9th February 2024
Circles and squares surround us on earth, in the heavens and in symbolism. Circles are endless and squares are so bloody normal that we probably don’t notice them that much. They are just there. But who really is the square and would she/he want to be a circle?
The carpenter’s square, that L shaped thing that you were given in maths class along with a pair of compasses, is Norma. In astronomy she resides in the southern skies as a constellation which looks like a sort of sort of square bathtub. Next to her is Circinus (the pair of compasses).
Norma’s got some credentials, being surrounded by Lupus (the wolf), Scorpius (the serpent/eagle and Ara, the Altar of Zeus which is also known as the incense burner.
Norma in Latin means both square and standard or rule (the norm). In English we derive our most meaningless word from it. No brownie points for guessing that word is ‘normal’. In Chinese the square and compass is called ‘Kuci chu’ which means ‘the way things should be’ (whatever that is). In American slang squares are a bit dull and ultra normie. Elvis sang about it. When the warden threw a party at the county jail he made it clear he wasn’t having squares around :
‘… hey buddy don’t you be no square
if you can’t find a partner use a wooden chair .. .
Let’s rock ….everybody let’s rock…’
The facts speak for themselves. Squares are a pain up the posterior because they don’t fit in round holes.
Why though? Round pegs in square holes don't seem to be much of a problem. They just slot in, no fuss, cutting the corners. But squares are a total pain. They stubbornly keep their ninety degree angles and that is a major problem for circles.
So why can’t we square the circle? It is an age old quest and people have been attempting ‘de quadrature circuli’ since at least the middle kingdom in Egypt when the scribe Ahmes produced the ancient Rhind Papyrus. Later squaring the circle became a problem for Greek civilisation to not solve properly. I am not supposed to say that because in the West we are required to believe the Greeks invented pretty much everything, despite the fact that they didn’t.
Squaring the circle means attempting to construct a square with an equal area to a given circle. Today we use Pi (the ratio of the circumference of a circle to its diameter) to get as close a value as we can, but this is not accurate and Pi is not resolved. It has seemingly infinite decimal places:
3.14159265358979323846264338327950288419716939937510 … etc. etc. etc. etc. etc…..
I believe attempts to square the circle are futile, as Pi is unlikely to ever resolve. The efforts are based on a misunderstanding of the nature of the circle.
Lets say you need to get from point A to point B as the crow flies. On a globe, for anywhere other than the poles, that means you need to mark out a trajectory on a curved surface. That will be continuous with no change of direction as below:
Travelling east or west, when you get to point B, if you carry on in the same direction you will eventually find yourself back at point A. So by continually travelling in the same direction, you go nowhere. You are back at square one (no pun intended).
‘Plus ca change, plus c’est la meme chose ..’
However for everyday purposes we do not need to factor in the curve if we are using small distances. So A to B can just be figured as
If when you get to point B, you decide you, say need some toblerone, and thus head to shop at C to buy some, and then want to return home to point A to eat your toblerone, you have to change direction twice. You have travelled in a triangle.
Triangular honey with triangular bees ……
Now let’s say, before you return home to A, you decide that despite having your toblerone on you, you need a good square meal and there’s a pub at point D. You will need one more change of direction and you will have travelled in a square. That’s 3 changes of direction.
The way we can recognise that we are travelling in a square shape is simply by the changes of direction at 90 degree angles. These changes of direction come in fits and starts. They take time and waste energy. Continuous travel in a circle is more streamline. More economical You can enclose more space with less effort. Cut out all the stops. This has real world uses. Say you are in a dispute with the neighbours over land and you want to grab as much as you can, you will get a bigger plot if you make your fence circular. To enclose a square plot of land, just as large, you would need more fencing.
But no matter what size you make your square plot of land, it will never be exactly the same area as the circle. The question is why? If you stake out your fence in a square, why can’t you enclose an area which is as great as the area you would get if you made the fence a circle? Why can you not have your cake and eat it? Why can’t a square be a circle?
Is it a man woman thing? As Professor ‘enry ‘iggins lamented:
Why can’t a woman be more like a man? Men are so honest, so thoroughly square ….
It’s more like man is missing the point. While he is concentrating on squaring the circle, he is trying to define continuous change by interruptions. If you draw a circle on a piece of paper with a compass your pencil is smoothly and continually changing direction without any need to stop, reverse, or slow down before returning to source. There is just one line. A square has four intersecting lines. There are abrupt and severe interruptions to direction in the form of angles. So squaring the circle is defining something by what it is not. It is an attempt to stuff a shape formed by starts and stops into a continuous smooth whole process. It’s like having to utter the eternal sound ‘ Ooooooohm’ with a stutter.
