Tilings of the sphere by congruent regular triangles and congruent rhombi
Abstract
All edge-to-edge tilings of the sphere by congruent regular triangles and congruent rhombi are classified as: (1) a 1-parameter family of protosets each admitting a unique (2a3,3a4)-tiling like a triangular prism; (2) a 1-parameter family of protosets each admitting 2 different (8a3,6a4)-tilings like a cuboctahedron and a triangular orthobicupola respectively; (3) a sequence of protosets each admitting a unique (2a3,(6n-3)a4)-tiling like a generalized anti-triangular prism for each n3; (4) 26 sporadic protosets, among which nineteen admit a unique tiling, one admits 3 different tilings, one admits 5 different tilings, three admit 2 different tilings, two admit too many tilings to count. The moduli of parameterized tilings and all geometric data are provided.
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