{\displaystyle \operatorname {MCG} _{\operatorname {Ho} }(\mathbf {T} ^{n})=\operatorname {Aut} (\pi _{1}(X))=\operatorname {Aut} (\mathbf {Z} ^{n})=\operatorname {GL} (n,\mathbf {Z} ).} Since the torus is an Eilenberg–MacLane space K(G, 1), its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); all homotopy equivalences of the torus can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism.
A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows:

  
    
      
        (
        x
        ,
        y
        ,
        z
        ,
        w
        )
        =
        (
        R
        cos
        ⁡
        u
        ,
        R
        sin
        ⁡
        u
        ,
        P
        cos
        ⁡
        v
        ,
        P
        sin
        ⁡
        v
        )
      
    
    {\displaystyle (x,y,z,w)=(R\cos u,R\sin u,P\cos v,P\sin v)}
  where R and P are positive constants determining the aspect ratio.