02 · First examples
G₁, G₂, G₃.
The segment and pentagon already display the self-dual pattern. In dimension three, the subdivision becomes genuinely spatial.
| Polytope | Geometry | f-vector | Maximal-cell picture |
|---|---|---|---|
| G1Segment | (1, 2, 1)2 vertices | one interval | |
| G2Pentagon | (1, 5, 5, 1)5 vertices | square + triangle | |
| G3Goodwillie polytope | (1, 12, 22, 12, 1)12 vertices | cube + 2 prisms + tetrahedron |
The vectors include the empty face and the whole polytope. The cases n = 1, 2 are the formal extensions of the coordinate formula.