Regular tetrahedron surface tours
by Jan Kristian Haugland

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Other polyhedra:   Cube   Octahedron   Dodecahedron   Icosahedron

Such a tour on the surface of a regular tetrahedron will never cross itself. Here is an example (a = 3, b = 2).

Also, 3D printed models of a couple of other examples are for sale at Shapeways.