The QR (Quick Response) Code
The square and the rectangle are probably the most powerful shapers of contemporary visual thought, but they were never seen on cave walls, never seen as accompaniments to lives embedded into nature. Squares and rectangles signify a non-organic world, one where humans have devised grids that they place over nature and use to own it. Rectangles form doors that allow us to think that we can step outside of the interconnectivity that we live within. These Platonic forms rely on straight lines for their construction and the earliest straight edge drawing devices would have probably been lines made by tautening strings of some sort, drawings would then be made in a similar way to how we now use chalk snap lines. We need a straight edge of some sort to construct a square or rectangle. We also need a way of ensuring that the edges of rectangular forms are at right angles to each other. This now usually means using a ruler and compass construction, a method that would have been preceded I'm sure by a taut string process. These complex constructional issues meant that the square and the rectangle were relative late comers to the arsenal of forms that human beings developed as ways to think about concepts using things that formed extended minds, objects that existed outside of their own bodies, such as drawings and other constructions.
How to draw a right angle
The stringing of areas of land allows you to divide the land up. The land along the edges of the Nile was very fertile and the extent of the annual flood of the river was measured by stringing, the results of which were recorded in some of the earliest records of land registry. Thousands of years later, when archeologists begin an Egyptian dig they too string the area to be dug beforehand, so that it and everything found within it, can be located and fixed into position. Once you get used to the techniques of measuring areas by using pegs and taut string it is a very simple step to begin string geometry. A taut line between two pegs is a straight edge and a taut line attached by a loop, stretched between a fixed peg and a rotating drawing tool (in the case below a lump of chalk) held firmly in a hand, becomes a compass.
Drawing a circle with chalk, string and a fixed point
String defining the edges of an archeologist's dig
From these basic processes was derived Euclidean geometry, 'geometry' being itself a word that is derived from the Greek words 'geo' = earth and 'metrein' = 'to measure'. Geometry developed at the same time that cities evolved, and they came into being alongside the development of agriculture; all three concepts associated with measuring out the land. Early cities were often laid out on grid systems and architecture which had previously been very organic, became concerned with structural principles based on right angled forms. The plumb line being another taut string device, this time using the action of gravity to find a vertical. By 2600 BC, Mohenjo-daro and Harappa, major cities of the Indus Valley civilization, were organised using blocks of buildings divided by a grid of straight streets.
By the time of the writing down of the Epic of Gilgamesh (2,100 BC), the square was understood as being central to the formal idea of a city.
"Go up on to the wall of Uruk and walk around. Inspect the foundation platform and scrutinise the brickwork. Testify that its bricks are baked bricks, And that the Seven Counsellors must have laid its foundations. One square mile is city, one square mile is orchards, one square mile is claypits, as well as the open ground of Ishtar's temple.Three square miles and the open ground comprise Uruk. Look for the copper tablet-box, Undo its bronze lock, Open the door to its secret, Lift out the lapis lazuli tablet and read."
The right angled square is a key formal device in the development of the modern world, and we still live in architecture dominated by it and now embed our lives into screen based technology that is also shaped by the rectangular form.
I was reminded to put up a post about the square by recently seeing a film, which in many ways carried ideas within it that were not very square like. 'The Square' was released in 2017 and without going into detail it is basically a dark comedy and satire on the hypocrisy of the art world. The film is about the publicity surrounding an art installation, and was partly inspired by an installation the writer and directer Ruben Östlund and the producer Kalle Boman had made a few years earlier for the Vandalorum Museum in Värnamo. In their artists' statement for the exhibition they wrote. "The Square is a sanctuary of trust and caring. Within it we all share equal rights and obligations." They had at that time obviously looked at the myths surrounding the square and used what they had found to determine their artists' statement, and then as they reflected upon their exhibition experience decided that the reality of the situation was nothing like what they had written. Out of such reflections ideas emerge and in this case a film was engendered.
My first contact with the mythology of the square, was via Bruno Munari's small book 'The Discovery of the Square', which was written in 1960. It came in a neat square format and by the time I began teaching on the foundation course in art and design at Leeds, was a standard text recommended to all students.
Bruno Munari: The Discovery of the Square
The Discovery of the Square is a delightful book and was accompanied on the library shelves by his other books which came in the same format, 'The Discovery of the Circle' and 'The Discovery of the Triangle'.
