         The picture shows an egg form constructed mathematically. The spirals are characteristic of the mathematics and are known as PATH CURVES. They were discovered by Felix Klein in the 19th Century, and are very simple and fundamental mathematically speaking. Geometry studies transformations of space, and these curves arise as a result. A simple movement in a fixed direction such as driving along a straight road is an example, where the vehicle is being transformed by what is called a translation. In our mathematical imagination we can think of the whole of space being transformed in this way. Another example is rotation about an axis. In both cases there are lines or curves which are themselves unmoved (as a whole) by the transformation : in the second case circles concentric with the axis (round which the points of space are moving), and in the first case all straight lines parallel to the direction of motion. These are simple examples of path curves. More complicated transformations give rise to more interesting curves. The transformations concerned are projective ones characteristic of projective geometry, which are linear because neither straight lines nor planes become curved when moved by them, and incidences are preserved (this is a simplification, but will serve us here). They allow more freedom than simple rotations and translations, in particular incorporating expansion and contraction. Apart from the path curves they leave a tetrahedron invariant in the most general case. George Adams studied these curves as he thought they would provide a way of understanding how space and counter space interact. A particular version he looked at was for a transformation which leaves invariant two parallel planes, the line at infinity where they meet, and an axis orthogonal to them. This is a plastic transformation rather than a rigid one (like rotation) and a typical path curve together with the invariant planes and axis is shown below. This will be recognised as the type of curve lying in the surface of the egg at the top of the page. If we take a circle concentric with the axis and all the path curves which pass through it then we get that egg-shaped surface. The construction is shown in the following animation: We can vary the transformation to get our eggs more or less sharp, or alternatively we can get vortices such as the following: In these pictures particular path curves have been highlighted. This particular vortex is an example of a watery vortex, so called by Lawrence Edwards because its profile fits real water vortices. It is characterised by the fact that the lower invariant plane is at infinity. If instead the upper plane is at infinity we get what he calls an airy vortex. Two parameters are of particular significance: lambda and epsilon. Lambda controls the shape of the profile while epsilon determines the degree of spiralling. Lambda is positive for eggs and negative for vortices, while the sign of epsilon controls the sense of rotation. This is illustrated below. < The top row shows lambda increasing from 1 (elliptical) to 10. When lambda reaches infinity the form becomes conical. The centre row shows lambda increasing from -0.616 to -0.1 for a vortex. The bottom row shows epsilon varying from 0.2 to 10, and when it reaches infinity the curves are vertical. If it is zero then the path curves become horizontal circles, and strictly speaking the profile is lost. The profile is thus controlled by a single parameter (lambda), and it is scientifically interesting that with such a restriction these curves fit very closely a wide variety of natural forms including eggs, flower and leaf buds, pine cones, the left ventricle of the human heart, the pineal gland, and the uterus during pregnancy. The watery vortex closely fits actual stable water vortices. Together with the airy vortex it also has significance for pivot transforms. The following shows approximately the way the left ventricle of the heart behaves as a path curve from diastole to systole: Lawrence Edwards spent many years finding out and testing the above facts experimentally, which he has described in Reference 7. In 1982 he started testing the shapes of the leaf buds of trees through the winter, and found that their lambda value (unexpectedly) varied rhythmically with a period of approximately two weeks. This was his main topic of research in his later years, and the evidence is now very strong - backed by thousands of measurements - that the rhythm corresponds to the conjunctions and oppositions of the Moon and a particular planet for each tree. This is a purely experimental fact and care should be taken in interpreting it. Download document Practical Path Curve Calculations for the basic algebra and formulae to work with path curves (pdf document). 