Notation. Let R denote the set of real numbers, which form a line. Write p, q, r, s, x, y, w, h, and t for real numbers. The Cartesian product AĂB of sets A and B is the set of ordered pairs (a,b) where a is in A and b is in B, and similarly for AĂBĂC. This means RÂČ = RĂR forms a plane, on which each point has standard coordinates (x,y). The closed interval from p to q is the set [p,q] of real numbers t with p †t †q. Just as [p,q] is a region of R, a rectangle is a region of RÂČ of the form [p,q]Ă[r,s].
Definition. A pixel gif GÂ having width W px, height H px, F frames, and colors in the set C, is a map G:AâC, where A is a 3D matrix of dimensions WĂHĂF.
Definition. A gif Î of height h and width w is a map Î:[0,w]Ă[0,h]Ă[0,1]âC.
Remark. The correspondence is given by G(i,j,f) = Î(iw/W,h-jh/H,f/F), where (i,j,f) is the pixel in the ith row (from the top) and jth column of frame f.
Definition. A gif Πloops if Î(x,y,0) = Î(x,y,1).
Definition. A function T:Â RÂČâC is called a tiling if T(x+1,y) = T(x,y).
Definition. A map T*:[0,1]ĂRâC from an infinite strip of width 1 is a tile.
Proposition.There is a 1-1 correspondence between tiles and tilings.
Proof. Every tile extends to a tiling and every tiling restricts to a tile.
Definition. A map F:RÂČâRÂČ is a transformation. Write F = uĂv for u,v:RÂČâR.
Theorem. For a tiling T and transformation uĂv, the gif Î(x,y,t) = T(u+t,v) loops.
Proof. Î(x,y,0) = T(u,v) = T(u+1,v) = Î(x,y,1).