This page records the method used for the related
output/gosset_projections/ collection.
The models are not 4-polytopes. They are strict orthographic zomeable projections of Gosset-family and adjacent D/E root polytopes:
| Polytope | Natural dimension | Models |
|---|---|---|
5-orthoplex (2_11) |
5 | 3 |
Rectified 5-orthoplex (t1 2_11) |
5 | 3 |
5-demicube (1_21) |
5 | 3 |
6-orthoplex (3_11) |
6 | 6 |
6-demicube (1_31) |
6 | 9 |
2_21 |
6 | 2 |
E6 root polytope (1_22) |
6 | 2 |
7-orthoplex (4_11) |
7 | 5 |
7-demicube (1_41) |
7 | 6 |
3_21 |
7 | 5 |
2_31 |
7 | 5 |
1_32 |
7 | 5 |
8-orthoplex (5_11) |
8 | 18 (R=3 confirmed) |
4_21 |
8 | 3 |
2_41 |
8 | 3 |
1_42 |
8 | 3 |
9-orthoplex (6_11) |
9 | 8 |
10-orthoplex (7_11) |
10 | 9 |
10-demicube (1_71) |
10 | 15 |
Scott Vorthmann’s Gosset’s Polytopes (2005) is prior vZome/zometool work on Gosset-polytope models:
https://vorth.github.io/vzome-sharing/2005/09/18/gossets-polytopes.html
The sweep documented here was run from scratch, but this page should be cited as earlier public work on the Gosset-polytope subject in vZome.
The successful unified sweep searches the raw columns of a projection matrix
P = [c_0 ... c_7], c_i in Z[phi]^3.
For a coefficient range R, each coordinate of each column is
a + b phi with |a| <= R and |b| <= R.
The sweep is not targeted at the H3 cases. It uses only the E8 root constraints and strict orthographicity:
+/- e_i +/- e_j, require c_i + c_j and
c_i - c_j to be parallel to zometool axes or to collapse.P P^T = c I_3 in Z[phi].2_21/3_21/4_21, this means all vertex-figure embeddings inside the
E8 model. For 2_11, 1_21, 1_31, 1_41, and the rectified 5-orthoplex, the
standalone source polytopes are checked directly; their edge directions are
only of the +/- e_i +/- e_j type, so there is no E8 half-root condition.
The 6-demicube and 7-demicube are 1_31 and 1_41 in Gosset notation and
use the same standalone D-type demicube edge constraint as 1_21, with six
and seven projection columns instead of five.
The rectified 5-orthoplex is the D5 root polytope with vertices all
permutations of
(+/-1,+/-1,0,0,0), so it has the same D5 root edge-direction constraints
as 1_21. For 2_41, the edge directions form the E8 root system with
the odd spinor parity, so the direct sweep uses the same type-A constraints
but checks odd-parity half-root sums. For 1_42, the edge directions are
the full E8 root system, so the direct sweep checks both spinor parities and
the +/-2e_i +/-2e_j roots. For the E6 root polytope (1_22), we realize the polytope as
the E8 roots orthogonal to an A2 subsystem and use the gauge representative
annihilating that A2 complement.| Range | Runtime | Leaves | Result |
|---|---|---|---|
5-orthoplex (2_11), R=1 |
1.2 s | 433,289 | 3 models |
5-orthoplex (2_11), R=2 |
29.3 s | 3,702,503 | same 3 models |
5-orthoplex (2_11), R=3 |
673.9 s | 48,536,391 | same 3 models |
6-orthoplex (3_11), R=1 |
3.6 s | 1,370,293 | 6 models |
6-orthoplex (3_11), R=2 |
64.4 s | 14,183,831 | same 6 models |
7-orthoplex (4_11), R=1 |
