Existence of a rigidly foldable reflector gadget in origami

Determine whether there exists a rigidly foldable origami reflector gadget that replicates the Bern–Hayes reflector’s functionality of taking a single incoming pleat-encoded binary signal and producing two outgoing signals—one equal to the input with a wider pleat width and one negated with the same pleat width—while permitting a continuous isometric folding motion with all faces rigid. Either construct such a rigidly foldable reflector or prove that no such gadget can exist under these constraints.

Background

The paper constructs an origami NAND gate using pleat-encoded binary signals and several gadgets adapted from Bern and Hayes, including a reflector gadget that is used to build rotation, duplication, and NOT operations. The standard reflector accepts one incoming signal pleat and outputs two pleats: one equal to the input but with a wider pleat, and one negated with the same width.

To pursue a fully rigidly foldable origami computer, the authors note that certain components used in their design are not rigidly foldable (e.g., the triangle twist required for a 3-variable NAE clause). Although a rigidly foldable NAE-4 clause gadget can be constructed, the existence of a rigidly foldable reflector gadget remains unresolved. Without a rigidly foldable reflector, critical signal-routing and duplication operations necessary for a rigidly foldable computational system may be infeasible.