path curve

The path curve C (or pathcurve) is formed by repeated application of a projective map P to some starting point a.
The curve was discovered in the 19th century by the mathematicians Felix Klein and Sophus Lie.
Felix Klein called it a W-Kurve, which was (incorrectly) understood as Weg-Kurve, which was translated to pathcurve.

Lou de Boer ²⁾ found 7 continuous types of path curves in the projective plane.
For some eigenvalues of the projective map the curve degenerates to a more common curve:

• straight line
• conic
• exponential
• logarithmic spiral

Relations have been made to several shapes of living things: buds, pine cones or eggs, human embryo, aorta, and the pineal gland.
Rudolph Steiner saw these relations (1920), George Adams studied the curve in the 1930's, and Lawrence Edwards after World War II ¹⁾.

In 3-dimensional projective space, space pathcurves can be found.

notes

1) He wrote: the vortex of life (natures patterns in space and time), published by Floris Books.

2) Boer, Lou de, Classification of real projective path curves, version 2, 2002.