The paper asks how a purely mechanical circuit can pass a binary signal through connected logic and memory so that it travels only from input to output (non-reciprocal) yet can be undone by reversing the input (reversible). The authors design compliant cross-pivot flexure memory bits with equal, symmetric energy wells, chain them with a step-wise decrease in energy barrier along the signal path, and add a purpose-built AND gate whose rotation suppresses the output unless both inputs agree; a compression spring impedance-matches the gate to downstream bits. A generalized dynamics model (simulat

Materials Horizons · 2025 · Lawrence Livermore National Laboratory & UCLA

Could a material sense, decide and remember using nothing but its own bending parts? Researchers built mechanical circuits whose signals travel one way down a line of snapping springs — and undo themselves when you pull the first one back.

Try it The circuit here is made only of snapping springs and levers. Push all four inputs to make the output 1. Then pull one back, and watch the output reset itself.

Most computers move electrons. A growing field called digital mechanics asks whether moving parts can do the same job: hold a 0 or a 1, pass it along, combine it with other bits. The authors picture mechanical integrated circuits — logic, memory, inputs and outputs shaped into one piece of material — as the basis for “embedded, powerless sensors and robotic actuators.”

The obstacle is getting a mechanical signal to travel through connected logic and memory in one direction only, while still being able to take it back. The paper calls controlled, reversible propagation through interconnected mechanical circuits “a long-standing challenge in mechanical computing.” To see why it is hard, start with a toy everybody knows.

Push it. Now pull it back.

A line of dominoes is a mechanical signal: knock over the first and the “message” races to the end. Beside it runs a line of ten flexure memory bits — springy mechanisms modelled on the ones in the paper, whose sliding shuttle sits either left (0) or right (1).

A standing domino stores energy; a fallen one has spent it. Falling is easy, getting back up is not — so someone’s hand has to put every piece back. In the language of the paper, the two states sit in unequal energy wells, and such systems “require an energy input to reset.”

The researchers wanted a mechanical signal with three properties at once:

The flexure line seems to manage it. But how can something both stay put and be easy to undo — and is it really one-way? Open up a single bit.

Inside one bit: two valleys and a hump

Each memory bit in the paper is a small frame whose central shuttle is guided by eight cross-pivot flexures — thin crossed blades that bend rather than slide, so there is no friction. Squeezed by a preload spring, the shuttle has two comfortable resting places, left (0) and right (1), with an awkward squeezed position in between.

Engineers draw this as an energy landscape : a curve with two valleys separated by a hump, the energy barrier . The ball marks where the shuttle is. Drag the shuttle sideways, or use the sliders.

mechanism · drag the joint

energy landscape · faint curve = no push

Try this

1Push until the joint snaps across.Past a critical push the left valley vanishes and the ball has nowhere to stay — the bit snaps.

2Let go. Does it come back?It stays: the other valley holds it. That is memory, with no power needed.

3Lower the preload below 1.The two valleys merge into one: no squeeze, no memory. The paper’s bits are made unstressed and then “energized” by setting the preload.

4Make one valley deeper.Now one direction is downhill and the way back costs more — domino-like. Equal valleys are what make the paper’s bits reversible.

Those are the two ingredients of a reversible memory: bistability (two valleys, so there is something to remember) and symmetry (equally deep valleys, so flipping back costs no more than flipping forward). The paper’s bits get both from bending flexures — “beam bending, not buckling” — whose springiness stores energy and gives it back.

E(x) = x⁴ − 2(P − 1)·x² − a·x − F·x

x is the shuttle position (−1 is “0”, +1 is “1”), P the preload, a the asymmetry and F your push. With P below 1 the x² term holds the shuttle in the middle (one valley); above 1 it pushes the shuttle out and two valleys appear. At P = 2 the valleys sit at ±1 and the hump between them is 1 unit high; a push larger than about 1.54 units erases the valley the ball sits in. This is a simplified model in the spirit of the paper’s pseudo-rigid-body diagrams, not the authors’ equations.

Ten bits, one wave

Now link ten bits with small diamond-shaped springs, as in the paper’s chains. Push the first shuttle over its hump and its spring tugs the next one; that one snaps, tugs the next, and so on — a transition wave . At every snap, stored spring energy turns into motion and back again: watch each ball in the energy row climb its hump and drop into the other valley.

Look closely at that energy row. Each hump is slightly lower than the one before it. That built-in energy step is the paper’s trick for setting a direction.

Forecast for these settings — calculating…

Experiments

1Send a 1 all the way down the chain.Each snap releases stored spring energy that drags the next shuttle over its hump.

2Undo it from the input.Equal valleys mean a front of 0s rolls down the same staircase as a front of 1s. No hand reset.

3Try to send a signal backwards from the output end.Downstream humps are lower, so a push at the far end has to climb ever-higher humps and dies out. One-way.

4Set the energy step to 0 and push from each end.Without the step the chain treats both ends the same.

5Find both edges of the window: make the coupling so weak the front stalls, then so stiff a push from the output leaks all the way back.Too soft: the next bit never feels enough pull. Too stiff: the chain moves like one rigid rod and loses its sense of direction.

