Return to Free Library

Return to Number Menu

Previous Article                                                                         Next Article


In this article we will discuss a very important and symbolic aspect of the square and circle – that of squaring the circle.

gilchrist squared circles

Credit: Charles Gilchrist




There are Four Relationships between the Circle and Square.

  1.  Square containing circle
  • Area of square = 4
  • Area of circle = π

circle in square2



  1. Square contained in circle
  • Area of square = 4
  • Area of circle = 2π

circle around square



  1. Square and circle with equal perimeters – “Squaring the Circle”
  • Area of square = π
  • Area of circle = 4

squared circle perimeter



  1. Square and circle with equal areas – “Squaring the Circle”
  • Side of square = √π
  • Diameter of circle = √4

squared circle area



Square Contained in a Circle

The circle in a square symbolizes the ‘square’ Earth within the circle of ‘heaven’.

Or conversely, the square represents the physical reality and the circle represents the metaphysical reality.

square in circle earth circle squared by playful geometer




The Circle in a Square and the Mystery of 273

Inscribed Circle in a Square:

Area of square = 1.273

Area of Circle

circle in square2

The four grey corners make up 27.32% of the total area of the square.

This is reached through the formula: (4 – π) / π = 0.2732.


When a square’s perimeter equals the circumference of a circle, the circle’s diameter is 27.3% longer than the edge of the square.



The ratio of Earth’s diameter to Moon’s diameter is 0.273.

That is, the moon is 27.3 % the size of the Earth.

The relationship of the Great Pyramid’s height to half its base is 1.273:1 (or 4:π) and thus ‘squares the circle’.




-273.2° Celsius is the temperature of Absolute Zero (0 Kelvin).

27.32° is the freezing point of water on Kelvin scale (K).

Water liquefies at 273°K.

Therefore, Absolute zero of water is 273.2% colder than the temperature it takes to boil.

The triple point of water is defined to take place at 273.16 K.

The triple point refers to the temperature where the solid, liquid and gas phases can coexist in equilibrium.

triple point



273 days = average length of pregnancy (10 sidereal months).

27.3 days = the average human menstrual cycle.

27.32 earth days is the sidereal period of the moon (Moon’s orbital period).

woman 438403 1920


1/273.2 per degree Celsius is the expansion/reduction of gas (Gasses expand by 1/273 of their volume with every degree on the Celsius/centigrade scale).

Sunspots revolve about the Sun’s surface in 27.3 days. This is the sunspot rotational period.

There are 273 days from the summer solstice to the vernal equinox.

2,730,000 is the circumference of the Sun in miles.

sun 11582 1920

The Cosmic Microwave Background Radiation is 2.73 K.

The Earth and Moon orbital periods are reciprocals.

1/27.32 = 0.0366 (366 days in a sidereal year)

(1/366 =.002732) 27.32 days in one month.

273 m/s2 = acceleration of the Sun.

.273 cm/s2 = acceleration of the moon along its path around the Earth.



Circle in a Square in a Circle

A circle in a square in a circle defines a heavenly ring whose area is equal to the enclosed circle.

In other words, the area between the two circles (the red ring) is equal to the area of the smaller white circle.

circle in square in circle



“Squaring the Circle”

The Squaring of the Circle is a method of understanding π that has fascinated mathematicians for centuries.

In ancient Greek times it was a puzzle posed by the Delphic Oracle.



It consisted in the challenge of constructing 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.

In other words, the perimeter of the square and the circumference of the circle must be equal.

Or, the area of the square must equal the area of the circle.

The circle is based on π, an incommensurable number, so it is impossible to draw a square more than approximately equal to it.

Robert Lawlor writes, “When a near-equality is drawn between the circle and square, the infinite is able to express its dimensions or qualities through the finite.”2


The challenge is of great significance:

The circle represents pure, unmanifest spirit-space; the “Heavens”.

I 7 globular cluster 597899


The square represents the manifest and comprehensible world; the Earth.

cartography 1801579


The ‘Squaring of the Circle’ combines Heaven and Earth, or Consciousness (Spirit) and Matter.

This is sometimes called “wedding flesh to spirit” or the “coming of the bride and bridegroom” – a wedding metaphor in the Bible.

It is a symbolic representation of the transmutation of a human.

