An Aperiodic Set of Four Edge-Marked Equilateral Triangles

We present Q, an aperiodic tileset consisting of four reflectible equilateral-triangle prototiles with standard nearest-neighbor edge markings, obtained by cutting the ten-kite Turtle aperiodic monotile of Smith, Myers, Kaplan and Goodman-Strauss along its own internal kite-grid lines. Aperiodicity is established via mutual local derivability (MLD) with the Turtle itself. Unlike the translation-only convention of Wang tiles, every prototile of Qis a priori free both to occupy any of six rotations and to be reflected; the Turtle’s own heterochirality – every one of its tilings mixes both orientations – is inherited directly by Q, and the proof of Section 6 shows this costs nothing: each Turtle contains exactly one T0 piece, whose edge labels record the Turtle’s position, rotation and chirality, so a tiling by Q decodes to a tiling by the Turtle with no ambiguity at all. We also prove a Reduction Lemma: for any reflectible regular m-gon, m∈{3,4,6}, every geometric edge marking reduces without loss to a finite combinatorial alphabet (class, flavor, pairing and partnership) on which matching is a single involution, so that the tilesets of each size form a finite space. Using it, we give a computer-assisted proof of minimality: an exhaustive classification of all tilesets of at most three edge-marked triangles, under every assignment of prototiles as reflectible or chiral, finds each of them (over 176 million in all) to be periodic or unable to tile the plane, with every verdict independently certified. Hence no aperiodic set of edge-marked equilateral triangles has fewer than the four prototiles of Q.