in order to focus attention on the 4-sided objective.

    But let’s not leave your finely presented progression hanging; for your mandalic meditations I include 5’s description here:

    The pentagon “is the figure ascribed to the physical and vital aspects of man [i.e., Life], who, through the five senses, perceives the natural world and thus brings it into existence. The star pentagram formed by the diagonals within the pentagon symbolizes transformed or perfected humanity, because all the line segments of the star pentagram are derived from the Golden Proportion” (p75).

  3. Mirco Says:

    I see…the pentagon, as the seal of the fifth element, is rightly inscribed in the ORIGINAL

    It is the 1 AND the 5.

    What Lawlor says is correct, albeit a bit abstract: the fifth seal indeed symbolizes the PERFECTED MAN, i.e. the first (and lower) degree of IMMORTAL.

    I will not spoil other’s meditations on this remarkable construction, but I will add one more thing.

    There are TWO kinds of pentagrams: one pointing UPWARD (lifting up the elements), and the other pointing DOWNWARD (the well known mark of the Dark One). When Man, passing beyond the impermanent material plane, reaches the world of the Force, the Path bifurcates….

  4. Geometry of the Opus Alchymicum « Unurthed Says:

    […] the alchemist performs the squaring the circle [see earlier post], thereby turning the two sexes into one. The motto repeats a saying of the ‘Rosarium’: […]

  5. tony Says:

    Members of the WordPress community may like to see some truly ancient sacred geometry:

  6. Stephen Fitz-Gerald Says:

    A most interesting exercise.
    I have made a sculpture of this construction,albeit in THREE DIMENSIONS.
    I have noticed a very subtle yet palpable vibratory resonation (for lack of a better or more rational term),the closer the two VOLUMES are to congruency…

  7. Anonymous Says:


  8. Bruno’s mathesis « Unurthed Says:

    […] Figura Mentis, p307. C.f. Cusanus’s paradigmatic diagram and the mouth of Ra. […]

  9. rsj reddy Says:

    sir , i have squared a circle exactly with a new value of pi. i will send my article.your postal address please.
    rsj reddy

  10. 7 Mystical Seals – Phosphenes – Rock Art Symbols « Alternative Thinking 37 Says:

    […] Figura Mentis, p307. C.f. Cusanus’s paradigmatic diagram and themouth of Ra. […]

  11. Rod Says:

    New perspective on the geometry of a circle, once squared: http://aitnaru.org/thekingdom.html

  12. Rod Says:

    Still running in squared circles, but the new “Pivotal Confirmation” corral offers out-of-the-box perspective on this “impossibility”: http://www.aitnaru.org/images/Pi_Corral.pdf

  13. sarva jagannadha reddy Says:

    Professors of the world
    Till now we have been accepting Exhaustion method of Archimedes as perfect and error – free in computing pi is less than 22/7. Hence pi is 3.14159265358.. But it is proved his method is wrong .
    I would think this news would be discussed in the departmental seminar .

    ———- Forwarded message ———-
    From: Mathematician Vitthal Jadhav
    Date: Sun, Aug 6, 2017 at 9:31 PM
    Subject: Re: 117 Method on Cosmic Pi
    To: Sarva Jagannadha Reddy
    Cc: bikasbikashchakraborty.math@yahoo.com

    Mistakes in the #Archimedes #pi value computed by using Method of exhaustion.
    (Issues with Traditional Pi (#?) )
    Assume circle, now draw the square inscribed in given circle,
    the perimeter of square is clearly less than circumference,
    Now if we increase side, draw regular hexagon with the given circle then difference between perimeter of hexagon & circumference will be reduced. Archimedes by using basic instrument like compass bisected each side of hexagon & obtained 12 point equally spaced on curved path of circle, join them & constructed regular dodecagon, in this way he continued this process. He calculated perimeter of regular polygon with 96 side & came to conclusion that the value of pi lies between
    3 + 10/71 < ? < 3 + 10/70,
    i.e. 3.140845 < ? infinity)
    then delta tends to 0,

    Now perimeter = n * (side of polygon i.e. AB)
    Circumference = n * (Arc AB)
    Error = n * ((Arc AB ) – (Side AB))
    = n * delta
    we clearly see, as n -> infinity , delta -> zero
    but error doesn’t tend to zero as
    it is “n * delta” becomes indeterminate

    Archimedes method assumes error become insignificant.

    This gives the reason – why cosmic pi = 3.146 given by Indian Professor R S Reddy (Sarva Jagannadha Reddy – We can call him as “PI Man” ) differ from traditional ? = 3.1459 approximately by 0.004

  14. Jan-Martin Wagner Says:

    The comment of Vitthal Jadhav is based on a false statement with respect to the limiting value of the error term, and it is systematically flawed because he considers a lower bound for pi only and neglects the upper boundary. When asked about these problems, Vitthal Jadhav replied (https://www.researchgate.net/publication/319165096_Mistakes_in_Archimedes%27s_Method_of_Exhaustion_in_Computing_Pi_number_314159265358_by_Prof_Vitthal_Jadhav/comments):

    >> J. M. Wagner , you are talking right. I already detected & accepted defects in my analysis. In my opinion it is not at all cosmic as it is not accurate value. Somewhere there is very minute error In R.S. Reddy’s analysis of Pi. For that the expert mathematician have to verify every step & assumption involved in RS Reddy’s method.
    Here is whole conversion that occurred on this topic from my side:
    **** Start of conversations *********
    One more analysis that will really confuse & exactly contrary to previous, Respected Sir, I had not proven value of pi = 3.146446 is correct. Just shown as As n -> infinity, n * (delta x) will be in indeterminate form doesn’t mean computing area by dividing into infinite pieces gives wrong result. for x -> 0 , tan(x) / x is in indeterminate form but its limit exist & it is slightly greater than 1. Similarly, For x ->0, sin(x) / x = 1 (slightly less than 1) In this case, sin(x) can be represented as the function of x. sin (x) = (x) – (x^3 / 3!) + (x^5 / 5!) – (x^7 / 7!) +…. Clearly sin(x) is multiple of x ; sin (x) / x = 1 – x^2 / 3! + (x^4 / 5! ) Now, Area of n sided regular polygon inscribed in circle n * (1/2) r^2 sin(2?/n) = n* (1/2) * r^2 * (2?/n) as n -> infinity clearly it tends to ? * (r^2) that means error becomes insignificant as n-> infinity which is quite “contrary” result to previous. Now question is which one is correct ? *** <> Sorry for misinterpretation, But this result is not contrary to my previous analysis but it is consistent. In previous analysis, I showed that Error = n* (delta x) will be in indeterminate form as n -> infinity & delta -> zero. It doesn’t mean limit doesn’t exist. Also concluding n* (delta x) is significant or insignificant at this stage is “illogical”. Whether error become significant or insignificant can be seen by recent analysis i.e. Area of n sided regular polygon inscribed in circle n * (1/2) r^2 sin(2?/n) = n* (1/2) * r^2 * (2?/n) as n -> infinity clearly it tends to ? * (r^2) Yes, error become insignificant as n -> infinity. Also by Inscribing regular polygon in the circle, increasing n to infinity even if we won’t get accurate value of pi up to all decimal place, that doesn’t mean it won’t gives accurate values at least up to five decimal places. ****** End of conversion *********** <<

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