“The medieval squaring of the circle through the pentagon is meant to symbolize the harmonization of intuition (indicated by the square); or the idea that the infinite (the circle) communicates with the human intelligence through the laws of harmony.”3

milkyway lowres



“Religious symbols and cosmological patterns the world over, from the mandalas of Eastern mysticism to the rose windows of Gothic cathedrals, are versions of the squared circle.”4

Basilica of Saint Denis Rayonnant Rose Window

“Squaring the Circle can be taken to mean the establishment or maintenance of heaven on earth.”5

This entails the spiritual transformation of individuals which results in the spiritual transformation of the Earth as a whole.

“In another sense it recapitulates the essential truth that time depends on the timeless and that manifestation depends on its motivating principle. Principle formulates matter and matter expresses principle.”6

This entails idealism – that consciousness forms matter.

It also entails the two-way universe with continual reciprocity between time/space and space/time, the metaphysical and physical, consciousness and matter.



The Squared Circle also represents Spheres within Cubes

See Articles 104-105. This concept is a reflection of Dr. Harold Aspden’s work and his model of Aether units (Quons) that are rotating spheres within cubes.

FF Aspden cubesphere



This also reflects the cube-sphere of Walter Russell’s physics.

russell cubesphere


Therefore, the geometric exercise of Squaring the Circle can be a way of understanding the intrinsic geometry of which all life, matter, and consciousness arise from.

“Squaring the circle therefore represents the achieved equilibrium between heaven and earth.

It is a perennial philosophical symbol: it is the establishing of heaven on earth and an act which informs as well as enforms that aspect of the cosmos which is furthest or most obscured from its archetypal source.”7

enform – to form or fashion

inform – to give knowledge or information


John Michell writes, “The circle and the square, made commensurable with equal perimeters, form a diagram of the fusion of matter and spirit and together illustrate human nature and the nature of the universe.”

This means, the Squaring of the Circle represents both the establishment of ‘heaven’ on Earth, and also the establishment of ‘heaven’ within each person.

In either case, establishing ‘heaven’ refers to living in harmony and unity within our minds and emotions, in our relationships, with our world, with our environment, and with our higher selves.

other purposes they sensed




Squaring the Circle – Two Ways

As we saw above there are two ways to square the circle: equal areas and equal perimeters.8

squaring circle Squaring the circle

Equal Perimeters                                     Equal Areas



Squaring the Circle Methods

Equal Areas:

squared circle area

“The relationship between the diameters of the large circle of the moon’s path and circle of the earth come close to the ratio of 9:8, the basic interval of the “tone” on which the musical scale is built.”9


A Square with area of 4 has width of 2.

A Circle with area of 4 has diameter of 2.256758…

A 9:8 ratio squares the area of the circle to 99.70% accuracy.
In music theory, a 9:8 ratio is the whole step, otherwise known as the whole tone, or major second.

9/8 = 1.125 (Whole step)

2.256 / 2 = 1.128379 (Circle diameter/2)
1.125 / 1.128379 = .9970 (Whole step/Circle diameter)



Equal Perimeters:

squared circle perimeter

A Square with perimeter of 4 has width of 1.

A Circle with circumference of 4 has width of 1.2732396…


Interestingly, our Moon takes 27.32166 days to make 1 revolution about Earth.
This is a 99.998% accurate correlation according to NASA.


“At other times [the squaring of the circle] is used to symbolize the virtually impossible task of making one order of things conform to another.”


The Squaring of the Circle is about reconciling paradoxes, unifying opposites, transcending polarities and finding compassion for those different than yourself:

  • conservative/liberal
  • masculine/feminine
  • rational/intuitive
  • parent/child
  • Republican/Democratic
  • Rural/urban
  • science/spirituality
  • North/South
  • East/West
  • Muslim/Christian
  • Atheist/Religious…etc.

“Its real importance is that it works on many levels of both practical and psychological significance.”10

heartlight lowres




Squaring the Circle – Earth & Moon (11 & 3)

The square fits around the Earth.

The moon is brought down to the surface of the Earth. The circle goes through its center.

The number 11 is used.

A circle measuring 7 across will measure 11 halfway around.

This relates to the Egyptian measurement of pi = 22/7.

The size of the Moon relates to the size of the Earth as 3 relates to 11.

Moon : Earth :: 3 : 11

If we draw down the Moon to the Earth, as shown, a heavenly circle drawn through the center of the Moon will have a circumference equal to the perimeter of a square around the Earth.

