Homotopy Types of Squared Grids in the Torus with the Word Metric

The Vietoris–Rips complex of X at scale r, denoted VR(X; r), is the simplicial complex with vertex set X that contains a finite simplex σ ⊆ X if and only if diam(σ) ≤ r. A persistent diagram, a well-known algebraic topology technique, is used to understand the nature of  Vietoris-Rips complexes (at different scales r). These diagrams give a clue to what the homotopy types of a metric space are. Classifying homotopy types for a given metric space is a popular task in algebraic topology (with many unsuccessful cases - open questions).
 
Now, let X be a grid of dimension n by n equipped with the word metric. Embed this metric space in the torus. What are VR(X,r) for different scales r?