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. 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. <
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.
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. |