The Diagonal and the Reciprocal.
p.34
As diagonals seem to play an important part in generating related shapes, this suggested an investigation of them in a different context: the diagonal and the reciprocal within various shaped rectangles.
The reciprocal of a rectangle is a figure similar in shape to the original rectangle, but smaller in size.
Take a rectangle which has a randomly chosen shape of length and breadth, then draw a diagonal: bottom left to top right. Place a set-square along the diagonal, so that the right-angled line meets the bottom right hand corner of the rectangle. Extend this line upwards to meet the top line, and drop a perpendicular from it down to the bottom line.
There are now two rectangles within the original one, but the two shapes are not the same as each other. The rectangle on the right is the reciprocal shape : the same shape as the whole rectangle. The end of the original rectangle becomes the length of the reciprocal.
Moving down the page there are two diagrams. In the first, the diagonals of any rectangle divide that rectangle in half. In the second, the diagonals of the whole rectangle and of half the rectangle divide it into thirds.
It is easy to see that divisions of an area can be more readily divided up geometrically than arithmetically -which uses measure For instance here, diagonals of a rectangle give the centre point; and the diagonals of the whole and half define the third.
The Root Rectangles and their Reciprocals.
p.35
Uniquely, the reciprocals of the root rectangles divide exactly into their original shape. This can be confirmed using the same method on the previous p. 34. Also in the revolving rectangles round the pole.
The exact meaning of reciprocal means: Function or expression so related to another that their product is Unity. In this case : 1/5 x 5/1 = 1. The relationship is an expression of perfect harmony, as the Whole is reflected in every part, and every part is a reflection of the Whole.
This understanding is of great importance; and has been used widely, by architects, designers, and by practitioners in other fields to bring their designs into harmonic relationships.
As an example, as seen at the bottom of the page in the two √3 rectangles, many interesting and complex abstract designs can be found using the reciprocal divisions, visible and invisible.
The Paradox of Sameness and Difference in the Making of a Design
pp. 36-37
I realised that I had encountered one of the vital laws of design: that it’s not just what an element is that matters - it is also what it does and how it is behaving in the design as a whole. See the Introduction ( A Personal Journey).
Though I had been aware of this paradox for some time, it was only gradually and later on, that I fully understood its pervasive influence in every ingredient of all calligraphic work: scale, colour, emphasis, position etc. In this case, because of the subject matter, I was using this law to bring the drawings and text into harmony on the page, though the same law could be used to create dissonance, if that were needed.
In the case of this double-opening I needed to adjust the design because of the too-eye -catching red sentence in the centre of the piece. However - by chance or good luck, the added red italic at the top somehow brings the whole page to life! An unexpected bonus.
In the paragraphs mentioned earlier every element has its own integrity and flexibility encoded within itself, the Absolute and the Relative. It is a paradox, and a paradox is a very profound way of opposing factors being held together. Thus a composition is created with an understanding of not only the actual colour or note played by an instrument (visual or aural) but of all the possibilities inherent within each and how they play together: visual art on a spacial surface, music in time.
Quantum Physics
In my understanding of an aspect of Quantum Physics, every thought or action has multiple possibilities both conscious and unconscious, which are expressed by its wave function. This is expressed in any mundane activity of the day: should I go out shopping now or later? Makers in any art form are familiar with this situation as multiple choices float around in the mind when considering any piece of work. In the quantum world it is suggested that a favouring of any one choice - a specific focus, an intention - causes extra energy in the brain that results in a more coherent firing of the cells, and the other possibilities fall away.
Relationships and Intention: A Philosophical Example
Though we are individual people, each one of us has within ourselves multiple sub-selves: we behave differently to our parents, children, cousins, friends, and to each of those differently in different contexts; and the subtle adjustments made in each situation, if sensitively handled goes to create a more harmonious society. Integrity and flexibility are encoded within the same being. And the reverse is also true! So the same principle is found described in many situations.