Thu, 22 Jan 2004

Lawrence Edwards

The mathematician Lawrence Edwards died on 3 January 2004. Nick Thomas gives an appreciation of his life and work.

London, 22 January (NNA) - Lawrence Edwards died on 3 January 2004 after a very full working life. He was born in 1912, was an inspired teacher and mathematician, and was also decorated with the Military Cross for bravery in the 1939-45 war. He was a member of the Anthroposophical Society and both taught and did research on the basis of the insights received from Rudolf Steiner. He was married to Barbara and had five sons.

Lawrence Edwards was widely known in scientific circles within the Anthroposophical Society for his work in applying projective geometry to botany and physiology. He was also known in wider circles, but the timidity of materialism prevented his work from receiving the recognition it deserved. He was gifted at mathematics which he taught at the Edinburgh Rudolf Steiner School for many years, and this gift led him to study projective geometry with George Adams.

Every year he would spend a whole week with Adams, and learned about some remarkable curves discovered by Felix Klein in the nineteenth century which are now called path curves. Adams was researching them and had lifted them out of their academic limbo into the “grubby” practical world, having found that some of them closely resemble eggs and vortices. This is true not only in outline but also for the spiralling forms we see for example on pine cones.

Lawrence told us that for many years he taught this to the children, until one day a question took root in his soul: are the forms in nature actually path curves, or do they merely resemble them? He found he could no longer teach this without further investigation! Such a question is beset with philosophical overtones of course, but we might also phrase it thus: are there processes in nature which inevitably give rise to path curves per se, as a parabola is essentially related to the path of a thrown stone, or is their appearance merely fortuitous?

It was in this spirit that he set to work scientifically in the hope of answering this question. His approach was based firmly on the search for process rather than mere form-fitting, a fact which should be borne in mind by pedants who are inclined to dismiss his work or point to “better” methods of analysis. George Adams had laid a firm mathematical foundation for Rudolf Steiner’s discovery of negative space, and the processes we have in mind here weave between the two spaces.

Apparently the house was transformed into a laboratory, with egg profiles being projected onto walls and measured! The initial findings were so promising that better methods were devised, both for measuring and for mathematical analysis. He developed a method for determining the deviation of an egg form from the ideal, and photographic methods for studying tiny plant buds (which are also egg-shaped). The analysis was based on the path-curve process for reasons already stated, which proved to be more sensitive than conventional methods of error analysis. Do all rose buds, for example, have the same type of path curve? That type is characterised by a number which is named ‘lambda’ after the Greek letter used to symbolise it.

He studied many species but found that lambda varied within each, so although the ranges of variation were characteristic, it was not possible to relate species directly to lambda values. But, variation or no, they fitted the ideal mathematical forms in general very well. He also found that the profiles of the uterus during pregnancy and the left ventricle of the heart are path curves, the latter indeed a varying one! Together with collaborators he showed that water vortices have a path curve profile.

In the early 1980’s he was occupied with the question: does the closeness of a form to the ideal relate to the strength of the life forces at work in it? For this he measured the sap content, and used leaf buds on trees through the winter as they are dormant until spring, and thus should provide a stable basis for comparison. To his surprise they proved to be less dormant than expected, the lambda value varying with an approximately two-weekly rhythm. He was quick to relate this to the two-weekly cycle of conjunctions and oppositions of the Moon and planets, which led to an investigation lasting until the end of his working life.

Early on he described the investigation to us at a conference, and I remember asking how he could be sure it was one planet rather than another e.g. Mars for oak trees. Fortuitously the alignments of the Moon coincided with the peaks and troughs of the graph showing the variation in lambda, but as the years went by there was a ‘slippage’ which he called the ‘phase shift’, whereby the rhythm remained the same but the alignments did not occur at peaks or troughs of the graph. We did not know at the time how long it would take for synchronisation to be re-established; it turned out to be a seven-year cycle. Interestingly it was the same for all trees measured, different ones relating to different planets. Here was hard scientific evidence for the traditional relationship between trees and planets!

Lawrence measured thousands of buds in the course of this investigation, which initially were picked in groups of ten from each tree and photographed, the lambda being taken as the average of the ten. We bought him a video camera and computer, which with suitable software made the task much easier and quicker. The results of all this work were firstly to confirm the existence of the rhythm as an objective fact, secondly to show that the phase shift had a seven year cycle, and thirdly to answer the question quite definitively as to which planet related to which tree. His faithfulness in pursuing the work every day when possible is reminiscent of the work of Tycho Brahe who likewise faithfully measured the courses of the planets over very many years, yielding a factual basis indispensable for the work of Kepler.

One tree, however, did not show the rhythm, and it turned out that it was very close to a large electrical distribution transformer. It would not be easy to confirm this scientifically by planting more trees near transformers! Instead he worked with Knapweed and those growing under electrical cables also had a suppressed rhythm. In 1994 comet Shoemaker-Levy crashed into Jupiter, and as Knapweed is related to Jupiter he took many extra measurements that year and found that its behaviour was quite different from the norm, a most significant discovery. The results seemed to show that the buds anticipated the collision, as their abnormal behaviour started well before it.

Perhaps his most brilliant discovery was a way of describing the shapes of seed-pods, which are not path curves. He invented the ‘pivot-transform’ for this with remarkably successful results. He also used it to transform vortices into embryonic forms.

Lawrence Edwards inspired many people to take up the work he started, and it is to be hoped that it will continue. A group has been formed for that purpose. His lectures and courses were among the best I have ever heard: warm, inspiring and informative. Indeed he gave his first lecture, on astronomy, at the age of 16 and at that early age showed his genius for communicating science to others. He also answered letters, and helped many of us with his apposite replies full of insight.

He wrote up his results in two books: The Field of Form in 1982, and then The Vortex of Life in 1993. He also wrote Projective Geometry, and as a very young man a book on astronomy with a forward by the then Astronomer Royal. In addition he wrote seven Supplements and Sequels to The Vortex of Life giving invaluable practical details and results of his research and that of collaborators.

Finally perhaps it is worth mentioning that for the last decades of his life he lived and worked in Strontian in Scotland, where strontium 90 was discovered, taking its name from the place. His spiritually-based research was perhaps an antidote to the kind of science associated with radioactivity.

END/cva

Nick Thomas is general secretary of the Anthroposophical Society in Great Britain

Item reference number: N040122-01EN

Date: 22 January 2004

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