A short proof of the best known upper bound $3.6108$ on the growth of polyiamonds

TL;DR

This paper addresses the asymptotic growth of polyiamonds by establishing an elementary recurrence-based upper bound. It develops a recurrence framework on a dual hexagonal lattice, converts it to generating functions, and derives a cubic equation $2z^3+z^2-1=0$ whose positive root $z$ determines the bound $\lambda_T \le 1+2z+3z^2$, with $x=1/(1+2z+3z^2)$ as the radius of convergence. The resulting bound yields $\lambda_T < 3.6108$ while remaining simple and verifiable by hand, avoiding heavy computer-assisted arguments. The approach provides a near-best-known upper bound through transparent recurrences, enhancing accessibility and tractability for similar lattice-growth problems.

Abstract

We provide a short and elementary proof that the growth rate of polyiamonds is at most $1+2z+3z^2$ for the only real root $z$ of the equation $2z^3+z^2-1=0$, which is nearly identical to, but slightly below, the best known upper bound $3.6108$. Unlike the previous proof of this bound, which relied on computer-assisted technical arguments and the counts of polyiamonds with up to 75 triangles, our method is based on a straightforward recurrence that can be verified by hand with minimal effort.

Figures (3)

• Figure 1: Triangular lattice and polyiamonds with up to 3 triangles.
• Figure 2: Hexagonal lattice and the representation of $6$ polyiamonds of size $3$.
• Figure 3: Marked vertices and the surroundings.

Theorems & Definitions (6)

• Theorem 1
• Proposition 1
• proof
• Proposition 2
• proof
• proof