Symmetrical combinations of three or four hollow triangles (Q1345646)

A ``hollow triangle'' is the planar region bounded by two homothetic and concentric equilateral triangles (homothetic ratio \(r > 1)\). The author describes, analyzes and compares two striking constructions (1), (2) in Euclidean 3-space made from such flat ``triangular rings''. (1) The Australian sculptor John Robinson assembled three such congruent hollow triangles \((r = (2 \sqrt 6 + 1)/3 = 1.9663..)\) to form a structure in which certain points on two outer edges of each ring fit into two inner corners of the next, in cyclic order; the symmetry group of the assembly is \(C_ 3\); not two of Robinson's three hollow triangles are interlocked, but paradoxically the whole assembly is inseparable in the manner of ``Borromean rings''. (2) The American artist George Odom assembled four congruent hollow triangles \((r = 2)\) to form a rigid structure in which the midpoints of the 3 outer edges of each ring fit into inner corners of the three remaining rings; the four rings are mutually interlocked, and the symmetry group is the octahedral group 0 or \(S_ 4\). The author's investigations use results about the polyhedral groups of the regular tetrahedron, octahedron and icosahedron (Felix Klein), Max Brückner's five tetrahedra, and symmetrical lattices.