Start with the cone generated by revolving a equilateral triangle about
a median.
Take three such cones.
Arrange them so their points form the vertices of a equilateral triangle
and their bases intersect in a line normal to this equilateral triangle.
Does the resulting solid have 3 vertices or 5?
If the triangle used to generate each cone has edge length 1, what is
the edge length of the triangle formed by the points of the cones? I
suspect it might be 1.5 as their bases have dihedral angles of 60
degrees, so their diameters in the plane of the large triangle form the
long diagonals of a hexagon, implying an overlap equal to the radius of
the cone bases.
An infinite family can be formed by n cones formed by revolving
isosceles triangles with an unequal angle equal to the internal angle of
a regular n-gon... however, my suspicion is that 4 90-degree cones would
have bases that reach a neighbor's apex, and that 5 or more would leave
central gaps...
Take the bicone formed by revolving a square about a diagonal. What do
you get by intersecting two whose axes are at 90-degrees? 3 bicones? 4
or more with non-orthogonal axes?
What does the family of solids formed by revolving regular n-gons and
regular star polygons about all of their lines of symmetry and taking
the intersection? Is their any pattern to the shapes?