Circling Squares and Pentads in a Finite World

Apparently when you
Square a Circle in a Pentagram

you get a woman over the moon.

The marriage of heaven and earth,
sometimes drawn by artists, geometers and conspiratorials
as a Circle Squared with Equal Circumference,
or Area – whichever you prefer.

And, with this same centre point,
set comfortably within the Pentagram… it leaves a gap
for the Moon… and the proportions
of a Woman Fivefold.

The Squared Circle and Proportional Woman

G. Barnes (2013) Squared Circle, Woman, and Moon in Pentagram [Pencil and tempera wash on paper]

I think. I am yet to confirm the perfect shape of woman.

And not only that, but a Moon that is in fact, in an exact ratio to the Earth.
As per in space – real space – as measured by students studying ASTRONOMY 101.

if Leonardo had have spent less time on war machines,
less time trying to fly and failing (flailing)
less time butchering,
and brotheling.
Less time positioning Man’s navel as the center of everything,
that strange herniac creature, that cadavar of ‘perfect proportion’.

He would have seen what is obvious to all cooks and cleaners and cheesemakers
– and artists with half a brain making vague scribbles in a fog of maths.

That the Squaring of a Circle, when in perfect alignment around a perfect center.
And, when within that fated shape of Platonic perfection.
The Pentad Polyhedra
Ridiculed since the Renaissance and forced into the service
of fantasy for demons
and devils of fiction
with witches and warlocks,
for fear,
that it maybe the key,
to her and she,
together beyond patriarchy.

This final and mysterious shape /a perfect solid –
Could it be the shape of a universe emasculated?
(I mean encapsulated)

Beyond and before, this icon of misguided Humanism.

With legs astride, in extension; the universe embraced in limbs.
Centered in pleasure. Squared in pain.
Engaged to the Moon in Birth and Death.
Married to Earth.
Transfigured with Heaven.

In constant Expansion and Contraction through the Universe.

G. Barnes. November, 2013

G. Barnes (2013) God Measures 1 Micron of Gesso [Photoshop Study]


The Platonic Solids.

Wenzel Jamnitzer (1958) Perspectiva Corporum Regularium. Pentad polyhedra: The Platonic regular Dodecahedron is the first one.


Seen most obviously as the five-pointed star (pentagram) or a polygon of five sides (pentagon). In a regular pentagon, all sides are equal in length and each interior angle is 108°. A regular pentagon has five lines of reflectional symmetry, (five fold symmetry) and rotational symmetry also in order of 5 (through 72°, 144°, 216° and 288°). The diagonals of a regular pentagon are also, beautifully, in golden ratio to its sides. These diagonals can be calculated based upon the golden ratio φ and the known side T:

frac {D}{T} = varphi = frac {1+ sqrt {5} }{2} ,

When mapped into three dimensions the pentad creates the dodecahedron – which is Plato’s last and final perfect solid. [There are only 5 regular solids in the world that can be constructed with regular faces.]  It is composed of 12 regular pentagonal faces, with three meeting at each vertex, and has 20 vertices, 30 edges and 160 diagonals.

The dodecahedron was the last of the Platonic solids to be discovered. In Plato’s dialogue Timaeus (c. 360 B.C.) he associates the other four platonic solids with the four classical elements, adding that there is a fifth figure – the dodecahedron which “God used in the delineation of the universe” – to shape the universe – and which is associated with the element ‘aether’ – ie cosmic dust?

Recently, various models have been proposed for the global geometry of the universe. In addition to the primitive geometries, these proposals include the Poincaré dodecahedral space, a positively curved space consisting of a dodecahedron whose opposite faces correspond (with a small twist). This was proposed by Jean-Pierre Luminet and colleagues in 2003. (Physics World) They predict a model in which space consists of 12 curved pentagons joined together in a sphere. Their ‘small’, closed universe should be about 30 billion light years across.

“Our work really addresses this ancient question of whether the universe is finite or infinite, the exciting point is that this is no longer pure speculation – we now have real data.” (Jeff Weeks, Physics World)

The pentad was revered by many early societies: it is found throughout nature – from flowers, to apple cores, to the very shape of the universe. It is a symbol of life and regeneration. It is the five wounds of Christ, it is a symbol for truth.


Earth | Moon. Ratio

Earth | Moon. Ratio

Circle Circumference Formula: CC = πd
Squared Circle Circumference CC/4=W
Circle Area Formula: AC = πr2
Squared Circle Area: √AC = w

The smaller circle (moon) is a circle with the radius of the gap between the squared circle and the square. It also has the diameter of the gap between the polygram and the square/circle.
The Moon is 3,476 km wide, the Earth is 12,742 km wide.
12742/3476 = 3.665 ratio
My picture: 1258 /297 = 4.11 ratio
NB. My pentagram was not perfect to start with! Therefore the squaring was a bit rough… The transcendence of pi implies the impossibility of exactly “circling” the square, as well as of squaring the circle. It is at best an approximation of the ‘truth’, as is all art and life!


Leonardo Da Vinci

Leonardo Da Vinci “The Viturvian Man” 1564 [watercolour]

Leonardo’s Vitruvian Man is NOT an accurate squared circle. Many commentators have tried to apply this geometric concept to it, but it is not a squared circle by area or circumference. I measured it! His circle was merely inscribed within the square from the navel – although it is close.

This image does provide an example of Leonardo’s interest in the proportion of ‘man’. In addition, this picture represents a cornerstone of Leonardo’s attempts to relate Fivefold Man to Nature. He conceived of the human body as a cosmografia del minor mondo (cosmography of the microcosm), to be an analogy for the workings of the universe.

It is called the Vitrivius Man because Leonardo applied Vitrivius’ proportions of man from his architectural treatise. Man = temple. I wonder what a temple designed around a woman’s body would look like?

“Similarly, in the members of a temple there ought to be the greatest harmony in the symmetrical relations of the different parts to the general magnitude of the whole. Then again, in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centred at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square. [Vitruvius, pg.73.]


Dumé, Belle. (2003) PhysicsWorld. Retrieved from

Critchlow, Keith.(1999)  Islamic Patterns: An Analytical and Cosmological Approach. Inner Traditions, rochester, Vermont.

Hemerway, Priya. (2008) The Secret Code. The Mysterious Formula that rules art, nature and science. Springwood; Lugano, Switzerland.

Lundy, Miranda. (1998) Sacred Geometry. Walker & Company; New York.

Pastorello, Tom. Leonardo Squared the Circle! — Da Vinci’s Secret Solution in the Vitruvian Man Decoded. Retrieved from:

Vitruvius. The Ten Books on Architecture. Translated by Morris Hicky Morgan, Harvard University Press, Cambridge (1914). Retrieved from:


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