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.

12 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

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