Given β ∈ (1, 2) the fat Sierpinski gasket is the self-similar set in generated by the iterated function system (IFS) Then for each point there exists a sequence such that , and the infinite sequence (di) is called a coding of P. In general, a point in may have multiple codings since the overlap region has non-empty interior, where Δβ is the convex hull of . In this paper we are interested in the invariant set Then each point in has a unique coding. We show that there is a transcendental number βc ≈ 1.552 63 related to the Thue–Morse sequence, such that has positive Hausdorff dimension if and only if β > βc. Furthermore, for β = βc the set is uncountable but has zero Hausdorff dimension, and for β < βc the set is at most countable. Consequently, we also answer a conjecture of Sidorov (2007). Our strategy is using combinatorics on words based on the lexicographical characterization of .