The Wieringa roof stands a flat tiling up and both rhombs become the same golden rhombus. That is where this starts and not where it stops. Dietl and Eschenburg prove that the five golden isozonohedra — Coxeter's list, every face a golden rhomb — are all projections of cubes from six-dimensional space: K30 = π(C6), F20 = π(C5), B12 = π(C4), and the two rhombohedra at k = 3. The golden rhomb itself is π(C2).
So it is not five solids and, separately, a face. It is one operation with k turned up, and every rung is the face of the one above. The list has eight entries for seven rungs, because k = 3 is two solids and not one.
For Dietl's figure 35, set faces to none and edges to ball and stick: that is the Zometool build, each strut colored by the one axis it runs along.
| k | π(Ck) | vertices | faces | in the tiling |
|---|---|---|---|---|
| 2 | the golden rhomb | 4 | 1 | the face of all of them |
| 3 | O6, oblate — a chain of axes | 8 | 6 | a tile |
| 3 | A6, acute — an equilateral triple | 8 | 6 | the other tile |
| 4 | Bilinski's dodecahedron | 14 | 12 | the minilens |
| 5 | Fedorov's rhombic icosahedron | 22 | 20 | the lens |
| 6 | Kepler's triacontahedron | 32 | 30 | the window |
The vertex counts are not measured but forced. Whether a subset of the generators lands on the hull is a question about directions: the sum over T is a vertex exactly when some c is positive on T and negative off it, so vertices correspond to the open regions of an arrangement of k great circles — and such an arrangement has k2 − k + 2 of them. That is 8, 14, 22, 32. The faces are k(k−1), and Euler holds at every rung.
Exploding pulls the faces off, each along its own outward normal, so zero reassembles exactly. Thirty rhombs come off a triacontahedron, twenty off an icosahedron, six off a rhombohedron — and they are all the same rhomb, which is the k = 2 rung seen from the solids above it.
The two colorings say different things. A face takes the pair of axes that spans it — the bands grammar the tiles use in the plane, so a face says which two zones made it. A stick takes the one axis it runs along, six colors, which is Zometool's own convention.
And there are exactly two rhombohedra for a reason that is a group action rather than a measurement: the icosahedral group is transitive on the six axes and on their pairs, and not on their triples — a triple is a chain or an equilateral triangle, and those are the two orbits, ten each. Their volumes are 0.470228 and 0.760845, in the ratio φ, the oblate the smaller.
And the plane has most of this ladder already, built before any of it was read: §5.0's singularity solids are π(C3), π(C4) and π(C5) — the hexagon, the octagon and the decagon — and a Penrose tile is π(C2). What the pentagrid cannot reach is k = 6: that needs a sixth zone, a family of lines with no direction, which the plane has nowhere to put. The icosahedral hexagrid has it, and the rest of this section is about what it builds.