The circumference equals 44, the same perimeter as the square around the Earth.

square root 2 paper0

Credit: Michael Schneider



Squaring the Circle – Earth & Moon

The diameter of Earth = 7920 miles.

The diameter of Moon = 2160 miles.

The radius of the Moon circle (the center of Earth to center of Moon) = 5040 miles.

5040 = 1 x 2 x 3 x 4 x 5 x 6 x 7 = 7 x 8 x 9 x 10

7920 = 8 x 9 x 10 x 11

7920 x 2160 = 10,080 (number of minutes in one week)

square root 2 paper stonehenge

Credit: Michael Schneider


In Stonehenge, the values are in feet, not miles:

50.40 feet; 21.60 feet; 79.20 feet



Squaring the Circle and the Twelve

The square fits around the Earth.

The moon sits on top and the circle goes through its center.

squaring the circle earth moon quadrant numbers

Credit: Michael Schneider



This diagram by Michael Schneider illustrates the 12 moons, or 12 months of a year.11

All 12 fit around the Earth. Note that they do not fit over the corners. On a deeper level this symbolizes that this 12-around-1 process has to do with spiritual, metaphysical and cosmic evolution more so than earthly material evoution. The 12 follow the circle, not the square.

squaring the circle moons

Credit: Michael Schneider



This can also represent the 12 ages of the Zodiac during the Precession of the Equinoxes. The Precession is of the utmost importance in regards to cosmic spiritual evolution vs. earthly material evolution.

2160 years = 1 Great Month, or the number of years it takes to cycle through 1 House of the Zodiac.

2160 miles = diameter of the Earth

2160 x 12 = 25,920 years = total time for one ‘Great Year’ or Precession of the Equinoxes

zodiac lowres


Michael Schneider explains, this diagram “holds the canonical numbers seen worldwide in virtually every culture as well as the geometric plan of many twelvefold organizations from antiquity. It also holds proportions appearing in the architecture of diverse cultures from Stonehenge to the plan of the Bible’s Heavenly Jerusalem descending to Earth.”12

The canonical numbers he is referring to are the Precessional numbers we have discussed many times in Cosmic Core.



Squaring the Circle and the Vitruvian Man

The human body is a squared circle in its proportions.

“Leonardo da Vinci’s Vitruvian man…stands in a square and circle – uniting our two primary polar forms.”


As Protagorus said, “Man is the Measure of All Things.”



Leonardo da Vinci wrote, “The workings of the human body are an analogy of the workings of our universe.”


Da Vinci’s Vitruvian Man is “based on the correlations of ideal proportions with geometry described by the ancient Roman architect Vitruvius. Vitruvius described the human figure as being the principal source of proportion among the classical orders of architecture.”13

“The drawing itself is often used as an implied symbol of the essential symmetry of the human body, and by extension, the symmetry of the universe as a whole.”14

universe lowres



Much has been written about this diagram. We will only skim the surface here.

Fivefold Man exists between sixfold Heaven and fourfold Earth.

Leonardo da Vinci shows how a man’s span equals his height.

This measure also equals seven of his feet.

The bellybutton is the exact center of the circle.

The bellybutton symbolizes the 3rd chakra – the seat of will power and a symbol of choice. This chakra is the springboard to spiritual evolution – the 4th chakra of love/understanding. Although in order to spiritually evolve one must make a conscious choice to do so. Strong will power leads to the ability to make this choice and live accordingly.

Therefore, the circle represents the arena of spiritual evolution.

vitruvian circle


When the feet are apart and the arms are held out the body forms a Star. This star represents the cosmic side of humanity. Our home is in the stars. Spiritual evolution is the process that takes us ‘home’. A circle fits around the body in this case.

The groin is the exact center of the square.

The groin represents the reproduction center – the reproduction of physical life.

Therefore, the square represents the arena of material life and evolution. vitruvian square

When the feet are together and arms are held up the body forms a Cross. This position symbolizes Christ being crucified upon the cross. On a deep level this imagery represents the importance of forgiveness. Upon forgiveness of self and others a transmutation, healing and transformation can occur that puts the spiritual evolution process in motion. Christ on the cross exemplifies this important process that each and every person must willingly undergo. It is a sacrifice of grudges, anger and blame that sets one free…not necessarily a sacrifice of life. A square fits around the body in this case.



