Electrochromic multi-layer devices with spatially coordinated switching

A multi-layer device comprising:

• a substrate having a non-rectangular shape;
• and
• a first electrically conductive layer on a surface of the substrate, the first electrically conductive layer having a sheet resistance that varies as a function of position in the first electrically conductive layer, wherein a contour map of the sheet resistance as a function of position within the first electrically conductive layer contains a set of isoresistance lines and a set of resistance gradient lines normal to the isoresistance lines, and the sheet resistance along a gradient line in the set generally increases, generally decreases, generally increases until it reaches a maximum and then generally decreases, or generally decreases until it reaches a minimum and then generally increases.