Colin’s Polyhedra — Oddities

A curated selection of twelve mathematical forms, not a classification:
Szilassi, Császár, pyritohedron, Schönhardt, Jessen’s orthogonal icosahedron,
Stewart’s G3, Dürer’s solid, Bilinski’s dodecahedron, rhombic icosahedron,
rhombic hexecontahedron, rhombic enneacontahedron and the associahedron.

Szilassi / Császár / rhombic hexecontahedron / rhombic enneacontahedron:
Antiprism named resources, pinned revision
7805b34d3f8860857fb9ac10a1917d1bf4c7888d.
Original MIT source and full notice retained in data/antiprism/std_polys.cc;
the deployed MIT notice is in uniform-data-notice.txt.
Antiprism credits Tom Ace’s Szilassi coordinates and Császár’s coordinates:
https://www.minortriad.com/szilassi.html
https://dmccooey.com/polyhedra/Csaszar1.html
The Szilassi model uses Tom Ace’s realization with adjusted proportions.
The dual link between these two models is topological, not a polar reciprocal.
Both have genus one. Each Szilassi face shares one edge with each other face;
every pair of Császár vertices is joined by an edge.
The chiral Szilassi, Császár and Schönhardt models offer reflected forms.

Schönhardt / Dürer / Bilinski / canonical associahedron:
Mathematical coordinates and incidence published by David I. McCooey, retained
with original source URLs in data/oddities/mccooey-coordinates.json.
https://dmccooey.com/polyhedra/Schonhardt.html
https://dmccooey.com/polyhedra/Durer.html
https://dmccooey.com/polyhedra/BilinskiDodecahedron.html
https://dmccooey.com/polyhedra/Associahedron.html
Dürer’s solid is one interpretation, using a 540/7-degree apex angle and 4/7
truncation; no unique historical reconstruction is claimed.

Jessen’s orthogonal icosahedron:
Constructed from cyclic permutations of (0, ±1, ±2). The 24 sqrt(6) edges and
six length-4 edges define its twenty triangular faces, without a convex hull.
Independent metric reference:
https://dmccooey.com/polyhedra/Jessen.html
Adjacent face planes are perpendicular at both convex and concave edges.

Rhombic icosahedron:
Constructed as the zonohedron generated by five of the six icosahedral
unit equiangular axes. Extreme sign sums define 22 vertices and 20 golden
rhombi. Coplanar hull subdivisions are merged into complete rhombi.
Independent metric reference:
https://dmccooey.com/polyhedra/RhombicIcosahedron.html

Stewart’s G3:
Jim McNeill’s mathematical VRML mesh, copyright Jim McNeill 1999/2003,
retained unchanged in data/oddities/stewart-g3.wrl, with attribution.
https://www.orchidpalms.com/polyhedra/misc/g3.html
https://www.orchidpalms.com/polyhedra/misc/g3.wrl
The 8-digit source rounding is refined to exact regular-polygon chord
constraints. The largest vertex correction is approximately 8.05e-9 of an
edge. Its seven triangles, three squares and three pentagons are regular.
G3 is Stewart’s model name; this example has genus zero.

Pyritohedron:
Original implementation of Eq. (45), up to axis permutation, in Koca et al.,
Quaternionic Representations of the Pyritohedral Group, Related Polyhedra
and Lattices (2015), https://arxiv.org/abs/1506.04600 .
Vertices: (±1,±1,±1) and cyclic (0,±(1+h),±(1-h²)).
Supporting planes: cyclic (0,±h,±1)·x=1+h.
The slider covers h=−0.90–0.90 in steps of 0.01. The original positive face
incidence continues analytically to negative h; it is not replaced by a hull
or by re-sorting concave vertices. Negative h gives congruent concave
pentagons. Exact presets at h=±(sqrt(5)-1)/2 give the regular dodecahedron
and the equilateral concave endo-dodecahedron. At h=0 the actual solid is a
cube: six squares, twelve edges, eight vertices. Collinear edge midpoints
and coplanar seams are excluded at this degenerate value. Away from zero
there are twelve congruent pentagons, thirty edges and twenty vertices.
The h=−1 and h=1 degenerate limits are outside the displayed range.
Comparisons retain each parameter value as a distinct saved realization.

All twelve use complete planar polygon faces, original edges and vertices.
Both toroids retain their open handle. Concave faces use the same two-sided,
nonzero-winding depth renderer as the existing nonconvex collections.
Each-face colours and the numbered face inspector allow all faces to be
examined, including hidden faces. No instructional exercises are included.

Geometry tests verify closed orientable incidence, genus, planarity,
nonintersecting edges/faces, independent metrics and 3D ray depth/picking.
Szilassi’s complete face-adjacency graph, Császár’s complete edge graph,
Jessen’s orthogonality and Schönhardt’s exterior diagonals are checked.
All 71 slider positions and the exact regular preset are validated.