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arXiv · 2505.16629

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

Wen Chen·Jinjin Liang·Erxiao Wang
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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.

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Wen Chen, Jinjin Liang, Erxiao Wang. 2025-05-22. Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle. https://arxiv.org/abs/2505.16629

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