The square is associated with the number four and therefore is often linked to other things that come in fours. Such as the four ancient elements of the physical world, (earth, air, water, and fire), the four states of matter, (hot, dry, wet, cold) or the four seasons (winter, spring, summer, and autumn), the four Directions (north, south, east, west) and other more debatable fours such as the four stages of life (birth, childhood, maturity, and death).
It is in the construction of a square that you can effectively watch symbolism arise, because the process of a square's construction is a physical thing, you have to make it and in making it somehow you are much more deeply embedded into its life than simply 'understanding' it, or intellectually referencing it.
When constructing the square you can begin with a circle drawn using your compass or taut string centred with a peg in the ground. Fig 1.
Draw a straight line through the centre point of the circle and you have a diameter and at the same time a fixed length determined as that diameter. Fig 2.
Create another diameter length that goes through the centre of your circle at right angles to the first. (using the how to construct a right angle method already referenced) Fig 3.
The circle is now divided by four equal points that form a cross. By connecting the points of the cross in contact with the circumference, we obtain a type of square that is symbolically 'active', i.e. balanced on one corner. Fig 4.
By rotating the active square by 45°, we get a passive or more grounded square that sits solidly on its base. Fig 5.
This process is the method on which is then built further divisions of the geometry of a square, most importantly beginning with the square in its active position.
Polygons of space and matter
The polygons that can be built out of the square correspond to emerging ideas of space and matter. Which is for myself an important issue, as it begins the process of linking space and matter together, and eventually what flows from this enfolding of space into matter is the concept of energy fields lying beneath both. For instance, the tetragon represents the four cardinal points of Space, the four directions that we normally think of as north, south, east and west and in cartesian geometry the four quadrants.
The four quadrants of Cartesian Geometry
The octagon represents the four Elements and the four States of Matter as a combined idea.
In the octagon, direction (space) is embedded into the elements (matter), the diagram helping us to think about how the two could be combined. The only problem being that there is a contradiction between this need for order and the fact that everything is in reality in a state of entropy. Rudolf Arnheim wrote a classic essay on these issues; 'Entropy and Art: An essay on Disorder and Order', which was an attempt to reconcile this contradiction and as I extend my thoughts on squares and rectangles I shall attempt to keep Arnheim's observations in mind.
Going from a circle to a square is not quite as easy as deriving a square from a circle. It demands more esoteric knowledge. The squaring of the circle was seen as much as a geometric problem as a spiritual exercise, that symbolised the passage from the earthly (the square) to the celestial (the circle), from the imperfect to the perfect, it is a metaphysical problem. Going from square to circle could also diagrammatically locate a move towards the invisible from the visible, and make transcendent the passage from the sensible to the divine.
The symbol π is obtained on my computer by pressing the option key while I type 'p'. The value of Pi (π) is the ratio of the circumference of a circle to its diameter and is approximately equal to 3.14159. In a circle, if you divide the circumference by the diameter, you will get exactly the same number. But π is a mathematical idea, and although mathematical ideas can seem like universal truths, life isn't like that.
It was Plato who thought there were universal forms that underpinned all reality and similar ideas have continued to pop up whenever human beings have sat around and reflected on what it might be all about. An idea such as an omniscience God is similar, as it depends on a sort of fixed entity, an unchanging something against which all transient life can be measured. But the reality I live with is that everything changes, everything is in flux and you cant really find a point outside of existence to look at it. We are entwined into existence with everything else and myself and everything else will become what becomes as moments unfold out of the nowness of now, there being no pre-known future, only possibilities.
I have looked before at the Necker Cube, a shape that visually oscillates between viewpoints, therefore it is a more flexible diagrammatic form. The Bronshtein hypercube uses this format and can help to visualise the idea that even emotions are quantum processes and that they are as tied into the physical world just as securely as mountains.
The Bronshtein cube
From: Studio Tomás Saraceno: A Hybrid web
The symbol for the Jacob Kramer College Leeds 1968 to 1993
Hybrid forms such as the old Jacob Kramer red spot are perhaps more useful. It could also well be that the QR (Quick Response) Code squares will become a symbol of interconnectedness rather than ownership, as we use them to link site specific experiences with associated information, they operate as a type of invisible link back to whatever context the QR code instigator wants to direct you to. This could be to direct you to buy more, but it doesn't have to be. However I would like to think that it's the spiders that have it right, and that they might be able to devise a new organic geometry for us, one that consists of a fine web of connections that radiates out and takes energy in, that rhythmically responds to the patterns developing around it and that is flexible enough to be constantly re-shaped in the ever happening moment of the present.
A visual conversation with a spider about hypercubes. (Homage to Tomás Saraceno)
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