by $\left[ \mathbb{R}^{n+d}\right] _{\Sigma} $, that is, a $n+d$-torus. In the topic of point patterns the above factor map is called the *parametrization map*, and shows in particular that any model set of $\mathbb{X}$ is an inter model set in the sense of definition \[def:inter.model.set\]. In fact, the collection $\mathbb{X}$ of model sets of a given cut $\&$ project scheme and window is precisely the collection of *repetitive* inter model sets arising from this data (see for instance [@Sch]).

An explicit example {#subsection:explicit.example}
-------------------

A well-known example of model set is the vertex point pattern of the famous *Ammann-Beenker tiling*, from which an uncolored local pattern about the origin shows up as

(-10.22,-5.98)(7,5.96)

(0,0) (0.05,0.2) (1,0) (1.25,-0.2)[$e_1$]{} (0.7,0.7) (0.85,0.8)[$e_2$]{} (0,1) (-0.2,1.25)[$e_3$]{} (-0.7,0.7) (-1.2,0.8)[$e_4$]{}

(0,0)(1,0) (0,0)(0,1) (0,0)(-1,0) (0,0)(0,-1) (1,0)(1.71,0.71) (1,0)(1.71,-0.71) (0,-1)(0.71,-1.71) (0,-1)(-0.71,-1.71) (1.71,0.71)(2.41,0) (2.41,0)(1.71,-0.71) (0.71,-1.71)(0,-2.41) (-0.71,-1.71)(0,-2.41) (-1,0)(-1.71,-0.71) (-1.71,-0.71)(-2.41,0) (-2.41,0)(-1.71,0.71) (-1.71,0.71)(-1,0) (0,1)(-0.71,1.71) (-0.71,1.71)(0,2.41) (0,2.41)(0.71,1.71) (0.71,1.71)(0,1) (0,0)(0.71,0.71) (0.71,0.71)(1.71,0.71) (0.71,0.71)(0.71,1.71) (0,0)(-0.71,0.71) (-0.71,0.71)(-0.71,1.71) (-0.71,0.71)(-1.71,0.71) (0,0)(-0.71,-0.71) (-0.71,-0.71)(-1.71,-0.71) (-0.71,-0.71)(-0.71,-1.71) (0,0)(0.71,-0.71) (0.71,-0.71)(1.71,-0.71) (0.71,-0.71)(0.71,-1.71) (1.71,0.71)(1.71,1.71) (1.71,1.71)(0.71,1.71) (-0.71,1.71)(-1.71,1.71) (-1.71,1.71)(-1.71,0.71) (-1.71,-0.71)(-1.71,-1.71) (-1.71,-1.71)(-0.71,-1.71) (0.71,-1.71)(1.71,-1.71) (1.71,-1.71)(1.71,-0.71) (2.41,0)(3.41,0) (0,2.41)(0,3.41) (-2.41,0)(-3.41,0) (0,-2.41)(0,-3.41) (0,3.41)(0.71,4.12) (0.71,4.12)(0,4.82) (0,4.82)(-0.71,4.12) (-0.71,4.12)(0,3.41) (-3.41,0)(-4.12,0.71) (-4.12,0.71)(-4.82,0) (-4.82,0)(-4.12,-0.71) (-4.12,-0.71)(-3.41,0) (0,-3.41)(-0.71,-4.12) (-0.71,-4.12)(0,-4.82) (0,-4.82)(0.71,-4.12) (0.71,-4.12)(0,-3.41) (3.41,0)(4.12,-0.71) (4.12,-0.71)(4.82,0) (4.82,0)(4.12,0.71) (4.12,0.71)(3.41,0) (0,2.41)(1,2.41) (1,2.41)(1.71,1.71) (1.71,1.71)(2.41,1) (2.41,1)(2.41,0) (2.41,0)(2.41,-1) (2.41,-1)(1,-2.41) (1,-2.41)(-1,-2.41) (-1,-2.41)(-2.41,-1) (-2.41,-1)(-2.41,1) (-2.41,1)(-1,2.41) (-1,2.41)(0,2.41) (3.41,0)(3.41,1) (3.41,1)(2.41,1) (1,2.41)(1,3.41) (1,3.41)(0,3.41) (-1,2.41)(-1,3.41) (-1,3.41)(0,3.41) 