The grand antiprism is one of two non-Wythoff convex uniform 4-polytopes (the other is the snub 24-cell). It has 100 vertices, 500 edges, 720 triangular faces, and 320 cells (300 tetrahedra + 20 pentagonal antiprisms), with symmetry group of order 400 (the “ionic-diminished” subgroup of H₄ that stabilises a Hopf pair of great decagons in the 600-cell).
It is constructed by diminishing the 600-cell: remove 20 vertices that form two interlocking great decagons on mutually-orthogonal Hopf circles. The 20 pentagonal-antiprism cells fill in around the two removed Hopf rings.
Discovered by Conway and Guy (1965) by computer search.
Empirical search on a ℤ[φ]⁴ direction grid up to rng=4 finds exactly 2 distinct zomeable orthographic projections, each verified strict- orthographic by the M·Mᵀ = s²·I test:
| File | Kernel direction | 3D balls | 4D edges → 3D struts (collapsed) | Strut counts |
|---|---|---|---|---|
grand_antiprism_vertex_first.vZome |
(1, 1, 1, 1) | 71 | 500 → 342 (8 collapse) | B:102 Y:104 R:136 |
grand_antiprism_ring_first.vZome |
(1, 0, 0, 0) | 60 | 500 → 260 (10 collapse) | B:80 Y:80 R:100 |
Both projections use all three default zometool colors (blue, yellow, red) — the same rich signature as the snub 24-cell.
Saturation confirmed: same 2 shapes at rng=2 (1819 dirs, 30 hits), rng=3 (20474 dirs, 60 hits), rng=4 (135750 dirs, 100 hits).
Kernel: (1, 1, 1, 1). The unit vector (½, ½, ½, ½) is a vertex of the GA (a tesseract-vertex of the parent 600-cell that survives diminishing).
This is the antipodal-pair-vertex-first projection. Eight of the 500 edges collapse to zero length; the remaining 492 → 342 distinct 3D struts.
3D point group: D₂ₕ (order 8). The proper rotation group is D₂ (Klein 4-group with three perpendicular C₂ axes); central inversion lifts this to D₂ₕ. The 3-fold rotation around (1,1,1,1) that exists in H₄ is broken by the GA’s diminishing pattern — the 400-element GA stabilizer of (1,1,1,1) only retains the three coordinate-pair-swap C₂’s.
Where are the deleted ring vertices? Of the 20 deleted 4D vertices, 16 land exactly on kept-vertex shadow positions (hidden by overlap from the 4D antipodal collapse + dense 600-cell shadow), and only 4 visible holes remain. All 4 visible holes lie on the outer equator (axial coord along (1,1,1,1) = 0) at the maximum perp radius 1.0. The equator hosts 26 kept balls + 4 holes evenly spaced among 30 slots on the unit circle. So the only direct evidence of the two removed pentagonal-antiprism rings, in this projection, is 4 missing balls on the outermost equator.
The same camouflage effect occurs in the snub 24-cell vertex-first
projection (18 of 24 deletions hidden, 6 visible on the outer equator);
see snub_24cell/RESULTS.md. In general, vertex-first projections of any
diminished 600-cell hide most deletions because (i) opposite-hemisphere
vertices project onto each other and (ii) the 600-cell shadow is
densely degenerate (75 positions for 120 vertices). This makes the
diminishing pattern hard to read by eye in vertex-first models; cell-
first or ring-first projections expose deletions at multiple radii and
are visually more revealing.
Kernel: (1, 0, 0, 0). This direction is a vertex of one of the two removed Hopf decagons — specifically (1,0,0,0) was deleted to form the GA. So this is the axis of one of the two pentagonal-antiprism rings.
Looking down this axis, the projection inherits clean 5-fold symmetry because the two removed Hopf decagons sit in mutually orthogonal 2-planes (Wikipedia: “the two rings are mutually perpendicular, in a structure similar to a duoprism”). The 15 deletions from the 600-cell shadow are distributed:
This signature matches Scott Vorthmann’s 2006 vZome model exactly (see “External reference” below).
3D point group: D₅d (order 20 = pentagonal antiprismatic group: C₅ + 5 perpendicular C₂’s + inversion + S₁₀ + 5 σd). The proper rotation group is D₅ (order 10). All 15 holes are visible: 1 at origin + 2 + 2 + 10 laid out at 5 distinct radii.
Just as for the snub 24-cell, both GA shapes are subsets of the
600-cell’s unique 75-ball zomeable projection (600cell/600cell_H4_to_H3.vZome):
| Kernel | 600-cell balls | GA balls | GA-only | 600-only (deleted) |
|---|---|---|---|---|
| (1,1,1,1) | 75 | 71 | 0 | 4 |
| (1,0,0,0) | 75 | 60 | 0 | 15 |
The single H₄-orbit of zomeable kernel directions for the 600-cell splits under the order-400 GA symmetry into (at least) two orbits, giving the two shapes. There are no GA-only zomeable kernels (the empirical search saturates at 2 shapes).
