Icosahedra constructed from congruent triangles. (English) Zbl 1161.52307

Summary: It is possible to construct a figure in three dimensions which is combinatorially equivalent to a regular icosahedron, and whose faces are all congruent but not equilateral. Such ‘icosamonohedra’ can be convex or nonconvex, and can be deformed continuously. A scalene triangle can construct precisely zero, one, or two convex icosamonohedra, and each occurs. Demonstrated here are two explicit convex examples, the first of which is the unique such object constructed from scalene right triangles, proving a conjecture of Banchoff and Strauss.


52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B15 Symmetry properties of polytopes
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