We exploit duality considerations in the study of singular combinatorial 2-discs (diagrams) and are led to the following innovations concerning the geometry of the word problem for finite presentations of groups. We define a filling function called gallery length that measures the diameter of the 1-skeleton of the dual of diagrams; we show it to be a group invariant and we give upper bounds on the gallery length of combable groups. We use gallery length to give a new proof of the Double Exponential Theorem. Also we give geometric inequalities relating gallery length to the space-complexity filling function known as filling length.