The simplest mental construction:
600cell/600cell_H4_to_H3.vZome.For the vertex-first kernel (1,1,1,1): 4 balls are deleted.
For the ring-first kernel (1,0,0,0): 15 balls are deleted, including the origin (where ±(1,0,0,0) collapse to). The other 14 form the characteristic 2/2/10 pentagonal pattern described above.
Per Wikipedia: “20 stacked pentagonal antiprisms occur in two disjoint rings of 10 antiprisms each. The antiprisms in each ring are joined to each other via their pentagonal faces. The two rings are mutually perpendicular, in a structure similar to a duoprism.”
This is implemented in lib/polytopes.py::grand_antiprism() by enumerating
all 72 great decagons of the 600-cell and finding a pair satisfying
both:
Counts in the 600-cell:
The 36 valid pairs match the expected H₄-orbit size of inscribed GAs: |H₄|/|GA stabiliser| = 14400/400 = 36.
| Snub 24-cell | Grand antiprism | |
|---|---|---|
| Construction | 600-cell minus 24-cell verts | 600-cell minus 2 perpendicular Hopf decagons |
| Removed verts | 24 | 20 |
| Verts | 96 | 100 |
| Edges | 432 | 500 |
| Symmetry order | 1152 | 400 |
| #zomeable shapes | 2 | 2 |
| Shape A balls | 60 (cell-first) | 71 (vertex-first) |
| Shape B balls | 69 (vertex-first) | 60 (ring-first) |
| Both inherit from 600-cell? | Yes | Yes |
| Genuinely new directions? | No (all kernels are 600-cell H₄ kernels) | No (same) |
Both non-Wythoff polytopes give exactly 2 zomeable projections each, and in both cases all kernel directions are inherited from the 600-cell’s single H₄-orbit zomeable kernel — the smaller symmetry just splits that orbit into 2 sub-orbits.
All four 4D polytopes here are centrally symmetric, so the 3D shadows all contain the inversion -I. The proper rotation group (chiral 3D symmetry) and the full point group are:
| Projection | Proper rotations | Full point group | Description |
|---|---|---|---|
| Snub 24-cell cell-first | T (order 12: E, 8C₃, 3C₂) | Tₕ (order 24) | Pyritohedral. None of the four C₃ axes is aligned with the kernel; they pass through 4 pairs of icosahedral cells off-axis. |
| Snub 24-cell vertex-first | D₃ (order 6: E, 2C₃, 3C₂) | D₃d (order 12) | Antiprismatic; principal C₃ aligned with the kernel (φ²,φ,1,0). |
| GA vertex-first | D₂ (order 4: E, 3C₂) | D₂ₕ (order 8) | The H₄ 3-fold around (1,1,1,1) is broken by the GA’s two-decagon diminishing pattern; only three coord-pair-swap C₂’s survive. |
| GA ring-first | D₅ (order 10: E, 4C₅, 5C₂) | D₅d (order 20) | Pentagonal antiprismatic; aligned with the surviving ring (the perpendicular removed decagon is normal to the projection, so its 5-fold drives the symmetry). |
lib/polytopes.py::grand_antiprism().
Enumerates all 72 great decagons, picks the first orthogonal-Hopf pair
(0 cross-edges, uniform degree 10) — guarantees canonical GA.lib/search_engine.py (gen_dirs, search, group_by_shape).lib/emit_grand_antiprism.py → lib/emit_generic.py
with extra_scale = GF(3, 5) = φ⁵ (gives B2/Y2/R1/R2 standard
zometool sizes, matching the snub 24-cell models).classify_direction
(every strut on a default zometool axis).Scott Vorthmann (vZome’s author) published vZome models of the grand antiprism in 2006: https://vorth.github.io/vzome-sharing/2006/02/24/grand-antiprism.html
His ring-first model uses a kernel along a
pentagonal-antiprism ring axis, the same as our
grand_antiprism_ring_first.vZome. Verified equivalent: his 600-cell-
shadow deletion pattern (1@origin + 2@sin36° + 2@sin72° + 10@1.0 = 15
balls) matches ours exactly. He uses vZome’s built-in UniformH4Polytope
operation (with polytope.index = 1 for the 600-cell and a
left-isoclinic quaternion to reorient), then deletes the 20 ring vertices.
Our approach reaches the same configuration by direct enumeration of
orthogonal Hopf-decagon pairs in the 4D vertex set.
Vorthmann also explored the vertex-first Zometool projection in 2006.
The last model on that page is captioned “grand antiprism, alternate
Zometool projection”. Our grand_antiprism_vertex_first.vZome is an
independent reconstruction from the 4D polytope, with some differences
from Vorthmann’s model.
➡️ Open this page on GitHub Pages to interact with the 3D models below (the embeds only render when this file is served via GitHub Pages, not in github.com’s markdown preview).