The Triangle Sky (2 papers)
Triangle centers lifted to tetrahedral apices
Triangle Sky: stars, strands and sheets
Explore 16 triangle presets or move the base freely. Add prekite and symmetrical-apex strands, power-k and Eisenstein sheets with strands drawn on them, and relevant triangle centers. Inspect individual apices alongside the 167 constructive and six optimization stars.
The Triangle Sky
This living report develops the Triangle Sky: a framework in which triangle centers are lifted to tetrahedral apices and organized into triangle stars. It studies best-sphere-approximation and dihedral optimization stars, isotropic normal frames, hyperstars in higher dimensions, face-constrained tetrahedra, and the cascade of feet and traces that reproduce known and candidate centers in the Encyclopedia of Triangle Centers.
The Triangle Sky: Canonical Apices, Triangle Strands, and Orthocentric Completions
The pilot collects the proved core of the Triangle Sky report and adds global optimization, power-strand, defect-decomposition, and boundary results. Its theorem claims are proved in the text, while the remaining open questions are explicitly identified.