It is trivial that a cube has both a square and a regular hexagonal projection. We can also easily construct a polyhedron with three perpendicular projection, which are different regular polygons.

But, is there a maximal number of different regular polygon projections of the same 3-dimensional polytype?

As a possible base idea: in space, can we "attach" three regular polygons to a regular pentagon with the following conditions?

  • each attached polygon has different number of sides
  • each polygon's plane is perpendicular to the base pentagon's plane
  • one of each polygon longest diagonal overlaps with a diagonal of the pentagon (each with a different one)
  • any four of the attached polygons projected onto the fifth's plane falls entirely in the fifth polygon's area

If this can be satisfied: can we do it with a more-than-five-sided regular polynom as base?

If not: does an other method exist?

  • $\begingroup$ I'm not entirely convinced we can construct a polyhedron to have three perpendicular, arbitrary regular polygon projections, but I think this is a very interesting question! $\endgroup$ – pjs36 Apr 13 '16 at 17:59
  • $\begingroup$ Any three 4k sided regular polygon can form a suitable composition. $\endgroup$ – Dávid Horváth Apr 13 '16 at 18:19

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