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Cyclotomic points on varieties and all rational $a^3b$-monotiles

Jinjin Liang, Yixi Liao, Erxiao Wang

TL;DR

The paper addresses the problem of classifying edge-to-edge monohedral spherical tilings by congruent $a^3b$-quadrilaterals. It employs a cyclotomic-point approach, translating angle relations into root-of-unity (cyclotomic) equations and solving them with Bradford–Davenport and Beukers–Smyth-style resultant methods, constrained by vertex-type conditions. The main result is a complete list of rational $a^3b$-monotiles, revealing three infinite families plus sporadic cases, and confirming two new 36-tile monotiles while clarifying gaps in previous classifications. This provides a rigorous, independent verification and a comprehensive catalog for the spherical tiling problem in this class, with implications for related tiling classifications and number-theoretic methods in geometric problems.

Abstract

By computing all cyclotomic points on some algebraic varieties, we get an independent and efficient way to find all rational $a^3b$-monotiles for the sphere, thereby completing the classification of edge-to-edge monohedral quadrilateral tilings. Both of the previous classifications \cite{lw2} and \cite{cl} depended on many old works of different authors while quite a few typos and gaps were found.

Cyclotomic points on varieties and all rational $a^3b$-monotiles

TL;DR

The paper addresses the problem of classifying edge-to-edge monohedral spherical tilings by congruent -quadrilaterals. It employs a cyclotomic-point approach, translating angle relations into root-of-unity (cyclotomic) equations and solving them with Bradford–Davenport and Beukers–Smyth-style resultant methods, constrained by vertex-type conditions. The main result is a complete list of rational -monotiles, revealing three infinite families plus sporadic cases, and confirming two new 36-tile monotiles while clarifying gaps in previous classifications. This provides a rigorous, independent verification and a comprehensive catalog for the spherical tiling problem in this class, with implications for related tiling classifications and number-theoretic methods in geometric problems.

Abstract

By computing all cyclotomic points on some algebraic varieties, we get an independent and efficient way to find all rational -monotiles for the sphere, thereby completing the classification of edge-to-edge monohedral quadrilateral tilings. Both of the previous classifications \cite{lw2} and \cite{cl} depended on many old works of different authors while quite a few typos and gaps were found.

Paper Structure

This paper contains 7 sections, 13 theorems, 12 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Basic Facts
  3. An explicit example applying the number theory techniques
  4. All rational $a^3b$-monotiles
  5. Tilings with $\alpha\beta\delta$ or $\alpha\gamma\delta$ as a vertex
  6. Tilings with two degree $3$ vertex types (excluding $\alpha\beta\delta$ and $\alpha\gamma\delta$).
  7. Tilings with a unique degree $3$ vertex type (excluding $\alpha\beta\delta$ and $\alpha\gamma\delta$)

Key Result

Theorem 1

There are $15$ sporadic and $3$ infinite sequences of rational $a^3b$-monotiles for the sphere, as listed in Table Tab-1.1 and Tab-1.2 together with all of their different tilings.

Figures (2)

  • Figure 1: Some "almost equilateral quadrilateral" or simply $a^3b$-tilings with $10,10,16,16,36,36$ tiles.
  • Figure 2: Notations for an almost equilateral quadrilateral or simply $a^3b$-quadrilateral.

Theorems & Definitions (13)

  • Theorem
  • Lemma 1: lw1
  • Lemma 2: wy2
  • Lemma 3: lw2
  • Lemma 4: lw2
  • Lemma 5: lw2
  • Lemma 6: lw3
  • Lemma 7: Parity Lemma, wy2
  • Lemma 8: Balance Lemma, wy2
  • Lemma 9: lw2
  • ...and 3 more