Further Triangle tilings

Here we give a more complete reckoning of the conjecture discussed in “Regular Production Systems and Triangle Tilings ” [16] Here we discuss which triangles do, and which don’t, admit a tiling of H 2, E 2, and S 2. These notes are meant to pick up as “Regular Production Systems and Triangle Tilings ” [16] leaves off. We pause for a conjecture and some additional nomenclature: Let T be any triangle in X = H 2, E 2, S 2. A configuration by T is a collection of congruent copies of T, for each pair of which meet edge-to-edge and vertex-to-vertex (that is, each pair has disjoint interiors and the vertex of one is not in the interior of the edge of another). A tiling by T is a configuration that covers all of X. A triangle admits a tiling iff there exists such a tiling. A vertex arrangement of T is a configuration by T all meeting meeting at and surrounding a point. Inductively, we define n-nions (“onions”) of T: a 0-nion is a vertex arrangment by T. A n-nion is any configuration C by T that (a) contains an (n − 1) − nion C ′ in the interior of C and (b) every copy of T is either in C ′ or incident to both C ′ and the boundary of C. That is, an n-nion really is a kind of onion, the union concentric layers of triangles