Solvable Rectangle Triangle Random Tilings

TL;DR

This paper demonstrates that a tenfold symmetric rectangle triangle random tiling can be solved exactly using Bethe Ansatz, extending previous solutions for twelvefold and eightfold cases, and enabling precise calculations of entropy and elastic constants.

Contribution

It introduces the Bethe Ansatz solution for the tenfold symmetric rectangle triangle tiling, marking the third known solvable example of such tilings.

Findings

Bethe Ansatz provides an accurate estimate of entropy.

Bethe Ansatz yields elastic constants for the tiling.

Exact analytic expressions are obtained for twelvefold and eightfold cases.

Abstract

We show that a rectangle triangle random tiling with a tenfold symmetric phase is solvable by Bethe Ansatz. After the twelvefold square triangle and the eightfold rectangle triangle random tiling, this is the third example of a rectangle triangle tiling which is solvable. A Bethe Ansatz solution provides in principle an accurate estimate of the entropy and phason elastic constants. In the twelvefold and eightfold cases even exact analytic expressions have been obtained from the Bethe Ansatz solution.