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Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle

Wen Chen, Jinjin Liang, Erxiao Wang

TL;DR

The paper resolves the classification of non-side-to-side tilings of the sphere by congruent triangles when at least one angle is irrational. Employing tools such as the Irrational Angle Lemma, adjacent angle deduction, extended edges, and AVC matching, the authors perform a case-based analysis by vertex-degree, revealing three families of tilings: (i) one-parameter families with two-layer earth-map tilings and rotational variants for even counts, (ii) a one-parameter family yielding a unique tiling with 8 tiles, and (iii) a sporadic triangle with a unique tiling of 16 tiles. The results hinge on precise vertex-configuration constraints and careful geometric deductions, culminating in a complete irrational-angle classification and a framework for the rational-angle case to be addressed in a follow-up work. The work advances spherical tiling theory by clarifying non-edge-to-edge structures and providing explicit tiling data and angle-length relations for the identified families.

Abstract

We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families of triangles each admitting many 2-layer earth map tilings with $2n$($n\geq3$) tiles, together with rotational modifications for even $n$; a 1-parameter family of triangles each admitting a unique tiling with $8$ tiles; and a sporadic triangle admitting a unique tiling with $16$ tiles. Then a scheme is outlined to classify the case with all angles being rational in degree, justified by some known and new examples.

Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle

TL;DR

The paper resolves the classification of non-side-to-side tilings of the sphere by congruent triangles when at least one angle is irrational. Employing tools such as the Irrational Angle Lemma, adjacent angle deduction, extended edges, and AVC matching, the authors perform a case-based analysis by vertex-degree, revealing three families of tilings: (i) one-parameter families with two-layer earth-map tilings and rotational variants for even counts, (ii) a one-parameter family yielding a unique tiling with 8 tiles, and (iii) a sporadic triangle with a unique tiling of 16 tiles. The results hinge on precise vertex-configuration constraints and careful geometric deductions, culminating in a complete irrational-angle classification and a framework for the rational-angle case to be addressed in a follow-up work. The work advances spherical tiling theory by clarifying non-edge-to-edge structures and providing explicit tiling data and angle-length relations for the identified families.

Abstract

We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families of triangles each admitting many 2-layer earth map tilings with () tiles, together with rotational modifications for even ; a 1-parameter family of triangles each admitting a unique tiling with tiles; and a sporadic triangle admitting a unique tiling with tiles. Then a scheme is outlined to classify the case with all angles being rational in degree, justified by some known and new examples.

Paper Structure

This paper contains 7 sections, 10 theorems, 14 equations, 25 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Basic Facts
  3. Proof of the Main Theorem
  4. Tilings with a full vertex of degree 3
  5. Tilings with a degree 4 full vertex or a degree 2 half vertex
  6. Tilings with a full vertex of degree 5
  7. Conclusion

Key Result

Theorem 1

All non-side-to-side tilings of the sphere by congruent triangles with any irrational angle are: 1. A sequence of one-parameter families of triangles admitting two-layer earth map tilings with $2k\ge 6$ triangles, and their rotation modifications for even $k$, as shown in Figure 1.2. 2. A one-parame

Figures (25)

  • Figure 1: A geodesic triangle on the sphere.
  • Figure 2: One-layer and two-layer earth map tilings and their modifications.
  • Figure 3: Triangular subdivisions of Platonic solids and their rotation modifications.
  • Figure 4: Non-side-to-side tilings by congruent isosceles triangles.
  • Figure 5: Non-side-to-side tilings by congruent right triangles.
  • ...and 20 more figures

Theorems & Definitions (16)

  • Theorem
  • Lemma 1
  • Lemma 2: Balance Lemma
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5: Ref. ac1
  • Lemma 6
  • proof
  • ...and 6 more