From Fractals and Cellular Automata to Biology

The following sections are included:

• Preface
• Contents

“There is no science on singulars (Latin, scientia non est de singularibus”), because human science is a knowledge through universals. In fact the human intellect acquires its knowledge abstracting from matter the universal form organizing each singular “matter” body. So our mind, being “immaterial”, does not know singulars, while our senses do, being “material” as part of our “matter” body…

In this first chapter we intend to introduce some elementary notions about the increasingly meaningful role of information in the context of the biological sciences, starting from the early decades of the 21st century. A role which seems to involve both the question on the evolution of species and the matter of the emergence of life. In a wider sense information is playing a significant role for order emergence (self organization) of the structure and the dynamics of physical and biochemical complex systems…

In the first chapter we presented some introductory considerations on information in comparison with the Aristotelian-Thomistic notion of form. A special attention has been devoted to the notion of algorithmic information viewed as a suitable candidate to approach the logical/ontological notions of definition/essence regarding the structure of an entity and respectively its nature on the side of its dynamics according to which it can operate and especially the dynamics generating the entity itself…

In order to attempt to model biological structures like, e.g., organs, and their generating dynamics, we need, at present, to make a further step, i.e., the passage from 2D to 3D systems. As we will see one of the main technical problems one encounters, just at the first level of modeling the geometrical shape of 3D structures, is related to the matter of rendering their boundary surface in a reasonably realistic and satisfactory way…

The following sections are included:

• Python 3 rendering of fractal landscapes
  • Mesh plots
  • Contour plots
  • Scatter plots
• POV-Ray rendering of fractal landscapes
  • The functions “Mandel” and “Julia”
  • The “height field” function rendering
  • Sequential rendering
  • Random rendering

Landscape rendering, we saw in the previous chapter, is not the only way to extend 2D fractals to a 3D space. In fact a proper 3D fractal body should be generated in such a way that its boundary is a fractal surface and not only a curve on a plane with segments adjoined along the third dimension of space. But if one attempts to extend to 3D a complex fractal as a Mandelbrot, or a Julia or Newton’s method set one encounters soon at least three serious problems…

The 3D fractals we dealt with in the previous chapters either as landscapes or as structures with cylindrical symmetry, generated by revolution around a symmetry axis, possess a complexity level depending only on one complex parameter variation, just like the 2D fractals starting from which they are built. In fact Mandelbrot, Julia and Newton’s method set fractals are constructed employing complex numbers (z = a + ib) which are characterized only by two real components (a, b) identifying two space dimensions (Gauss plane)…

The following sections are included:

• Introduction
• Fractals generated randomly by cellular automata
  • 2D Mandelbrot, Julia and Newton’s method sets generated randomly by cellular automata
  • 3D Mandelbrot and Julia sets generated randomly by cellular automata

The following sections are included:

• Introduction
• Python 3 rendering of a heart-like external structure
  • The heart as a whole
  • Sequential rendering of a heart-like model
  • Random rendering of a heart-like model
  • Cellular automaton rendering of a heart-like model
• POV-Ray 3.7 rendering of a heart-like external structure
  • The heart as a whole
  • Sequential rendering of a heart-like model
  • Random rendering of a heart-like model
  • Cellular automaton rendering of a heart-like model

The following sections are included:

• Heart model based on POV-Ray 3.7 “smooth_triangle” object
  • Generating Hughes heart model by an ordered sequence of smooth triangles
  • Generating Hughes heart model by random smooth triangles
  • Generating Hughes heart model by cellular automata in smooth triangles
• Heart model generated by single cells in ordinary triangle objects
  • Generating a heart model by an ordered sequence of cells in ordinary triangles
  • Generating a heart model by random cells in ordinary triangles
  • Generating a heart model by cells born from cellular automata

The following sections are included:

• Conclusion
• References and bibliography
• Index