A catalogue of 4-dimensional polytopes that can be projected to 3D in ways that are physically buildable with zometool struts (Blue / Yellow / Red / Green) and modelled exactly in vZome.
For each polytope, we enumerate orthographic projections whose every edge lands on a default zometool axis. Hits are grouped up to rotation and uniform scale, then emitted as .vZome files that you can spin, zoom, and inspect right in the browser.
Pick a category and click View → to jump into its interactive page.
| Category | Polytopes | Zomeable polytopes | Zomeable projections | |
|---|---|---|---|---|
| Regular | 6 | 6 | 18 + 1 inf family ¹ | View regulars → |
| Uniform (non-prismatic) | 41 | 28 | 73 | View uniforms → |
| Duoprisms ({p}×{q}) | ∞ | 6 | 8 + 2 inf families | View duoprisms → |
| Polyhedral prisms (P × [0,1]) | 17 | 12 | 45 | View polyhedral prisms → |
| Antiprismatic prisms (A_n × [0,1]) | ∞ | 1 | 9 | View antiprismatic prisms → |
| Regular compounds and star polytopes | 62 | 21 | 21 | View models → |
| Total | ∞ | 74 | 174 + 3 inf families |
Related non-4D collections:
| Category | Source polytopes | Models | |
|---|---|---|---|
| Gosset orthographic projections | 19 | 113 | View projections → |
| A5 simplex family projections (all 19 A5 Wythoff polytopes) | 19 | 57 | View projections → |
| Approximate 3D polyhedron zomes (not projections; curated RGBY approximations) | 16 | 16 | View approximate zomes → |
Every category page lists the polytopes in scope and gives one-click access to a vZome 3D viewer embedded inline.
¹ Infinite families currently occur for the tesseract (8-cell) and the duoprisms {4}×{6} and {4}×{10}; other corpus entries saturate at small finite counts. See the 8-cell master theorem and duoprism category page for details.
Visualization option: some viewer pages include a second model immediately below the original labelled hidden-cell-removal view. This is a descriptive term used here by analogy with hidden-line/hidden-surface removal: cells facing away from the positive side of the 4D projection kernel are omitted, while equatorial cells are retained, so the model is easier to inspect. These variants are not new projections and are not included in the counts above. See the hidden-cell-removal index.
For each polytope P ⊂ ℝ⁴ and each candidate kernel direction n ∈ ℤ[φ]⁴, we form the orthogonal projection Q = I − n nᵀ / |n|² and project every vertex of P to ℝ³. The projection is zomeable when every projected edge points along one of the 31 zometool axes (B / Y / R / G families). We enumerate all kernel directions in a bounded coefficient box, dedupe up to rotation + uniform scale, and emit a .vZome file per distinct shape.
The catalogue’s counts are empirical: we report a number once enlarging the coefficient box stops finding new shapes. They are not formally proven complete; see docs/METHODOLOGY.md for the discussion.
grand_antiprism_vertex_first.vZome is an independent reconstruction
from the 4D polytope, with some differences from Vorthmann’s model.{p}×{q} produce zomeable rank-3 projections, including infinite families for {4}×{6} and {4}×{10}.The full source, methodology, generation scripts, and per-polytope deep-dive write-ups live in the GitHub repository. Python 3.10+ and NumPy; MIT-licensed.
Built on the zometool construction system and vZome (Scott Vorthmann).