An aperiodic tiling of variable geometry made of two tiles, a triangle\n and a rhombus of any angle

Aperiodic tiling is a well-know area of research. First developed by\nmathematicians for the mathematical challenge they represent and the beauty of\ntheir resulting patterns, they became a growing field of interest when their\npractical use started to emerge. This was mainly in the eighties when a link\nwas established with quasi-periodic materials. Several aperiodic tilings made\nof two tiles were discovered, the first one being by Penrose in the seventies.\nSince then, scientists discovered other aperiodic tilings including the\nsquare-triangle one, a tiling that has been particularly useful for the study\nof dodecagonal quasicrystals and soft matters. Based on this previous work, we\ndiscovered an infinite number of aperiodic tilings made of two tiles, a\ntriangle and a rhombus of any angle. As a result, a variable geometry, i.e.\ncontinuously transformable, aperiodic tiling is proposed, whose underlying\nstructure is dodecagonal. We discuss this limit case where the rhombus is so\nthin that it becomes invisible. At the boundary of this infinite space of\ntilings are two periodic ones; this represents a uniform view of periodic and\naperiodic tilings.\n