Periodic Functions

Abstract

The G _s may be regarded as tori. The 1-dimensional torus G₁ may be obtained by identifying two end-points of the unit interval 0 ≤ x₁ ≤ 1 and G₂ by indentifying 2 opposite sides of the unit square 0 ≤ x₁ ≤ 1, 0 ≤ x₂ ≤ 1. In general, G_s is obtained by identifying the 2s opposite surfaces of the s-dimensional unit cube, i.e., the points

$$ \left( {{x_1},...,{x_{v - 1}},0,{x_{v + 1}},...,{x_s}} \right) $$

and

$$ \left( {{x_1},...,{x_{v - 1}},1,{x_{v + 1}},...,{x_s}} \right) $$

are identified, where 1 ≤ v ≤ s.