Geometric Palindromic Closure

We define, through a set of symmetries, an incremental construction of geometric objects in $\Z^d$. This construction is directed by a word over the alphabet $\{1,\dots,d\}$. These objects are composed of $d$ disjoint components linked by the origin and enjoy the nice property that each component has a central symmetry as well as the global object. This construction may be seen as a geometric palindromic closure. Among other objects, we get a 3 dimensional version of the Rauzy fractal. For the dimension 2, we show that our construction codes the standard discrete lines and is equivalent to the well known palindromic closure in combinatorics on words.