20 icosagon
21 icosikaihenagon, icosihenagon
22 icosikaidigon
23 icosikaitrigon
24 icosikaitetragon
25 icosikaipentagon
26 icosikaihexagon
27 icosikaiheptagon
28 icosikaioctagon
29 icosikaienneagon
30 triacontagon
And, of course, in three dimensions the -gon ending should become -hedron.
Then again, I'm not sure how well the rest applies here either. The basic shape appears to be an icosahedron. If you count the flattened-off edges, that adds thirty* more sides, making it a pentacontahedron. If you count the triangular facets being indented like they are, each counts triple, adding another forty net sides for an enneacontahedron. If you count all the additional fractal bevels ... er ... what's the Greek word for infinity?

* Note that each of the twenty triangular facets borders three halves of flattened-off edges. There are therefore 20*3/2 = 30 full flattened-off edges.