07. Great Hexagonal Hexecontahedron – 18. Self-Intersecting Snub Quasi-Regular Duals Earrings & Necklace - Geometric Earring - Sacred Geometry Necklace Fashion Statement

Title: 18. Self-Intersecting Snub Quasi-Regular Duals Earrings & Necklace - Geometric Earring - Sacred Geometry Necklace Fashion Statement
Family: 18. Self-Intersecting Snub Quasi-Regular Duals
Motif: Geometry Polyhedra
Model Size: 1 in
Series: Geometry
Family List:
01. Medial Pentagonal Hexecontahedron
03. Great Pentagonal Hexecontahedron
04. Great Inverted Pentagonal Hexecontahedron
06. Medial Hexagonal Hexecontahedron
07. Great Hexagonal Hexecontahedron
08. Small Hexagonal Hexecontahedron
Self-Intersecting Snub Quasi-Regular Duals: Exotic Isohedral Pentagonal Hexecontahedra of Chiral Stars
Self-intersecting snub quasi-regular duals are a mesmerizing group of nonconvex isohedral polyhedra that arise as the duals of the chiral self-intersecting snub quasi-regular uniform stars, featuring 60 irregular or mirror-symmetric pentagonal faces that deeply penetrate one another, forming dense, self-crossing structures with high topological complexity. These face-transitive solids include the great pentagonal hexecontahedron (dual to the great snub icosidodecahedron), great inverted pentagonal hexecontahedron (dual to the great inverted snub icosidodecahedron), medial inverted pentagonal hexecontahedron, and others like the medial hexagonal hexecontahedron for related snubs. Portions of their concave pentagonal faces lie inside the core, creating hidden intersections and cavities visible only in transparent or wireframe renders, while preserving the underlying icosahedral symmetry.
Constructed via polar reciprocation of their vertex-transitive chiral counterparts, these duals showcase scalene or asymmetric pentagons with varying edge lengths, resulting in dihedral angles that evoke swirling, petal-like layers. In 3D visualization tools such as Stella4D, their self-penetrating facets produce stunning depth effects, with chiral pairs mirroring the handedness of the original snubs, bridging convex Catalan solids to extreme nonconvex forms.
For 3D printing cults, these hexecontahedra are ultimate treasures—challenging to print due to intersections but breathtaking in translucent resin that reveals internal glows, or as large-scale metallic sculptures capturing chiral twists. Makers produce enantiomorphic pairs for display, embed lighting for ethereal effects, or compound them for infinite-depth illusions, making them icons of advanced geometric artistry and topological wonder.
Originator of the Geometry
The self-intersecting snub quasi-regular duals, primarily the various pentagonal and hexagonal hexecontahedra, were identified as duals to the nonconvex uniform snub star polyhedra during the mid-20th-century enumeration of the 75 uniform polyhedra by H.S.M. Coxeter, M.S. Longuet-Higgins, and J.C.P. Miller in their 1954 paper, with completeness proven by John Skilling in 1975. Specific nonconvex isohedral forms, including the great pentagonal hexecontahedron and its variants, are extensively documented, named, and visualized in modern catalogs by David McCooey and others, extending the convex pentagonal hexecontahedron (dual to the snub dodecahedron) into the self-intersecting star domain.