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On the Evolution During Growth of Regular Boundaries of Bodies into Fractals

Vladimir Goldshtein, Reuven Segev

Abstract

Generalizing smooth volumetric growth to the singular case, using de Rham currents and flat chains, we demonstrate how regular boundaries of bodies may evolve to fractals.

On the Evolution During Growth of Regular Boundaries of Bodies into Fractals

Abstract

Generalizing smooth volumetric growth to the singular case, using de Rham currents and flat chains, we demonstrate how regular boundaries of bodies may evolve to fractals.
Paper Structure (9 sections, 100 equations, 7 figures)

This paper contains 9 sections, 100 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Smooth Growth in a Proto-Galilean Spacetime
  3. Proto-Galilean spacetime
  4. Smooth extensive properties and flux fields
  5. The balance equations in spacetime
  6. Body points
  7. Modeling Singular Growth by Currents: Asymptotic Version
  8. Fractals as Currents and Flat Chains
  9. Fractal Growth

Figures (7)

  • Figure 1.1: Fractal-like bacteria colonies: (a) Paenibacillus vortex sp. bacteria. (By Eshel Ben-Jacob, https://commons.wikimedia.org/wiki/File: Paenibacillus_vortex_colony.jpg.) (b) Surface morphologies of 14 days old WT Bacillus subtilis biofilm. Taken from Rafi2022 with permission of the authors.
  • Figure 4.1: The triangle and its boundary as polyhedral chains for Example \ref{['exa:1-1']}.
  • Figure 4.2: An illustration of the construction of the snowflake as a flat chain for Example \ref{['exa:2-1']}.
  • Figure 4.3: (a) A typical equilateral triangle in the construction of the snowflake. (b) A different snowflake. The type of snowflake depends on the vertex angle. Here (a) is a standard snowflake.
  • Figure 5.1: The triangles for the construction of the continuous construction of the snowflake.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Remark 4.1
  • Example 4.2
  • Example 4.3
  • Remark 4.4