o_194c354f0fne14b05mcimi6ppa.pdf

Haken concluded that “The concept of structural stability seems to play a fundamental role
in biology in a still deeper sense than in the formation of different species by way of
deformation. Namely, it seems that, say within the species, organisms exhibit a pronounced
invariance of their functions against spatial or temporal deformation. This invariance
property seems to hold for the most complicated organs, the human brain. For example, this
property enables us to recognize the letter “a” even if it is strongly deformed. From this
ability an art out of writing letters developed in China /and in old Europe/.”
Hermann Haken’s example is illustrated here with the original Thompson’s illustration of
the transformation of the fish Argyropelecus olfersi into the fish Sternoptyx diaphana by
applying a 70° shear mapping. The reverse transformation is possible simply with
manipulating the grid and shear mapping.
The example illustrated this transformation actually is a good example of homeomorphism.
Two objects are homeomorphic if they can be transformed /or deformed/ into each other by
a continuous inverible mapping, continuous one-to-one and having continuous inverse. The
two fish are two objects with the same topological properties. They are said to be
homeomorphic. There are properties that are not destroyed by stretching and desorting an
object.
After the distance of these twenty years, since my first exploration on Hermann Haken/
D’Arcy Thompson thesis what I could see is the growing relevance of D’Arcy Wentworth
Thompson, especially in support of my research on qualitative quantity. /14/
The non discursive transition of quality or the phase transformation and nonequilibrum
phase transition linked the quality of the quantity with evolution. There is a great example of
the quality of the quantity find in the evolution of biological spieces illustrating the non
discoursive qualitative change. This example is the fish Latimeria chalumnae or Coelacanth.
Coelacanth is like the fusion of life and time. It is a “living fosil”. Scientists had known
coelacanths only in fossils up to 400 million years old. Then in 1938 they discovered the
ancient fish were still living off the coast of Africa. A new population of coelacanths was
located in 1998 off the coast of Indonesia. The first living Coelacanths are discovered in 1938
by Marjorie Courtenay Latimer. The Coelacanth is named Latimeria after Marjorie Latimer.
Borislav G. Dimitrov 137