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It seems the only parts of a dishwasher that couldn't be folded down are the heating and washing elements. A folding-dishwasher could consist of a heat-resistant cover of waterproof fabric stretched over a collapsible frame. The frame would include tracks for racks to slide out of. The racks would also be collapsible. The cover would have a watertight slide-lock zipper to close up the dishwasher before running. Heating and jet elements and spaces for detergent would be on the bottom in a tray. Electric cords and water tubes would enter through sealed holes on the bottom in the back.
This would be good for people in small homes or RVs or for others to use as an auxiliary dishwasher when their dirty dishes won't all fit in their regular dishwasher.