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?