A FEW EXCEPTIONS TO THE USUAL RULES
Oddities.
Holes, twists, dents and unexpected faces. Twelve forms worth a closer look.
Some familiar.
Some surprising.
Five triakis-tetrahedron and three rhombic-dodecahedron cell layers, plus Wenninger W43–46 of the cuboctahedron. TT-L and RD-L are local layer labels, separate from Hart’s numbering.
Colours distinguish original face planes. Inspect a plane to see all its surface pieces together.
Choose a colour to inspect one constituent.
UC1–UC75 cover Skilling’s catalogue. UC20–UC25 are examples of infinite prism series; rotational entries show one fixed configuration. Eight selected dual compounds and the non-uniform compound of three disphenoids are also included.
All 53 duals of W67–W119, in parent order: 43 finite and 10 unbounded. DW69 means the dual of W69; UD13 means the dual of U13. Unbounded cases have adjustable, open-ended cut-off views. The four Kepler–Poinsot duals remain in their existing family.
Parallel arms continue in both directions. Dashed edges mark open display cuts; no caps are added. Core radius is the distance to the farthest finite vertex. The view fits the chosen length.
h = −0.90 to 0.90: inward dents below zero, a cube at zero. The two dodecahedron presets use exactly h = ±(√5 − 1)/2. Comparisons keep the chosen shape.
NM1–31 follow McNeill’s list. Later NM and NNM numbers are local to this collection; the boundaries are informal.
Drag to rotate · scroll or pinch to zoom · tap a face to inspectKeyboard: arrows · + / − · 0 to reset
SIDE BY SIDE
Connections in the numbers.
Near-miss labels are informal; this is a documented selection, not a complete classification. NM1–31 preserve McNeill’s ordering; some source realizations have been adjusted to planar faces, as stated beside each model. For unbounded duals, V separates finite and ideal vertices; opposite arm ends represent the same ideal vertex. Counts exclude artificial cut edges and corners. ¹ Unbounded edges have no finite length ratio. Near misses show the edge-length range. Other edge lengths show count × relative length, with the shortest = 1 (rounded); all Catalan faces are congruent. For boundary-mesh stellations, F counts connected surface pieces, with holes kept open; E and V describe their boundary mesh. The 59 Icosahedra use 20 original planes; other stellations use their parent’s 12 or 14 planes. For compounds, counts add constituents separately, including coincident vertices. For star polyhedra, crossings are not extra vertices, and V − E + F need not be 2. Models are fitted to the viewer, not shown at a shared scale.