Eden Model for Pentagons

Year: 2024 Authors: Érika Roldán; Claudia Silva; Rosemberg Toala-Enriquez

Core claim

Regular pentagon attachment in an Eden-style process yields two orientations, dense possible centers, and linear growth in perimeter and holes, unlike lattice-based models.

Topics

cell growth process, holes in tilings, participatory installation, generative art, first-passage percolation

Domains

topology, geometry, graph growth, first-passage percolation, generative installation, participatory art, physical fabrication, exhibition design

Methods

random edge attachment, simulation, geometric lemma, combinatorial counting

Media

regular pentagons, wood laser-cutting, engraved acrylic, installation surfaces

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2024 Conference Proceedings

Eden Model for Pentagons

Érika Roldán ${}^{1,2}$ , Claudia Silva ${}^1$ , and Rosemberg Toala-Enriquez ${}^3$

${}^1$ ScaDS.AI, Leipzig University; roldan@mis.mpg.de, callame@ciencias.unam.mx ${}^2$ Max Planck Institute for Mathematics in the Sciences; roldan@mis.mpg.de ${}^3$ George Mason University; rtoalaen@gmu.edu

Abstract

We introduce and study the topological and geometric properties of a cell growth process in the Euclidean plane, where the cells are regular pentagons. An artistic activation of this model enables a participatory and generative installation of random asymmetries and an infinite number of holes that remain uncovered.

The Model

The model involves randomly attaching regular pentagons along their edges, ensuring no overlap between any two pentagons. More precisely, let $P_0$ be a regular pentagon centered at the origin. Assume that $n$ -th pentagon $P_n$ has been positioned. Define $F(n)$ as the set of ‘Free sides’, comprising the pentagon sides $s$ where attaching a new pentagon to $s$ avoids overlap with any of $P_0,\dots ,P_n$ . Then, a side is randomly chosen from $F(n)$ and a new pentagon $P_{n+1}$ is attached along it, forming the $(n+1)$ -th instance of the model. Subsequently, the set of Free sides is updated: $F(n+1)$ includes $F(n)$ along with the new sides of the new pentagon, minus the sides that might now cause overlap with $P_0,\dots ,P_n,P_{n+1}$ . This random process is inspired by the Eden Model which is a First Passage Percolation Process (FPP) based on the regular square tessellation of the plane [2]. Recently [1, 3, 4], people have studied the topology of FPP processes based on lattices. Simulations of the Eden Model are available, see [3]. In the pentagon model, we do not have an underlying lattice and there will be holes that become impossible to cover at any future time. One of the consequences of this fact is that in the pentagon model, there is also no notion of a convex limiting shape. Moreover, we can observe from our simulations that the perimeter and the number of holes grow linearly with respect to the number of tiles, which differs from the behavior of the classical FPP models based on lattices.

(a) Cell growth model with 200 pentagons

(b) Model with 5000 pentagons Figure 1: The black pentagon is centered at the origin. The colors of the tiles reflect the stage at which that pentagon was placed. The underlying tree represents how the pentagons were attached.

Roldán, Silva, and Enriquez

Geometrical and Topological Properties

A simple non-intuitive fact is that pentagons come only in two orientations, the first one and its “reflection.”

Lemma 0.1. 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 that are multiples of $36^{\circ }$ .

Proof. Let $L$ be the set of five lines through the origin which are parallel to the sides of $P_0$ . The lines in $L$ meet at angles that are multiples of $36^{\circ }$ . When a new pentagon is glued, it is obtained by reflecting over the gluing side. The angles do not change under reflection, and one side remains fixed. Therefore, the sides of any new pentagon are still parallel to the lines in $L$ . We conclude that all the pentagons in the model have sides parallel to $L$ and therefore can only be in two possible orientations. Moreover, any two sides of any two pentagons must intersect at the same angles as two lines in $L$ , that is, at angles that are multiples of $36^{\circ }$ .

We proceed to analyze the locus of centers of possible pentagons within the model. The center of a new possible pentagon can be obtained by translating the center of the old pentagon along 5 vectors, see Figure 2. We denote these vectors by ${\vec{u}}_1,{\vec{u}}_2,{\vec{u}}_3,{\vec{u}}_4,{\vec{u}}_5$ . These can be assumed to be unit vectors, but all of our analysis is independent of scale. Let us regard $\{{\vec{u}}_1,{\vec{u}}_2\}$ as a basis for ${\mathbb{R}}^2$ . Then ${\vec{u}}_3,=-{\vec{u}}_1+\varphi {\vec{u}}_2,{\vec{u}}_4=-\varphi {\vec{u}}_1-\varphi {\vec{u}}_2$ , and ${\vec{u}}_5=\varphi {\vec{u}}_1-{\vec{u}}_2$ where $\varphi =2\cos (2\pi /5)=\frac{\sqrt{5}-1}{2}$ . Since $\varphi$ is irrational, the usual argument shows that the set of integer linear combinations of the vectors ${\vec{u}}_1,{\vec{u}}_2,{\vec{u}}_3,{\vec{u}}_4,{\vec{u}}_5$ form a dense

Figure 2: Orientations and directional vectors of possible neighboring centers.

subset of ${\mathbb{R}}^2$ . Let $r$ be the distance from the center of a pentagon to one of its vertices. We conjecture that the set of possible centers of the pentagons in the model is dense outside a disk of radius $2r$ . To prove this conjecture we need to consider the non-overlapping geometric constraint of the pentagons. This difficulty might be overcome by choosing an appropriate geometric realization of the linear combination by deciding a plausible order in which the pentagons are glued. Additionally, a linear combination remains unchanged when we add a term $+{\vec{u}}_i-{\vec{u}}_i$ . Geometrically, this corresponds to adding a pair of pentagons represented by opposite vectors ${\vec{u}}_i$ and $-{\vec{u}}_i$ , in this way the linear combination remains unchanged but the geometric realization has more freedom. These two choices should give us enough maneuverability to avoid overlapping when a linear combination is transformed back to pentagons.

