Finding relationships between incommensurable systems of geometry can be likened to the alchemist's search for a chemical process whereby a base metal can be changed to a precious one. A true alchemist knows that the search and transmutation isn't so much a physical one, but rather, a transformation of consciousness and spirit where a union takes place between his spirit and matter, in much the same way that prayer and ritual -hopefully - work for the religious faithful. The "philosopher's stone", that praised and elusive elixir, can be compared to the geometer's quest to draw a construction that will link him to a more spiritual existence, a deeper understanding of the universe of potentialities. The Seal of Solomon (the Star of David) in the Middle East, the horizontal and vertical axes of the Cross worldwide, and the symbol of the Great Goddess Yin -Yang in the East are but three examples of this concept. The mandala (Sanskrit for circle) is a universally found device that seeks to unite the circle with the square - what is above with what is below. Symbolically and traditionally, the mandala serves as a blueprint for mindfulness and proper actions. The circle is equated with heaven or the cosmos, and the square with the earth. The square usually suggests the four directions, the four seasons, or the four elements, all of which have four equal magnitudes. In geometry, one of the crowning glories of an effective transformation is quadratura circuli, or, in more modern terms, the squaring of the circle. The question it posits is this: Is there a construction, with compasses, a straightedge with no calibrations, and drawing tools alone, that can yield either the areas or the perimeters of the two shapes equal? [1] Although proven to be impossible to do in the 19th century (a solution nonetheless!) because of the irrationality of pi, a search on the internet will show that squaring the circle is still alive and well. The issue is that, as a rule, incommensurable systems - irrationals[2] - cannot be mixed with other irrational or rational systems into a cohesive whole, in much the same way that J.S. Bach cannot be fully appreciated while listening to the Beastie Boys or Perry Como. This is not to say that all incommensurables cannot be joined together. One need only look to the square root rectangle system as an example. This system is generated by the incommensurablity of the side and diagonal of the square . Using the Pythagorean Theorem, these two lengths cannot be rectified, for when the side is 1, which is rational, the diagonal will be v 2, the decimal places being eternally long and whose sequence is impossible to predict. This diagonal can be used to generate a new rectangle whose sides are in this ratio: 1 : 1.4142… The system then continues to use the diagonal of the previous rectangle to make a new rectangle, so that we have an arithmetic progression under the radical signs: Ö1, Ö2, Ö3, Ö4, …Ön, Ön + 1, … But when attempts are made to use this 'root system' with, say, the whole number ratios (e.g., 1:1, 1:2, 2:3, 3:4, etc.), inherent difficulties usually arise with alignment, structure, and eurythmy. In his Ten Books on Architecture, Vitruvius states that eurythmy is,

… beauty and fitness in the adjustments of the members. This is found
when the members of a work are of a height suited to their breadth, of a breadth
suited to their length, and, in a word, when they all correspond symmetrically...[3]

( the emphasis on symmetrically is mine).

I emphasize symmetry because there are various types of symmetry operations in geometry, and they follow different geometric paths. Very frequently, these systems clash…they just don't fit together without discord to both. However, a skilled geometer [4] can make the necessary adjustments to overcome some of these issues, but a marriage of incommensurables is, for all intents and purposes, a union where an entire geometric structure is generated into another that is totally unrelated to it. Now I realize that a mathematician will argue that an irrational like pi or the golden section is only an "approximation of the true magnitude", and because of this, any claims made to a fitting together of these measures to a rational measure isn't really truthful, mathematically. For me, it is the issue of what the geometry is suggesting to me when I am constructing the drawing, and what it may be indicating symbolically and philosophically.[5] If I am constructing a Tibetan Mandala, finding a center that I determine is that center, and I draw a circle with a predetermined radius, I do not concern myself with the irrationality of pi when I begin circumscribing it with a square. My task is to do the best job I can possibly do to make sure it is the best circle and square I can do today. My focus is on technique and process, symbolic and philosophical meaning and aesthetics. Another geometer may be more concerned with the shape's power and intensity. [6] Still another may be using the shapes as a contemplative device. In my work, I concern myself with exactitude, and I frequently use a calculator. I always use very sharp graphite and expensive tools, i.e., I don't buy compasses in drug stores. If I'm onto something important, like a new discovery, I increase the scale 5 to 10 times in order for the line thickness to be less influential in the final calculations. I do try to have scholarly approaches and professional concerns; and yet I also know that we live in an earthbound plane, complete with numerous limitations, including mathematically imperfect compasses. I still focus on what the geometric configurations may suggest to me. In the following group of drawings, my hope is that the beauty of the constructions may not be lost or under appreciated even under the scrutiny of the most brilliant of our mathematical readers.

For me, discovering new relationships between incommensurables, finding connections that are hidden from view, is deeply rewarding. It fascinates me that for all of the hard edged, technical aspects of geometry, its almost restricting and cold objectivity, all of these things give way to the organic nature that lies hidden under the appearances and preconceived notions about it. Some of us aren't aware of this living quality that geometry has. Nonetheless, geometry has this almost canny ability to bend and be flexible to our will. An open mind will also find that geometry can work with us, as well as the other way around. A symbiotic relationship develops. It responds to the creative energy that the geometer brings to it, and it seems to me that geometric structure resonates on a very deep level with human consciousness. When I work to find these unions between and among geometric systems, I sense that the relationship between the mind and geometry is almost instinctual. Geometry teaches us to look for visual relationships, to explore possibilities of what may potentially exist in its shapes, forms, and ratios. Geometry is the visual relationships between numbers. It is also a bridge between what is and what may be. Our civilization left two dimensional, Euclidean geometry long ago on its quest to understand the universe. Now there is geometry for spacetime, for quantum physics, and 0 - dimension space. It is interesting to note that it is still geometry, but now, it is all dressed up. However, I trust the column will demonstrate that 2 - dimensional geometry is still alive, and that it hasn't been fully explored and exhausted. There is still fertile ground. I hope that the constructions I show and discuss will motivate some of our readers to go exploring for themselves.

NOTES
[1] The other two classical problems are the trisection of the angle, and the doubling of the cube. See Sir Thomas Heath's two volume set, A History of Greek Mathematics, New York: Dover Pub., 1981, for an in depth study of these and other classical mathematical problems. return to text

[2] So called because the numbers in the decimal sequence are infinitely long and cannot be predicted. Even though 1/3 is infinitely long, 0.3333333333.…, we know that the next number is another 3. return to text

[3] Vitruvius, Pollio. Ten Books on Architecture. Trans. Morris Hickey Morgan, 1914 (rpt. New York: Dover Publications, 1960). To order this book from Amazon.com, click here. return to text

[4] Andrea Palladio had seven ratios that he suggested be used in building. All but one were whole number ratios. The seventh was a square and its diagonal. I refer the reader to a new edition and an outstanding translation of Palladio's great treatise, The Four Books on Architecture. Robert Tavernor and Richard Schofield, trans. (Cambridge, Massachusetts: MIT Press, 1997). To order this book from Amazon.com, click here. return to text

[5] I never stray far from humanist, Pythagorean/neo-Platonic, and Egyptian/Greek thought regarding the study of Geometry. return to text

[6] One need only look at the effects, beliefs, and consequences of a cruciform, a Star of David, a swastika, a crescent moon, or a pentagram to know the power of geometric shapes. return to text

Copyright ©2005 Kim Williams Books