9.4 s | 4,157,641 | 5 models |
7-orthoplex (4_11), R=2 |
190.1 s | 53,115,083 | same 5 models |
8-orthoplex (5_11), R=1 |
31.5 s | 12,101,671 | 11 models |
8-orthoplex (5_11), R=2 |
11.3 min | 190,459,868 | 18 models; R=3 still running |
9-orthoplex (6_11), R=1 |
77.7 s | 33,820,471 | 8 models |
9-orthoplex (6_11), R=2 |
36.7 min | 648,460,652 | same 8 models |
10-orthoplex (7_11), R=1 |
222.3 s | 90,870,809 | 9 models |
10-orthoplex (7_11), R=2 |
1.97 h | 2,092,743,206 | same 9 models |
5-demicube (1_21), R=1 |
1.2 s | 433,289 | 3 models |
5-demicube (1_21), R=2 |
30.4 s | 3,702,503 | same 3 models |
5-demicube (1_21), R=3 |
708.5 s | 48,536,391 | same 3 models |
6-demicube (1_31), R=1 |
4.4 s | 1,370,293 | 5 models |
6-demicube (1_31), R=2 |
70.3 s | 14,183,831 | 9 models |
6-demicube (1_31), R=3 |
55.3 min | 390,832,319 | same 9 models |
7-demicube (1_41), R=1 |
12.8 s | 4,157,641 | 6 models |
7-demicube (1_41), R=2 |
200.3 s | 53,115,083 | same 6 models |
7-demicube (1_41), R=3 |
6.45 h | 2,914,645,635 | same 6 models |
10-demicube (1_71), R=1 |
13.1 min | 90,870,809 | 13 even-coset models |
10-demicube (1_71), R=2 |
2.46 h | 2,092,743,206 | same 13 even-coset models |
Rectified 5-orthoplex (t1 2_11), R=1 |
1.3 s | 433,289 | 3 models |
Rectified 5-orthoplex (t1 2_11), R=2 |
32.7 s | 3,702,503 | same 3 models |
Rectified 5-orthoplex (t1 2_11), R=3 |
678.4 s | 48,536,391 | same 3 models |
1_22, R=1 |
1.5 s | 433,289 | 2 models |
1_22, R=2 |
17.0 s | 3,702,503 | same 2 models |
1_22, R=3 |
444.1 s | 48,536,391 | same 2 models |
2_31, R=1 |
8.7 s | 1,370,293 | 5 models |
2_31, R=2 |
79.8 s | 14,183,831 | same 5 models |
2_31, R=3 |
57.2 min | 390,832,319 | same 5 models |
1_32 |
reused 2_31 directions |
n/a | 5 labelled models |
2_21/3_21/4_21, R=2 |
0.98 h | 190,459,868 | all 10 labelled models |
2_21/3_21/4_21, R=3 |
47.49 h | 19,816,930,273 | same 10 labelled models; no new models |
2_41, R=1 |
48.7 s | 12,101,671 | 3 models |
2_41, R=2 |
13.5 min | 190,459,868 | same 3 models |
1_42, R=1 |
68.7 s | 12,101,671 | 3 models |
1_42, R=2 |
13.8 min | 190,459,868 | same 3 models |
The R=3 leaf estimator predicted 1.960e10 +/- 1.828e8 leaves, close to the
actual 1.982e10 leaves.
The public captions use full Euclidean point-cloud symmetry:
B3-symmetric means full octahedral symmetry, full order 48.D4-symmetric means full tetragonal/dihedral symmetry, full order 16.Th-symmetric means full tetrahedral symmetry with inversion, full order 24.H3-symmetric means full icosahedral symmetry, full order 120.D5d-symmetric means full pentagonal antiprismatic/dihedral symmetry, full
order 20.D2h, C3i, and D3d are lower-symmetry full Euclidean point-cloud
labels used by some D-family projections, including the 8-, 9-, and
10-orthoplex pages and the C3i 10-demicube (1_71) variants.The computed symmetry audit is in
output/gosset_projections/symmetry_analysis.json.
The B3-symmetric models are related through the usual vertex-figure chain
5-demicube (1_21) -> 2_21 -> 3_21 -> 4_21.
