Raymond Aschheim presents the simplex, by nature an elementary brick in each dimension, first metric-free and topological as a complete graph, then equipped with the trivial metric generating the family of regular simplices, and finally taking values in an algebraic ring of coordinates, becoming the perfect collection of shapes in discrete spaces, and the canonical bricks for tiling aperiodically the space as quasicrystals.
We understand why the golden ratio builds the best quasicrystals, thanks to the golden simplex.
He shows concrete examples and geometries, and also gives the algebraic rules for a canonical exploration.
Results are presented for 3D and 4D.
Applications to physics are shown, linking to loops, quantum groups, and group theory.