Tent Tile

A tent tile is a compact set defined by the contracting algebraic conjugates of the branches of an asymmetric tent map associated with a special Pisot number that is an algebraic unit (Scheicher et al. 2026). For a Pisot number , put . The number is special when is also a Pisot number, and the asymmetric tent map is

(1)

There are exactly 11 special Pisot numbers (Smyth 1999, p. 1). The exceptional value 2 is not an algebraic unit. The remaining ten values yield two one-dimensional, six planar, and two three-dimensional tent tiles (Scheicher et al. 2026).

In the planar case, let be the algebraic conjugate of with positive imaginary part, and put . The two contraction mappings and define the tent tile by the iterated function system

(2)

Each tent tile is the closure of its nonempty interior. The planar examples therefore have positive area, despite their fractal boundaries (Scheicher et al. 2026).

The six planar tent tiles are illustrated above in reading order. The table gives the corresponding minimal polynomials of , the Hausdorff dimension of the boundary, and the periodic tiling constructions established by Scheicher et al. (2026). A half-turn means a rotation through , as in , rather than a reflection in a line. The constructions using half-turns do not assert that half-turns are necessary. An unresolved row means that the paper establishes neither of these types of point lattice tiling.

figure	parameter	minimal polynomial	(approx.)	periodic construction
1			1.10026	translations
2			1.02952	translations and half-turns
3			1.37858	unresolved
4			1.70018	translations and half-turns
5			1.25074	translations and half-turns
6			1.92089	unresolved

The parameters in the second and third rows are the supergolden ratio and plastic constant, respectively. Tent tiles are related to Rauzy fractals through substitution systems, but a common algebraic number does not identify the same attractor. For example, the Rauzy fractal studied by Gjini (2001, pp. 305-306) also uses , but its boundary has Hausdorff dimension approximately 1.47131 (Gjini 2001, pp. 317-319). It is therefore not similar to either the second or the third tent tile.