Beginnings
pp. 14-15
This was an essay that I wrote in the early stages of writing the Journal, when I had just begun to realise some of the implications and scope of the whole subject. At first, I was aware of the importance of enactment, as it seems that the symbolic level only reveals itself in the act of drawing, as the shapes unfold themselves - stage by stage, each from the preceding one. Drawing brings a greater awareness as to what is happening on the page: the bisection of a line, the repetition of a shape; how complexity builds from very simple beginnings. And because the principles are universal, of Division, Evolution, Harmony, the enactment of these seems to key us in to the archetypal level.
The two different kinds of numbers: the whole numbers and the irrational numbers seem to play a significant part in the unfolding of processes. For instance, in the circle, if the diameter is one unit, its circumference is PI -=3.14159…. an irrational number ( that which continues for an unlimited series of never repeating digits after the decimal place. It is unknowable, unfathomable). In the square, if the side is one unit, the diagonal is √2, and if √2 is the side of the bigger square, and its diagonal is 2. I understood that the interplay of the two kinds of numbers was important as they represented different aspects: the irrational numbers represented the hidden dynamic forces of process, and the whole numbers the manifested form- the particular event in time and space generated by this energy.
The mathematical and geometric rules have to be learnt, and I put myself through a course of basic grounding in geometry early on in the development of this project. This, coupled with study and more practice, gave me the beginnings of a deeper insight into the whole subject.
Through reading, I began to understand that the knowledge and symbolism of philosophical geometry was widely used in the ancient world, in many sacred traditions. For example the Great Pyramid, the Parthenon, and many medieval churches and mosques embody a mystical enactment-in- stone of sacred principles, such as ‘squaring the circle’ - of Earth and Heaven; and the knowledge of the laws of harmonics and proportional relationships of the parts to the Whole.
Early Christian churches were often built quite simply with a domed roof placed over a square edifice, signifying the connection between heaven and earth. Later more elaborate buildings were created, using Geometry to organise the relationships between various parts of the structures, as Geometry rather that Arithmetic more easily facilitates the unfolding harmony of the parts to the whole. Geometry uses proportional units of quantity rather than individual measure.
I understood that it was the Muslims that translated Euclid (325-265 BC) from the Greek into Arabic in the 9th century. They were then able to combine Euclid’s treatise in Geometry with the arithmetic and algebra learnt from the Hindus and use it for their own purposes.
The Muslims were far more advanced in mathematics than the Europeans of this time, and they were particularly active in Spain where around the 10th century Muslims, Jews and Christians lived in harmony.
The knowledge they had acquired was shared, and gradually filtered up into Northern Europe over the next few centuries.
The Muslims had a sophisticated ideology of Number. Because God, they believed, was unknowable (never- to-be-known, only experienced) they designed their sacred buildings round this philosophy, using irrational (the unknowable) numbers as an integral part of the structure. In the great flowering of Christian church and cathedral building in Northern Europe carried out in the 11th - 13th centuries, the architects ( called the Masters of the Compasses) incorporated this number ideology into their designs. For instance, at the centre of the church, where the nave meets the transepts, a circle is described, out of which a √3 rectangle emerges, which then sets the width of the nave. Triangles could then be drawn out, which decided other features in the building. Ad triangulum as opposed to Ad quadratum, the latter being an alternative method of architectural planning based on the square.
The use of Geometry as a guide in creating complex architectural systems was in fact well established in the ancient world. It was used extensively in the Neolithic stone circles in Northern Europe, in the Great Pyramid at Giza, the Parthenon in Athens, and in many other important buildings in the past.
Mandala
pp. 16 - 17
I have suggested we could make our own contemplative ‘mandala’ : a personal geometric drawing using some of the understandings learnt on the previous pages. For instance, as we draw out the diagram and enact it, we learn that growth always comes from within, unfolding into further shapes, and understand that the disturbing fractures and expansions of the process are always held by the containing circles which are always in touch with the centre.
To clarify this statement I need to give closer description of what ‘enacting’ a drawing means. It means a requirement of a closer engagement, a heightened attention, and so we become more aware of the sequence and significance of each unfolding. It is only in participating in the process of making that we can truly understand its evolution and significance. In this picture there is the Principal of Growth at the top centre of the page, the diagonal in red divides the smaller black square in half - but it is also the foundation of the larger square. The square root of Two - √2 - is the root ( the foundation) of Square Two. If the square’s side is 1 unit. To continue. In drawing the original black square the lines enclose a particular space and shape: it has a specific wholeness and identity. All the corners are in turn connected to each other in defining that area - except at the diagonal. Taking pen and ink and slicing diagonally across this internal space to connect to the other side can feel brutal - especially as it severs the space at its maximum dimension. However, the severing line has awakened the two opposing corners and made them potent. These now provide the two well-springs for the next moves, where the compass roots itself in one of the points and stretches diagonally to include the other, and takes the energy from that second point which then swings upwards to define the limit of the of the vertical line rising from the first. The same thing takes place from the other side, and the square is completed at the top. This is vey cumbersome to write out in words, but it brings the drawing alive and it is then possible to begin to inhabit its world: understanding how the different points are awakened in turn and the energy is transmitted as the drawing builds. Three important things have come to light.
1. The necessity of process: subsequent elements are drawn from previous ones.
2. Economy of means: nature always takes the easiest route.
3. The use of paradox: The diagonal line seen above is the destroyer and creator simultaneously. This is also economical, as it means that one element can do two jobs. Paradox can be found in other worlds as well: in the natural world the seed-case is broken by the new shoot, the autumn fall of leaves provides nourishment for the spring growth. Psychologically, the crisis or doubt which shatters one’s present worldview can itself be the foundation of a larger world vision. Lastly in Myths: these themes of death and resurrection are celebrated all over the world.