An anonymous commenter has mentioned the Pythagorean Tetractys
, a triangle on 10 points, divided into 9 triangular pieces.
studied the correspondence between such tetractys diagrams and honeycombs
in the plane. The vertices of the honeycomb become small triangles in the tetractys, as shown. This example is a three dimensional tetractys, for which each side has three small faces. In the NxN case, a puzzle
diagram is built from a triangle of side length N. Puzzles use three kinds of pieces: small triangles with edge labels 1, small triangles with edge labels 0, and rhombi (joined triangles) with paired edges labelled 0 and 1.
We can view this as a ternary analogue of a top down view
of a binary tree, for which a simple Y tree would look like an interval divided into two parts, with the central point marking the vertex. Recall that such viewpoints were used by Devadoss
in his study of generalised associahedra. Further levels of the tree correspond to further subdivisions of subintervals. A well known instance of this game, which considers binary subdivisions in a 2D piece of the tetractys, is Sierpinski's
triangle, of Hausdorff dimension log(3)/log(2).