The Graph

In what follows, we study properties and the growth of the number of vertices, edges, and holes of the graph formed by the pentagons. We will distinguish between the sides of the pentagons and the edges of the graph. A side could be split into 2 edges. For example, in Figures 3 (b) and (c), the 4 new sides of the red pentagon contribute to 5 new edges of the graph. Summarizing, the graph of the model comprises the vertices and sides of the pentagons, with the convention that if a vertex lies on the side of another pentagon, then such side will be regarded as two distinct shorter edges. A hole is a bounded connected component of the complement of the union of all pentagons.

When a new pentagon is attached, besides the gluing edge, it can touch the existing structure in a combination of 4 different configurations, as shown in Figure 3. In each case, adding a new pentagon increases the number of vertices by at most 3, the number of edges by at most 8 (two per side of the pentagon), and the number of holes by at most 8 (one per new edge). We conjecture the following limits, supported by numerical simulations as shown in Figure 4.

Eden Model for Pentagons

(a) Edge-edge

(b) Partial edge

(c) Vertex-Edge Figure 3: A new pentagon (red) can touch the rest of the structure in a combination of 4 different ways.

(d) Vertex-vertex

Conjecture 0.2. Let $n$ be the number of pentagons in the model, $V(n)$ , $E(n)$ , and $H(n)$ the number of vertices, edges, and holes, respectively. Then, the expected value of these parameters satisfies the following limits:

During the simulations, $V(n)$ and $E(n)$ were kept track of, while $H(n)$ was determined using Euler’s formula for planar graphs: $V(n)-E(n)+(n+H(n)+1)=2$ . Where the number of faces, $F(n)=n+H(n)+1$ , accounts for pentagons, holes, and the unbounded face.

Figure 4: Growth of graph parameters with respect to the number of pentagons $n$ .

Description and Analysis of Holes

Recall that all the sides of the pentagons form angles that are multiples of $36^{\circ }$ between themselves. Thus, the holes are polygons whose angles are multiples of $36^{\circ }$ . Examples of holes discovered in the simulations are illustrated in Figure 5. Consider a hole with $l$ sides whose angles are $36^{\circ }{a}_i$ for some positive integers $a_i$ , $1\le i\le l$ . The interior angle sum of the hole polygon is $36^{\circ }{a}_1+36^{\circ }{a}_2+\dots +36^{\circ }{a}_l=180^{\circ }(l-2)$ . This provides a necessary condition for the $a_i$ ‘s, namely, $a_1+a_2+\cdots +a_l=5(l-2)$ .

Consequently, up to scale, only two distinct types of triangular holes are possible. These correspond to solutions $(a_1,a_2,a_3)=(1,2,2)$ and $(1,1,3)$ , resulting in triangles with angles $(36^{\circ },72^{\circ },72^{\circ })$ and $(36^{\circ },36^{\circ },108^{\circ })$ , respectively. Type $(36^{\circ },72^{\circ },72^{\circ })$ is shown in Figure 5(a). We have not observed a hole of type $(36^{\circ },36^{\circ },108^{\circ })$ , however it can be created by pentagons whose centers are in the locus of centers of the model, Figure 5(b).

The angle types of quadrilateral holes (up to rotation and reflection) can be $(a_1,a_2,a_3,a_4)=(1,1,1,7)$ , $(1,1,2,6)$ , $(1,2,1,6)$ , $(1,1,4,4)$ , $(1,4,1,4)$ , $(1,2,3,4)$ , $(1,2,4,3)$ , $(1,3,2,4)$ , $(2,2,2,4)$ , $(2,4,2,4)$ , $(2,2,3,3)$ , or $(2,3,2,3)$ . It remains unclear whether these configurations are feasible. Our simulations have shown one hole of type $(1,1,1,7)$ and two instances of type $(1,4,1,4)$ , Figure 5 (c), (d), and (e), respectively.

Roldán, Silva, and Enriquez

Figure 5: Except for (b), all images of holes were obtained in our simulations. The model is likely to generate infinitely many different holes that remain uncovered forever.

Towards Developing an Exhibition

Figure 6: Model with 200 pentagons.

Figure 6 shows a physical realization of Figure 1(a) using a laser cutter on wood with an engraved acrylic layer lying on top, showing the growing tree. We understand contemporary art through the working progress mind setting that gives more value to the artistic process over the final object that is usually prefixed and that reaches an end state. This inspired us to propose a hands-on and collaborative exhibition where the public will place the pentagons on a citizen simulated cell growth process on different surfaces at an exhibition room at DESFOGA 2024, Cambados, Spain. DESFOGA is a curatorial program in Cambados that questions powers, inequalities, and all forms of human rights violations through performances, installations, and exhibitions.

References

[1] D. M. Hua, F. Manin, T. Queer, and T. Wang. “Local behavior of the Eden model on graphs and tessellations of manifolds.” Journal of Applied and Computational Topology, Springer, 2023. [2] E. Murray. “A two-dimensional growth process.” Dynamics of fractal surfaces, 1961. [3] F. Manin, E. Roldan, and B. Schweinhart. “Topology and local geometry of the Eden model.” Discrete & Computational Geometry, Springer, 2023. Simulations: https://www.erikaroldan.net/topology-eden-model. [4] M. Damron, J. Gold, W. Lam, and X. Shen. “On the number and size of holes in the growing ball of first-passage percolation.” Transactions of the American Mathematical Society, 2024. [5] B. Cuquejo, and M. Daporta. DESFOGA, Plataforma cultural. 2023. https://desfoga.eu/