In the last few days, I noticed something interesting about the maximal projections of the n-cross.
The maximal projection of the n-cross into (n-1) dimensions, as far as I can tell, is always when projecting to the hyperplane orthogonal to (1,1,1,1,...1). The projection envelope is always the convex hull of an (n-1)-simplex superimposed over its dual.
For example, the maximal projection of the octahedron is the projection orthogonal to (1,1,1), which is the convex hull of two dual triangles (i.e., the compound star 6/3), which is the regular hexagon. The maximal projection of the 16-cell is the projection orthogonal to (1,1,1,1), which is the convex hull of two dual tetrahedra (i.e., the stella octangula), which is the cube.
The maximal projection of the 5-cross is the convex hull of two dual 5-cells... as far as I can tell, this figure should be vertex-transitive as well as cell-transitive, although it doesn't appear to be uniform. If I'm not mistaken, this is the 4D catalan which is the dual of the bitruncated 5-cell.
Now, the question I'm grappling with right now is, is the maximal property transitive? That is, is the maximal (n-2)-dimensional projection of the n-cross equal to the maximal (n-2) projection of the maximal (n-1) projection of the n-cross? Is this true of any maximal (n-k)-dimensional projection of the n-cross for k<n? If so, what is the maximal 3D projection of the maximal 4D projection of the 5-cross? Is there a pattern to which projection is maximal in the series of maximal projections of decreasing dimensions?
Why am I interested in this, you ask? That's because ... the maximal 3D projection of the 6-cross, if I'm not mistaken, is none other than the regular icosahedron. I'm very curious to know, if the transitivity property is true, what is the 4D object whose maximal projection is the icosahedron. What is the series of projection planes that eventually results in the icosahedral projection of the 6-cross?