02 · First examples
G₁, G₂, G₃.
The segment and pentagon already display the self-dual pattern. In dimension three, the subdivision becomes genuinely spatial. The exact face-number pattern and its face lattices continue below.
| 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.
Faces ordered by inclusion
Read the f-vector as a lattice.
The number of nodes in each horizontal rank is the corresponding entry of the complete f-vector. A line means that the lower face is covered by the upper face, with no intermediate face between them.
Exact incidence data
The face lattice of G3
Every face and every cover relation is shown. Select a node to highlight its immediate neighbors and connecting covers.
Ranks run from whole to empty.
Certified from exact supporting facets, face intersections, affine ranks, and adjacent-rank inclusions.
Exact face enumeration
The pattern beyond dimension three.
For 0 ≤ d ≤ n − 1, the proved formula is
| Polytope | Complete f-vector | Total faces |
|---|---|---|
| G1 | (1, 2, 1) | 4 |
| G2 | (1, 5, 5, 1) | 12 |
| G3 | (1, 12, 22, 12, 1) | 48 |
| G4 | (1, 28, 73, 73, 28, 1) | 204 |
| G5 | (1, 64, 215, 304, 215, 64, 1) | 864 |
| G6 | (1, 144, 591, 1070, 1070, 591, 144, 1) | 3612 |
| G7 | (1, 320, 1551, 3412, 4360, 3412, 1551, 320, 1) | 14928 |
| G8 | (1, 704, 3935, 10178, 15764, 15764, 10178, 3935, 704, 1) | 61164 |
| G9 | (1, 1536, 9727, 28912, 52528, 63616, 52528, 28912, 9727, 1536, 1) | 249024 |
| G10 | (1, 3328, 23551, 79086, 164784, 233856, 233856, 164784, 79086, 23551, 3328, 1) | 1009212 |
| G11 | (1, 7168, 56063, 209900, 493260, 801984, 939456, 801984, 493260, 209900, 56063, 7168, 1) | 4076208 |
| G12 | (1, 15360, 131583, 543466, 1421860, 2604888, 3494304, 3494304, 2604888, 1421860, 543466, 131583, 15360, 1) | 16422924 |
| G13 | (1, 32768, 305151, 1378280, 3973112, 8097056, 12233232, 14006784, 12233232, 8097056, 3973112, 1378280, 305151, 32768, 1) | 66045984 |
| G14 | (1, 69632, 700415, 3434470, 10815688, 24268816, 40769872, 52564512, 52564512, 40769872, 24268816, 10815688, 3434470, 700415, 69632, 1) | 265246812 |
| G15 | (1, 147456, 1593343, 8429540, 28793492, 70537376, 130418288, 186929600, 210477696, 186929600, 130418288, 70537376, 28793492, 8429540, 1593343, 147456, 1) | 1064175888 |
Every vector includes the empty face and the whole polytope. The table uses exact integers, is symmetric by self-duality, and satisfies Σfi = 4n − 2·3n−1 + 2.