Squaring the Circle – Earth, Moon, Stonehenge & Great Pyramid

As we saw above, Stonehenge has the exact same proportions as the Earth/Moon.

stonehenge full


Stonehenge encodes the squared circle through its bluestones (Sarsen stones).

If we draw a square around this circle of stones it will have the same perimeter as the sarsen ring of stones on the very outside.

It is a calculator to determine positions of the sun and planets in relationship to where we are in the year (astrological/astronomical calculations).

The inner circle of stones is the square.

The outer ring of stones makes the squared circle.

These are shown in red in the diagram below.

stonehenge squared circle


The Great Pyramid’s height is to its base lengths as a circle’s radius is to its circumference – thus, it squares the circle.

egypt lowres


In other words, the perimeter of the base (the square) equals the circumference of a circle whose radius equals the height of the pyramid (drawn through the capstone).

Pyramid Earth Moon Geometry copy squaring the pyramid


This is due to the slope angle being 51º 51′.

This 51º 51′ is naturally derived from a vesica piscis nested within another vesica.

great pyramid and vesica



Squaring the Circle and Chartres Cathedral

Chartres Cathedral in France was built and designed by the Knights Templar who brought this new science from Egyptian mystery schools and Persian/Sufi mystery teachings after the Crusades.

They used the Squared Circle sacred proportions.

chartres cathedral 1120139 1920


There are two pillars on front of the cathedral. One is Gothic, one is Romanesque. These are the Sun Tower and Moon Tower that symbolize the two pillars, Joachim & Boaz, of Freemasonic/Egyptian symbolism. We discussed these pillars in Article 20.

1200px Chartres Cathedral


There is a beautiful mandala in the cathedral – a Rose window – in the center.

chartres rose


Stained Glass windows such as this came from Persia and Islamic art.

The building itself is shaped like a Crucifix. The two pillars are at the base of the cross.chartrescathedral floorplan blog2 fig6


The top third of the floor plan (the high altar point) looks exactly like Stonehenge. The image below right is part of the ground plan of Chartres. The image on right is Stonehenge. Notice the same proportions down to the same horseshoe shaped structures.

stonehenge chartres


The roof inside is composed of pyramid shaped arches.

chartres 663730 1920


John Mitchell describes in detail the sacred geometry relationships of the Chartres Rose window.

Images by Michael Schneider can be seen here: Chartres Rose Window

chartres rose windows


Remember, these cathedrals were designed to be ‘crucibles for human transformation’.

They would feel the energetic effects from the roof shape as they burnt frankincense and myrrh to activate the pineal gland (6th chakra/3rd eye) while they listened to Gregorian chants and gazed upon the light illuminating the Sacred Geometry mandalas.

The whole point was to activate and raise the consciousness of attendees. The combination of sacred art and sacred proportions operated on a spiritual, meditative, and emotional level.

See Article 15 for more information on Rose windows.



Cubing the Sphere

Scott Onstott teaches that the Cube Volume equaling the Sphere volume is related to the golden ratio.15

onstott cubing sphere

Credit: Scott Onstott, Tripartite


The length of the side of the cube has the ratio of phi (1.618) to the radius of the sphere.

Or: side length of cube = 1 and radius of sphere = 1/phi.

Volume of a cube = s3

Volume of a sphere = .333333…x πr3 + πr3

Here Onstott has rewritten the formula for the volume of a sphere to emphasize the important pattern of repeating threes. This suggests a tetrahedron in a sphere.


Reference Construction Lesson #38: Squaring the Circle.


  1. Michell, John, The Dimensions of Paradise, Inner Traditions, 2nd edition, 2008, page 74
  2. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982
  3. ibid.
  4. Michell, John, The Dimensions of Paradise, Inner Traditions, 2nd edition, 2008
  5. Critchlow, Keith, Time Stands Still, Brecourt Academic, 2nd edition, 2007
  6. ibid.
  7. ibid.
  9. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
  10. Critchlow, Keith, Time Stands Still, Brecourt Academic, 2nd edition, 2007
  11. Schneider, Michael,
  12. ibid.
  14. ibid.
  15. Onstott, Scott, September 2014 AOM: Measuring the Mystery, 27 August 2014,


Return to Free Library

Return to Number Menu

Previous Article                                                                         Next Article