The same raw-column projection can produce compatible strict images for several members of this chain. Under such a projection, vertices of the larger Gosset polytope may coincide in 3D, so a larger source polytope can produce a degenerate image whose visible ball arrangement agrees with a smaller source polytope’s image.
In the computed R=2/R=3 data, the B3 cases split into two projection branches:
| Branch | 2_21 image |
3_21 image(s) |
4_21 image |
|---|---|---|---|
| A | 15 balls | 15 balls | 27 balls |
| B | 19 balls | 19 balls and 14 balls | 33 balls |
Branch A is the simplest inherited family: the same B3 projection gives the
15-ball 2_21 and 3_21 images and the 27-ball 4_21 image.
Branch B explains the apparent extra B3 model for 3_21. The same projection
branch gives the 19-ball 2_21 image, the 19-ball 3_21 image, and the
33-ball 4_21 image; but after choosing this B3 projection, the 3_21
vertex figures inside 4_21 split into more than one orbit under the remaining
B3 symmetry. One orbit gives the inherited 19-ball image, while another orbit
gives the 14-ball cube-hull image. Thus the 14-ball 3_21 model is not an
unrelated projection direction; it is another degeneration within the same B3
projection branch.
The regular 5-orthoplex (2_11) adds three labelled models: two B3-symmetric
models with 7 and 9 balls, and one D4-symmetric model with 7 balls.
The higher orthoplexes 3_11, 4_11, 5_11, 6_11, and 7_11 use the same D_n root
edge-direction constraints as the matching n-demicubes: orthoplex edges join
non-antipodal pairs among +/-e_i, so their directions are exactly
+/-e_i +/- e_j. A dedicated n-orthoplex sweep verifies the projection
relation directly. The 6-orthoplex (3_11) has six labelled models through
R=2, the 7-orthoplex (4_11) has five labelled models through R=2, the
8-orthoplex (5_11) has 18 labelled models through R=2 with R=3 still running,
the 9-orthoplex (6_11) has eight labelled models through R=2, and the
10-orthoplex (7_11) has nine labelled models through R=2. Some visible point
clouds agree with lower-dimensional orthoplex images plus collapsed vertices,
but the source edge sets differ, so the gallery keeps them as separate labelled
models.
The
5-demicube (1_21) member adds three models. Its 15-ball B3 model is the
same visible point-cloud geometry as the 15-ball branch above; the 8-ball B3
model and 12-ball D4 model are specific lower-dimensional degenerations of the
5-demicube and do not match the current 2_21/3_21/4_21 gallery models.
The rectified 5-orthoplex fits the notation as the first rectification of
2_11, so a compact label is t1 2_11. It is the D5 root polytope /
rectified pentacross (t1{3,3,3,4}), while unmodified 2_11 is the regular
5-orthoplex. Since its 240 edges use the same D5 root directions as the
5-demicube, the direct R=1/R=2/R=3 sweep gives three labelled models with 19,
21, and 26 balls.
The E6 root polytope (1_22) contributes two labelled B3 models. Their
visible point-cloud geometries agree with the 4_21 27-ball and 33-ball B3
models, but the source polytope and source edge set are different, so the
gallery keeps them as separate labelled models.
The 2_31 member contributes five labelled models. Three are B3-symmetric
degenerations with 19, 27, and 33 balls; two are H3-symmetric 93-ball models
with different convex hulls. As with 1_22, some visible geometries agree
with existing smaller entries, but the 2_31 source polytope and its edge set
are distinct, so they are listed separately.
The 1_32 member has the same E7 root edge-direction constraints as 2_31,
so no separate projection-direction search is needed. Evaluating the completed
2_31 projection directions on the 576-vertex 1_32 source gives five
labelled models: three B3-symmetric models with 38, 59, and 81 balls, and two
H3-symmetric 384-ball models.
The 6-demicube (1_31, demihexeract) is the next demicube after 1_21.
