Kuperberg’s SL(3) non-elliptic web basis consists of certain trivalent planar graphs. Fontaine--Kamnitzer--Kuperberg showed that their duals may be realized as subcomplexes of a corresponding rank 2 affine building. The result is a collection of CAT(0) triangulated surfaces related to the geometric Satake correspondence. Recently, an SL(4) web basis was introduced by Gaetz--Pechenik--Pfannerer--Striker--S. which comes with "moves". We show the moves may be understood geometrically as forming "pockets", certain highly structured 3D simplicial subcomplexes of the corresponding rank 3 affine building. These pockets have extraordinarily rich combinatorial structure. Special cases correspond to plane partitions, alternating sign matrices, tilings of the Aztec diamond, and more. Joint with Christian Gaetz, Jessica Striker, and Haihan Wu.