‘Uh …uh ..uh…….O …. o … o .…… u h …... m … um uh… uh …um ………er …
It doesn’t work. It’s like trying to define water by its ‘dryness’. It’s all pre-frontal cortex. It’s the Hadron particle collider mentality. The same kind of mind that pursues the goal of finding the smallest possible particle by making the blasts bigger and bigger. The mind that misses the wood for the trees.
Archimedes is usually cited as the hero of squaring circles. The claim is that although you can't do it with a square and compass you can draw fancy spirals and right angled triangles and this solves it. It doesn’t though. It is merely an assumption since you aren’t gonna find the area of a triangle with pi.
Even modern computers have not been able to resolve the problem. Back in 1882 a French guy, Von Lindemann, showed that pi is transcendental. What he meant by that was that it is impossible to draw a line of length ‘pi’. In fact is we don’t actually know what the area of any circle is and can only approximate with pi multiplied by the radius squared.
But why did anyone use the term squared to measure the area of a circle? After all squared means what it says on the tin. Squared. If I take 120 wooden stakes which I could space out at one foot intervals as a circular plot. What is the point with fluffing about with pi and measuring the circle in ‘x’ amount squared? Surely the measure should be ‘x’ amount ‘circled’? I could measure the perimeter of my plot and define the space enclosed as 120 ft circled.
0
120 ft
What’s wrong with that? We don’t try and define the measurement of squares in circles, so what’s the deal with spending thousands of years trying to define circles in terms of squares?
In nature the shortest point from A to B looks to us like a straight line (the sun’s rays, the spider’s silk) and there are what appears to be straight lines in nature, for example the edges of crystals. However when it comes to enclosed spaces nature prefers curves. Rain lives in cloud-shaped houses which morph into different shapes. When they venture out into raindrops they also morph and change shape. Modern man has for some reason developed a tendency to build his nest with straight lines forming box rooms. Not that many people choose igloos these days and most people don’t wanna go back to being cavemen.
But at least caves aren’t boxes. Box rooms do not feel natural, because mother nature tends to work in curves, foxes and badgers don’t make square holes, moles don’t pile their earth in cubic mole hills. Trees don’t grow one hundred percent straight like rigid telegraph poles.
Living in a box is not conducive to great mental health, some would rather live on the streets than in four walls. Baroque and gothic architects understood the importance of harmonics in building and this has been forgotten in modern structures. The same harmonics that apply in music , also apply in gothic churches and cathedrals. Looking at modern ‘affordable housing’ it seems to have got stuck in the military march which is probably the most tedious and unenlightening form of music. It is too even, too exact, too square. The epitome of conformity. Norma. Never misses a beat. Reliable though. You can fix your widescreen TV flush to the wall. Contrast that with the lazy sound of jazz floating freely on the evening breeze. That’s your spooky castle , the kind of venue you come across in the Rocky Horror Picture Show. Not so reliable. Anything can happen.
Looking at why the musical March has such a boring feel, it is not due to the repetition. Both boring tunes and exciting tunes have repetition. Repetition is really what makes a song. But exciting or interesting has a more diverse pattern. The nature of the pattern creates the melody and rhythm. And how those patterns are layered (or how they relate to each other) is what increases the level of interest (harmony or counterpoint). Jazz and swing use syncopation, more complex rhythms and varied tones from different instruments and all these things create interest. If you close your eyes and sing, or listen to music you will see shapes according to the sound. I’ve never seen static squares to Gershwin’s ’summertime’. Sound comes out of holes in instruments too. And they aren’t usually square holes.
The Aristocats were right on it …
‘ … now a square with a horn can make you wish you weren’t born every time he plays
and with a square in the act you can set music back to those caveman days … cha cha bo dam bo day
well I’ve hear some corny bird that tried to sing
still a cat’s the only cat, really know’s how to swing
who wants to dig a long-haired gig or stuff like that
when everybody wants to be a cat …..
I mean can you imagine any self-respecting cat bothering itself with squaring circles?
But before we condemn all squares, it is important to remember the ying and yang. Where does the light come from? It doesn’t come from light does it?