Its direct D6 demicube sweep gives nine labelled models through R=3: four
B3-symmetric projections with 8, 13, 14, and 15 balls; one D4-symmetric
projection with 12 balls; two Th-symmetric projections with 32 balls; and two
H3-symmetric projections with 32 balls.
The Th- and H3-symmetric 1_31 projections use the following exact 6x3
matrices, shown up to overall scale as the raw sweep matrices multiplied by
1/(3phi); the emitted .vZome files apply their physical post-scales.
1_31_Th_32_balls_a:
[
[-phi, 0, -1/phi ],
[-phi, 0, 1/phi ],
[-1/phi, -phi, 0 ],
[-1/phi, phi, 0 ],
[ 0, -1/phi, -phi ],
[ 0, 1/phi, -phi ],
]
1_31_Th_32_balls_b:
[
[-phi, 0, -1/phi ],
[-phi, 0, 1/phi ],
[-1/phi, -phi, 0 ],
[-1/phi, phi, 0 ],
[ 0, -1/phi, -phi ],
[ 0, -1/phi, phi ],
]
1_31_H3_32_balls_a:
[
[-phi, -1, 0 ],
[-phi, 1, 0 ],
[-1, 0, -phi ],
[-1, 0, phi ],
[ 0, -phi, -1 ],
[ 0, phi, -1 ],
]
1_31_H3_32_balls_b:
[
[-phi, -1, 0 ],
[-phi, 1, 0 ],
[-1, 0, -phi ],
[-1, 0, phi ],
[ 0, -phi, -1 ],
[ 0, -phi, 1 ],
]
The 7-demicube (1_41) is the next demicube after 1_31. Its direct D7
demicube sweep gives six labelled models already at R=1 and saturates through
R=3: three B3-symmetric projections with 8, 15, and 27 balls; one D4-symmetric
projection with 12 balls; one Th-symmetric projection with 64 balls; and one
H3-symmetric projection with 64 balls.
Both 64-ball 1_41 projections have convex hulls with 30 rhombic faces. The
H3 case has the regular rhombic triacontahedron as its hull; the Th case has a
pyritohedral 30-face rhombic hull with two rhombus shapes, slim and broad.
The Th- and H3-symmetric 1_41 projection matrices are 7x3 matrices in the
same 1/(3phi) normalization as the 1_31 matrices. The final row is zero,
so these two 1_41 models are inherited from the corresponding 6-demicube
patterns.
1_41_Th_64_balls:
[
[-phi, 0, -1/phi ],
[-phi, 0, 1/phi ],
[-1/phi, -phi, 0 ],
[-1/phi, phi, 0 ],
[ 0, -1/phi, -phi ],
[ 0, -1/phi, phi ],
[ 0, 0, 0 ],
]
1_41_H3_64_balls:
[
[-phi, -1, 0 ],
[-phi, 1, 0 ],
[-1, 0, -phi ],
[-1, 0, phi ],
[ 0, -phi, -1 ],
[ 0, -phi, 1 ],
[ 0, 0, 0 ],
]
The 10-demicube (1_71) uses the same D10 demicube constraints as the lower
demicubes. Its R=1 sweep already finds 13 labelled even-coset models, and the
R=2 sweep confirms saturation at the same 13 even-coset signatures. The C3i
and H3 matrices also contribute distinct variant-b models obtained from the
opposite demicube parity coset. The projections include B3,
D4, Th, H3, D2h, D3d, and C3i symmetry types, with ball counts from 8 through
364.
The 2_41 member uses the E8 root directions with odd spinor parity. The
direct parity-correct sweep gives three labelled models through R=2: two
B3-symmetric projections with 65 and 93 balls, and one H3-symmetric projection
with 921 balls.
The 1_42 member uses the full E8 root directions. The direct sweep gives
three labelled models through R=2: two B3-symmetric projections with 169 and
251 balls, and one H3-symmetric projection with 5936 balls.