🗝️ 04. Truncated Dodecadodecahedron – Self-Intersecting Truncated Quasi Common Polyhedra Pendant and Keychain – 3D Print File for Geometric Keychain & Pendant

Title: Self-Intersecting Truncated Quasi Common Polyhedra Pendant and Keychain – 3D Print File for Geometric Keychain & Pendant – Lady’s Trend Equipment
Household Checklist:
01. Cubitruncated Cuboctahedron
02. Nice Truncated Cuboctahedron
03. Icositruncated Dodecadodecahedron
04. Truncated Dodecadodecahedron
05. Nice Truncated Icosidodecahedron

Self-Intersecting Truncated Quasi-Common Polyhedra: Deep Quasitruncations of Archimedean Stars

Self-intersecting truncated quasi-regular polyhedra symbolize a specialised subset of uniform star polyhedra the place quasitruncation—a deeper truncation that inverts faces into retrograde polygons—is utilized to quasi-regular bases just like the cuboctahedron and icosidodecahedron, yielding nonconvex, densely interwoven buildings. Key examples embrace the nice truncated cuboctahedron (quasitruncated cuboctahedron), truncated dodecadodecahedron (quasitruncated dodecadodecahedron), and nice truncated icosidodecahedron (quasitruncated icosidodecahedron), that includes mixtures of squares, hexagons, decagons, and decagrams that cross via one another in complicated patterns. These kinds keep vertex-transitivity with three or extra common face sorts alternating round vertices, producing larger density and mesmerizing self-intersections that reach the Archimedean truncation collection into star territory.
In contrast to normal truncations of Platonic solids, these come up from quasitruncation operations on quasi-regular Archimedean solids or their star extensions, leading to retrograde faces and central-passing edges that create optical depth in renders. In 3D modeling software program, wireframe or translucent views spotlight their layered intersections, typically sharing vertex preparations with different unique uniforms like rhombihexahedrons or ditrigonal variants. Their Euler traits alter for density, revealing topological richness fitted to superior symmetry research.
For 3D printing fanatics, these polyhedra present difficult but rewarding fashions—secure in strong kind however spectacular in clear resin to showcase penetrating sides, or as giant hole lattices for illuminated shows. Makers mix them into compounds, animate quasitruncation sequences, or craft jewellery that captures their dense, starburst aesthetics, making them staples in mathematical artwork and geometry cult collections.

Originator of the Geometry
The particular kinds inside self-intersecting truncated quasi-regular polyhedra, such because the quasitruncated cuboctahedron and quasitruncated icosidodecahedron, have been documented as a part of the uniform star polyhedra enumeration by H.S.M. Coxeter, M.S. Longuet-Higgins, and J.C.P. Miller of their seminal 1954 paper. Earlier particular person discoveries hint to Nineteenth-century mathematicians, with the quasitruncation idea refined within the twentieth century. Completeness of the 75 nonconvex uniform polyhedra, together with these truncated quasi-regular variants, was confirmed by John Skilling in 1975. Fashionable categorizations and visualizations are extensively featured by David McCooey in his polyhedra catalog, highlighting their place in prolonged Archimedean and star households.