The basic form of the square is integral to the beauty of Hindu temple architecture. This square repeats and repeats in one continuous form. It’s obvious that Hindu architects of the Vedic tradition knew their sacred geometry.
11th century Temple in Khajuraho which means ‘the great god of the cave’
The origin of the Vedic square mandala lies in seeing the square as representation of the divine form. That is because it symbolises inertia, stability and presumably dependability. Brahma was at centre and other Gods relegated to outer positions in the square. The square was shown to accommodate a yogic form
Leonardo Da Vinci , influenced by the Roman architect Virtruvius, also represented the human figure as the basic proportional measure, But in a circle.
Vitruvian Man
The circle in Vedic tradition represented the world or the cosmic egg. The problem of squaring the circle for them is one of equating the earth and sky altars. So it is a cosmic problem. It is not something that can therefore be tackled in just two or three dimensions.
So this is most likely why we can only approximate to the unsatisfactory Pi value. I believe it is because for Pi to resolve and for a square to have exactly the same area as a circle, you would need to divide the circumference of the circle into the smallest possible parts in order to measure it as tiny straight lines. Here lies the crux of the issue. There are no bits of truly straight line in a circle unless you literally divide it into points. But a point on a line has no measurable length in geometry and therefore there is nothing to measure. Going smaller (dividing within the point), we then go into micro measurements, and eventually the world of atoms with fuzzy edges which lies in the realm of quantum physics and invisibility which is not helpful in trying to measure a visible polygon. At that level the ruler itself and the act of measuring interferes with the calculations and you never get a truly objective result.
The real issue with trying to force Norma to be a circle is that …… she is not.
There is still a debate as to whether the Great Pyramid of Giza ‘squares the circle’. An imaginary circle drawn with a radius that is the height of the pyramid does have a circumference approximate to the perimeter of the base and if you divide the base perimeter by the height you get very close to 2 pi. But the Greek pi does not accurately square the circle. It is a mere approximation. Similar results could be got by just dividing 22 by 7. And the Egyptians were well versed in the correspondence between harmony and measure. Seven embodied growth and harmony. Seshat (meaning 7) was the female counterpart of Thoth as Mistress of Measure. The Egyptian cubit had 7 palms each with four fingers and the gradient( seked) was worked out by how many palms in length along the base of a pyramid were used for each cubit of vertical. Egyptian art and architecture did use a grid of 22 squares in the late kingdom, but previously ancient Egyptians used a grid of 19 squares for their architecture. For them everything is process and relationship. Numbers to them (neters) were living entities, actually gods. They were functions, not fixed immutable values. Nothing in ancient Egypt was considered separate from the whole which is why their fractions could not have more than 1 as a numerator (apart from 2/3 which was considered an integral part of the whole).
Schwaller de Lubicz did in depth studies of Egyptian mathematics and their techniques of using 19 squares (as did the Mayans). He doesn’t come up with a reason for the choice of 19, but I myself have come up with a theory. I believe it may have been due to movements of the moon. The Egyptians used proportion in their art based on ratio of 18:19. There is an 18.6 year moon cycle between lunar standstills and the moon ends up in the same position from our perspective every 19 years. In the present time we are just about at lunar standstill in this cycle which is due 2024/5, which is an indicator that the fiat financial collapse is coming soon. Egyptian architecture was based on an understanding of the importance of these cycles. So 18 divided by 19 may have been important in Egyptian harmonics and building due to an attempt to mirror the moon on its 18.6 year cycle.
The Great pyramid is, regardless, an incredibly impressive scale model of earth, built with knowledge of both the dimensions of the earth and the movements of the stars and planets. Robert Bauval theorised that the Giza pyramids have a correlation with the three stars of Orion’s belt. It is interesting that the layout of the pyramids at Teotihuacan in Mexico also seem to correspond to Orion’s belt. Although it is not certain who built those pyramids, the Mayans also used a grid of 19 squares for their buildings. This is not coincidence.
Man will probably carry on trying to square the circle, willing that Pi value to stop spilling out endless decimal places. The more so, now he’s got digital toys to play with. Norma is his ideal woman. She who spills out the edges will just have to be stuffed in, coz one size fits all.
Did someone vote for Norma, or is she just God?
I enjoyed your meditation on squares and circles. Circles appear all over nature, but squares not so much...really only in the human-made world. It strikes me, after reading your article, that PI is a square-centric view of circles, declaring that the circle is irrational. Makes we wonder what a circle-centric view of squares might be.