
Path Curves George Adams suggested in 1950 that one variety of the path curves discovered by Felix Klein could apply to natural forms, particularly the spirals found on the eggshaped surfaces of plant buds and cones, and also birds eggs. Lawrence Edwards, while teaching this to the children in his mathematics class in Edinburgh, began to wonder whether it was just a nice story or whether it was actually true. This is not an easy matter as it is all too easy to dismiss practical research to test it as mere "curve fitting". However that is far from the truth as it lives within a thought context or paradigm from which it derives its meaning, namely the application of Rudolf Steiner's discovery of Counterspace to a deeper understanding of the subtle aspects of Nature. Edwards set about making practical measurements in 1964 to see whether the theoretical forms were really manifest in natural forms. He started with hens eggs, and went on to plant buds and then to the shape of the heart, and then the uterus in pregnancy. It became clear that the geometry described such forms well. His method was to photograph the objects, measure the curves in a systematic way, and then apply a simple statistical test to check the goodness of fit. This test was based on the paradigm in use rather than the mean radius deviation (although both approaches are now used), namely on the consistency of the parameter lambda describing the profile of the bud. The following diagram shows the meaning of this parameter: The numbers are the lambda parameter, tending to infinity for a cone. The negative values yield vortex forms instead of buds. The bottom row shows eggs with the same lambda but with differing steepnessofascent of the spirals. This is controlled by a second parameter designated 'epsilon', which is 0 for horizontal circles and infinite for vertical profile path curves (see Reference 1 or Practical Path Curve Calculations for the maths). An example profile fit for Kerria Japonica is shown below: This led to certain kinds of question:
The conduct of the research depends upon the question being asked. If one is trying to establish that buds are path curves then certain disciplines are necessary to avoid only picking the best examples and the most propitious points. However, even if the hypothesis is true it cannot be expected that the bud will fit the form where it is obvious that near the mathematical singularities it cannot (e.g. the top of the Kerria bud shown above). Common sense cannot be avoided. The following example also illustrates this sort of consideration: It is a bladder campion and it is clear that at the top the horizontal cross extruding from the form (as seen from the top) cannot belong to the path curve. A fitted mathematical curve is shown in red. Had it been attempted blindly to include the outline of the cross then the fitted curve would be way off. On the other hand a fair sampling method e.g. by always using a standard number of equally space levels for measurement, is one way of achieving scientific objectivity. On the other hand if the hypothesis has been tested and accepted, then it may be part of another project to find the best possible fit in order to find out the actual lambda of the bud. Then "outlyers" caused by wind and weather, or obvious blemishes, should obviously be avoided. The question being asked determines what is scientifically acceptable. It was while Lawrence Edwards was testing a hypothesis about the vitality of buds in 1983 that perhaps his most significant discovery was made. He wanted to test buds over a period of time, and he wanted them to remain the same as far as possible during the test. This is difficult with flower buds which develop progressively and then open. On the other hand the leaf buds of trees form in the autumn and remain dormant through the Winter until they open in Spring . They seemed the best candidates for his project. However he was surprised to find that they are not as dormant as he had supposed! The lambda value, which increases systematically above 1 as the bud becomes ever more "pointed", was found to vary with an approximately twoweekly cycle during the whole of the Winter (other things being equal). His project switched to an investigation of this unexpected phenomenon, and as suggested by a cycle of two weeks he looked into the conjunctions and oppositions of the Moon with various planets. This proved plausible, the planet depending upon the tree involved, as the cyclic variation was in phase with the those lunar aspects. One tree failed to show the effect, and upon checking he found it was very close to a large power distribution transformer. So it seemed that the magnetic fields might shield the tree from the cosmic rhythm otherwise in evidence. To test this idea he worked with knapweed (rather than trees which take a long time to grow!), comparing those growing under electric cables with others, and the effect (a Moon and Jupiter aspect in this case) was confirmed: those under the cables had their rhythms greatly damped. As an aside, the impact of the comet Shoemaker Levy with Jupiter in 1994 was reflected dramatically in the lambdas that year (see also the Volatile page in Projective Geometry). An example of a graph showing the lambda variation for an oak tree in Sydney is shown below: and the following illustration shows the change in bud shape typically implied by the change in lambda: A red egg of higher lambda is compared with a green one of lower lambda. This is the reality lying behind the lambda variation. As the years went by Lawrence Edwards found that a slippage in the phase between the graph and the lunar aspects occurred, the cause of which has puzzled us all to this day. It was interesting that the phaseshift, expressed in days, was the same for all the trees under test i.e. regardless of which species and planet was involved, showing that some objective factor is at work for them all. After 7 years the phase shift becomes zero again, and the graph below shows the results of thousands of measurements taken to establish what is happening: It can be seen that the project was conducted continuously from 1983, when fortuitously the phase shift was zero, to 1999 (but in fact until 2001 when Lawrence Edwards retired). John Blackwood has been working in Sydney, and it seems from his work that the phase shift was different but consistent there. More recently work by Richard Katz has started in USA as a dependence on longitude is suspected (but only suspected), and we need researchers on as many longitudes as possible to test that suspicion. An early question when the phase shift was discovered was: how can we be sure which planet is involved, especially in view of the phase shift? After a sufficient number of years Lawrence Edwards combined his results in what he called Aggregate Graphs, or AGraphs. Because he systematically used the same method for selecting the number and relative positions of the levels at which measurements of buds were made, they could be combined into one overall aggregate showing the mean behaviour of a particular species. These show the characteristic variation if the correct planetary aspects are correlated with them, but not if the wrong planet is chosen. All of this is described in detail by Lawrence Edwards in his book The Vortex of Life together with his Supplement and Sequals to that. This aspect of the research has been dominant because it is surely of great interest to find objective scientific evidence for a relationship between rhythms in plant life and in the cosmos. Pivot transforms Can the shape of the gynoecium of plants also be understood in this sort of manner? They are not path curves. Lawrence Edward's discovery of the pivot transform is described in The Vortex of Life and coloured illustrations are available on the relevant page of Projective Geometry. He found that this transformation applied to a vortex does indeed yield a close fit to a seed pod, and again it must be stressed that this "fit" is based on the underlying counterspace paradigm; it is not just a "let us see what fits" approach which would be scientifically uninteresting. An example for a rose bud gynoecium is shown below:
This, like many real buds, is not symmetrical so it is sensible to try finding the form for one half only, as shown. The left side looks more promising, and now the full import of the earlier remarks on scientific method become practically important, not just philosophically. This is because the practical method for deriving the gynoecium form from the bud requires an accurate lambda value i.e. we must assume for this that the left side of the bud is a path curve and find the best possible fit, just as we assume without needing to prove it that shoes are intended for feet and find the best possible fit!! Based only upon a profile in this case, we do not know the epsilon of the bud, the parameter which controls the steepness of the spiralling, so instead we estimate where the bottom of the gynoecium intersects the axis (see Annex 3 of Pivot Transforms) which implies a value of epsilon; indeed we may find the otherwise unknown epsilon this way. This will fail if the lambda is not determined accurately, so we must choose the best points and perhaps try several times to find the best lambda value. Then we may find the theoretical gynoecium form, with the result shown. The calculated value of epsilon is 2.65 in this case, and the lambda of the vortex which is transformed into the gynoecium is 3.17. While Lawrence Edwards did much work on this his detailed results are no longer available, and a rich field of research is open here! Nick Thomas 