An origami Universal Turing Machine design

TL;DR

This paper investigates whether origami can support universal computation by building a UTM using a NAND gate and memory. It presents a method to realize a NAND gate on a folding pattern by reinterpreting Bern and Hayes' NAE gadget, introducing rotation, duplication, and NOT gadgets built from a reflector element, and wiring three NAE gadgets with an auxiliary constant-zero signal. Scaling this gate alongside clocked D-type flip-flops enables memory and a CPU-like architecture, suggesting that, given an infinite sheet of paper and memory, a Universal Turing Machine could be implemented in origami. The discussion compares efficiency and rigidity considerations with a related recent result (Hull 2023), highlighting design trade-offs and limitations in rigid-foldable implementations.

Abstract

It has been known since 1996 that deciding whether a collection of creases on a piece of paper can be fully folded flat without causing self-intersection or adding new creases is an NP-Hard problem (Bern and Hayes). In their proof, a binary state was implemented as a pleat, with the state corresponding to the pleat layering order; states then interact via pleat intersections. Building on some of the machinery of their result, we will present a method for constructing an origami NAND logic gate, leading to a theoretical origami Universal Turing Machine.

Figures (9)

• Figure 1: Definition of the binary states 0 and 1 on the left and right, respectively.
• Figure 2: The NAE clause of Bern and Hayes with three incoming states A, B, C, defined on the left. The clause is foldable when the signals A, B, C are not all equal and for $\theta>30^{\circ}$. On the right is the clause NAE(0,1,0) that can be folded flat.
• Figure 3: The NAE clause can be re-interpreted as two incoming states and one outgoing negated state, shown on the left. On the right the clause NAE(0,1,0) with the outgoing negated state, $\neg$A=1.
• Figure 4: Two signals A and B can pass through each other unhindered when intersecting at right angles (left), otherwise, only the pleat that zig-zags, A, can pass through unhindered (right).
• Figure 5: The reflector gadget of Bern and Hayes on the left, with one input signal and two output signals, one equal to the input but with wider pleat width and one negated of equal pleat width. It folds flat for $\theta>90^{\circ}$. On the right our drawing convention, with black for signals of interest, gray for intermediate signals, red for signals to be ignored, and the reflector gadget as a filled circle.