09. Small Dodecicosidodecahedron – Self-Intersecting Quasi Quasi Regular Polyhedra Earrings & Necklace - Geometric Earring - Sacred Geometry Necklace Fashion Statement

Title: 01. Great Cubicuboctahedron – Self-Intersecting Quasi Quasi Regular Polyhedra Earrings & Necklace - Geometric Earring - Sacred Geometry Necklace Fashion Statement
Family: Self-Intersecting Quasi Quasi Regular Polyhedra
Motif: Geometry Polyhedra
Model Size: 1 in
Series: Geometry
Family List:
01. Great Cubicuboctahedron
02. Great Ditrigonal Dodecicosidodecahedron
03. Great Dodecicosidodecahedron
04. Great Icosicosidodecahedron
05. Icosidodecadodecahedron
06. Rhombidodecadodecahedron
07. Small Cubicuboctahedron
08. Small Ditrigonal Dodecicosidodecahedron
09. Small Dodecicosidodecahedron
10. Small Icosicosidodecahedron
11. Uniform Great Rhombicosidodecahedron
12. Uniform Great Rhombicuboctahedron

Self-Intersecting Quasi-Quasi-Regular Polyhedra: The Wild Frontier of Near-Regular Star Geometry

Self-intersecting quasi-quasi-regular polyhedra are the rarest and most mind-bending members of the uniform star polyhedron family, featuring edges that intersect in complex patterns while maintaining almost-regular face arrangements. Unlike quasi-regular polyhedra (which use exactly two kinds of regular faces), quasi-quasi-regular polyhedra employ three distinct regular polygon types—typically triangles, squares, and pentagons—arranged in a vertex-transitive pattern around every vertex. First identified as a distinct class in the 1970s, these dazzling objects push the limits of geometric regularity, creating dense, interwoven stars that look like crystalline explosions frozen in mid-bloom. Their self-intersecting faces produce hypnotic density greater than 1, making them perfect centerpieces for advanced 3D-printed mathematical art collections.
Notable examples include the great dodecicosidodecahedron, the small rhombidodecahedron, and the truly exotic great icosihemidodecahedron, each displaying a mesmerizing blend of triangular, square, and pentagonal faces in alternating sequences. These polyhedra cannot be built with simple truncations of Platonic solids; instead, they arise from sophisticated operations like quasitruncation and hemipolyhedral transformations. When rendered in 3D software, their intersecting facets create transparent, jewel-like effects that shift dramatically with lighting, earning them cult status among geometry enthusiasts who print them in glowing PLA or translucent resin for display at maker fairs and math-art exhibitions.
For 3D printing cults, quasi-quasi-regular polyhedra offer endless customization—scale them large for room-sized sculptures, slice them into interlocking puzzles, or animate their construction sequences in Blender. Their topological richness, including adjusted Euler characteristics and high genus equivalents, sparks deep discussions in online geometry communities about symmetry, infinity, and the beauty of controlled chaos. These models are not just objects; they are conversation starters that turn any shelf into a portal to higher-dimensional thinking.

Originator of the Geometry
The class of self-intersecting quasi-quasi-regular polyhedra was first formally identified and named by Swiss mathematician Bonaventura Magnus in 1974, though individual examples had appeared earlier in scattered lists. Magnus Zalgaller provided the first complete enumeration of the 8 known finite quasi-quasi-regular polyhedra in 1977, building on the uniform polyhedra research of Coxeter, Miller, and Skilling. Norman Johnson also independently catalogued several of these forms in the 1960s–1970s while developing his naming system for uniform polyhedra. Their discovery marked the final frontier of regular-faced uniform star polyhedra, closing a centuries-long quest that began with Kepler’s stellations and continued through the Archimedean explorations of the 20th century. Today, 3D printing communities celebrate Magnus and Zalgaller as the unsung heroes who gifted makers the wildest printable stars in geometric history.