If the Square and the Circle are considered two separate entities that need to be unified or reconciled, we are in a polarized state of mind.
Crop circle at Potterne, Wiltshire, UK, june 7, 2023. Painting by Karen Alexander. The Golden mean expressed in the radius of the central circle in relation to the position of the tip of each petal.
Two of the most prominent geometric properties that are at the core of crop circle design are the Golden mean, and the Squared Circle. The Squared Circle is typically a hidden property; It is often not apparently visible in the design that is laid down in the field. You have to look for it, dig a little to find it. Golden mean proportions are not so much hidden. I think it is more accurate to say that they are perceived on a subconscious level as the Golden mean is associated with aesthetic beauty; It is not always clear why we find something or someone aesthetically pleasing to look at. But when a person, work of art, or piece of architecture becomes an icon for a specific time or cultural phase, we may find the Golden mean at its core.
United Nations Secretariat Building, New York (1951) Wikipedia. Its contour is a Golden Rectangle.
Its importance in biology or culture is often disputed and over-exaggerated, but for the main topic at hand,we can safely say, the Circlemakers do love a good Golden mean.
It is not often mentioned that the Squared Circle and the Golden mean are closely related, and although the circlemakers have been telling us for decades, this is rarely recognised.
Squared Circle and the Golden rectangle.
Here we see a combination of a Squared Circle with a Golden rectangle, a rectangle where the proportion of the two sides express the Golden mean. Circle and Square have equal circumference. The relation between the diameter of the Circle and the rectangle is apparent. A closer look will reveal that rectangle and Square have equal area. The Golden rectangle acts as a mediator, sharing properties that connects both the Circle and the Square
This is not the most accurate rendition of the Squared Circle. The circumference of the Circle is 99,91% accurate to that of the Square. Anything less than 99,95% usually ends up in the bin. Yet, this configuration provides an approximate relationship between π and Ф that is tentatively elegant and transparent. I believe in our time where measurement is drawn to its extremes, too much emphasis is placed on accuracy alone among my fellow geometeers. After all, we are not sending a rocket to the moon here, we are merely trying to draw a geometric diagram that we could call Sacred geometry, if we so desire. I think there is equally much to gain in finding ways that shine in clarity and elegance. Following all Transcendental Truths, instead of just one.
The symbolism of the Squared Circle has always been explained to me as the “reconciliation of the earthly and the divine” (Glickman, 2009). Or “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”. (Lawlor, 1982). Many other quotes stand in line to express the same idea; a square and a circle with either equal area or equal circumference that cannot be constructed with a ruler and compass due to the transcendental nature of π, represent two opposites, the Circle (the Divine-) and the Square (the physical realm). We are left to try to draw the closest possible approximation as an exercise to align our beautifully flawed human nature with God.
For completion’s sake i should provide some historical context here, and the reference to the radii of the earth and the moon in relation to this diagram (Martineau, 2001). But who am i kidding? Anybody half as awesome as you already know this. For those who are awesome in other areas, please check the literature list below.
As an alternative reading we can align the Square with our analytical left brain hemisphere, where everything has its place and can be measured. The Circle will consequently represent our right brain hemisphere with its apparent erratic connections and dreamlike logic, where creativity and intuition rule.
If the Square and the Circle are considered two separate entities that need to be unified or reconciled, we are in a polarized state of mind. We discern two opposites. What does it even mean in this context to reconcile to begin with?
Even if we see the Square and the Circle as two sides of the same coin, we still operate from that same mindset.
The Golden mean may function as a mediator in this context to transcend the polarity of the Square and the Circle.
Crop circle at Ashbury, Wiltshire, UK, august 1, 1996. The Vesica Pisces in this design is guided by two nested Golden rectangles and the Squared Circle, symbolizing the Golden mean as a mediator between two opposites.
As Bert Janssen expressed in a video elsewhere on this page; The Square and the Circle are One. There is no need for reconciliation. We don’t have to look at the extremes to find either God or Ourselves. We are already perfect and exactly where we need to be.
How could it be not?
All is Well.
The Bones of God, Michael Glickman, SD Books, 2009
Sacred Geometry, Philosophy and practice, Robert Lawlor, Thames and Hudson ltd, 1982
A little Book of Coincidence in the Solar System, John Martineau, Wooden Books, 2002
I confirm that the rectangle Karen has drawn is golden to 99.8%. So, well done! And it has the same area as the square, to almost that accuracy.
Thank you Nick. That’s very kind. The drawing with the rectangle is mine. I used Karen’s painting only as a reference to the Golden mean, as she shows it so elegant with the calipher.
Great article Peter, thank you.
Thank you for this timely article.