Below are the two icosamonohedra described in E. N. Miller's paper, "Icosahedra Constructed from Congruent Triangles," Discrete and Computational Geometry 24: 437-451 (2000). An icosamonohedron is a polyhedron topologically equivalent to an icosahedron (shown left), all of whose faces are congruent triangles. Miller's paper constructs two examples. The first was discovered by Banchoff and Strauss in 1979. Its protoface is a right triangle of side lengths {Sqrt[7], Sqrt[3], 2}. The polyhedron is show in the top left figure below. Several triangles are coplanar, in groups of four, and of two. The figure to its right shows another view of the polyhedron, with two opposite pentagons removed, leaving an "equitorial band" of ten triangles.
The second example constructed by Miller is new. Its protoface
is a nearly-right triangle of side lengths {3Sqrt[7], 2Sqrt[11],Sqrt[23]}.
The polyhedron is shown in the bottom left figure below, and again an equitorial
band is displayed to its right. Several pairs of adjacent triangles
are coplanar.
Banchoff & Strauss polyhedron: | Equitorial band of B & S's polyhedron: |
Miller polyhedron: | Equitorial band of Miller's polyhedron: |
It remains open to make a complete list of all icosamonohedra, as well as to classify the combinatorial types of all monohedra.