In the Euclidean space R n with coordinates ( x 1 , … … , x n ) , the set V of points for which x 1 + x 2 + ⋯ ⋯ + x n = 0 forms a (hyper)plane, an (n − 1) -dimensional subspace. For every pair of distinct elements i and j of { 1 , … … , n } and every integer k, the set of points in V that satisfy x i − − x j = k forms an (n − 2) -dimensional subspace within V, and there is a unique reflection of V that fixes this subspace. Then the affine symmetric group S ~ ~ n can be realized geometrically as a collection of maps from V to itself, the compositions of these reflections.