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Eden model for Pentagons

Claudia Silva, Erika Roldan, Rosemberg Toala-Enriquez

Abstract

We study topological and geometric properties of a cell growth process in the Euclidean plane, where the cells are regular pentagons. To explore the aesthetic aspects of this model, we employ a laser cutter on various materials to create physical representations for some simulations of the model.

Eden model for Pentagons

Abstract

We study topological and geometric properties of a cell growth process in the Euclidean plane, where the cells are regular pentagons. To explore the aesthetic aspects of this model, we employ a laser cutter on various materials to create physical representations for some simulations of the model.
Paper Structure (3 sections, 1 theorem, 2 equations, 7 figures)

This paper contains 3 sections, 1 theorem, 2 equations, 7 figures.

Table of Contents

  1. The Model
  2. Geometrical and Topological Properties
  3. Laser Cut Model

Key Result

Lemma 2.1

Let $P_0$ be the original pentagon and $P_1$ a reflection of $P_0$ over one side. Then, all pentagons in the model are translations of $P_0$ or $P_1$. Moreover, the sides of all pentagons intersect at angles multiples of $36^{\circ}$.

Figures (7)

  • Figure 1: The black pentagon is centered at the origin. The combination of rainbow hues indicates the stage at which that pentagon was placed. The underlying tree represents how the pentagons were attached.
  • Figure 2: The two orientations of the pentagons and the directional vectors of possible neighboring centers.
  • Figure 3: A new pentagon (red) can touch the rest of the structure in a combination of 4 different ways.
  • Figure 4: Growth of parameters concerning the number of pentagons $n$.
  • Figure 5: Images of holes (at the risk of being psychoanalyzed, the authors couldn't resist the temptation of finding familiar objects in the shape of the holes). Just pentagons with edges that bound holes are shown.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 2.1
  • proof
  • Conjecture 2.2
  • Conjecture 2.3