Why one way? A front moving toward the output always meets a slightly lower hump, so it keeps going; it even gets a little energy back at each step to make up for what is lost to internal friction. A push from the output end meets higher and higher humps and fades. In the paper’s words, “a unidirectional gradient of the energy barrier enables stable non-reciprocal signals and inhibits backpropagation.”

Why undoable? Every bit still has two equally deep valleys. So pulling the input back launches a front of 0s that travels down the same staircase, in the same direction, and the whole chain resets itself.

Why does the spring matter? The authors call the ratio of coupling stiffness to a bit’s own stiffness the pulse stiffness ratio. Weak coupling gives compact pulses that need little energy but travel only a few bits; stronger coupling spreads the front over more bits and needs more energy to launch. In their detailed model, signals propagate for ratios between about 0.5 and 3.5. Our ten-bit toy has its own, narrower window.

Each cell below runs two hidden experiments with the toy chain — push from the input, push from the output — for one combination of coupling and energy step. The ring marks your current sliders.

Map of this 10-bit teaching model. The authors’ detailed model finds a working pulse-stiffness region of roughly 0.5–3.5 (their supplementary Fig. S6a); our ratio is defined differently, so only the pattern — a bounded window — carries over.

Where two signals meet: the AND gate

A chain only carries a bit along. To compute, signals have to combine. The paper’s AND gate outputs 1 only if both inputs are 1. It is not assembled from smaller units: it is one purpose-built flexure mechanism — a bar held at each end by an input, with a hinged stem from its middle to the output.

The paper writes the state as (input A, input B, output). Flip the inputs in any order and fill in the truth table.

1 / 4 rows found

Experiments

1Fill all four rows of the truth table.Only the (1, 1) combination slides the bar far enough to snap the output.

2From (1, 1, 1), get the output back to 0 without touching it.Pulling either input back tilts the bar again, and the shortened stem drags the output back to 0 — reversible logic, as in the paper’s (1,1,1) → (0,1,1) → (0,1,0) simulation.

Inputs become different — a disjunct move. Only one end of the bar moves, so the bar rotates. Its hinges soak up the motion and the tilted stem reaches less far, like a leaning ladder that no longer reaches the window — the paper calls this “cosine length attenuation.” The output stays at 0, and the bent hinges store energy: an energy sink.

Inputs become equal — a conjunct move. The bar slides as a whole and pushes the stem straight into the output, which snaps and launches a wave down the output chain. The stored energy is released: an energy source.

Switch off the shortening in our teaching model and replay the experiments. A single input still cannot flip the output — half a push is not enough — but from (1, 1, 1), pulling one input back no longer drags the output down: it stays stuck at 1. In this model, the rotation’s shortening is what lets the output follow the inputs both ways.

The paper also adds a compression spring at the gate’s output so that its stiffness matches the chain downstream (“impedance matching”); without that match, the pulse would not propagate out of the gate. That spring is the subject of the next chapter.

Efficient, or forgiving?

Not all the energy that enters the gate comes out — some stays in its hinges. The paper defines the gate’s efficiency as energy out ÷ energy in, and maps which output-spring stiffnesses let a pulse through. The working region is shaped like a horn: wide at low efficiency, pinching to a point at high efficiency. Their best design reached 63% — yet the team built lower-efficiency gates “to ensure wider tolerances, accommodating variations in manufactured flexure thickness.”

Test that reasoning. Pick a design point (drag the ring or use the sliders), choose how precisely flexures can be made, and fabricate a batch.

Boundary redrawn approximately from the paper’s Fig. 3h (differential stiffness ratio rkad = 0.15). The fabrication scatter is an illustrative assumption: a flexure’s bending stiffness grows with the cube of its thickness, so a small thickness error becomes a bigger stiffness error. Useful energy = efficiency × working gates.

From meters to microns: what was built and measured

Everything you have played with so far runs on simplified models written for this page. Here is what the researchers actually built, filmed and simulated. The same flexure architecture works in polymers, metals and ceramics — slide down the scale.

A 10-bit memory-bit chain

Ten macro-scale bits, each module about 200 mm tall, filmed with high-speed video; markers tracked every shuttle. The measured position-vs-time curves match the simulation, including the small wobbles after each snap.

The front crossed all ten bits in roughly 0.3 s (read from Fig. 4f), slowing slightly along the chain.

Logic in a chain

Three input bits → AND gate → three output bits. One gate input was set in advance; pushing the other completed the gate and the output bits cascaded in turn.

Measured and simulated shuttle motions agree (Fig. 4b). In the simulation, the gate adds a short delay between the bits on either side of it and slightly weakens the signal.

Printed at the micro scale

Chains and AND gates were 3D-printed by two-photon polymerization, with 200 µm scale bars (a whole gate circuit is under half a millimetre). Chains switched from 0 to 1 bit by bit, and gates were actuated through logic states.

Because flexures bend instead of sliding, there is no friction to fight at this size.

A general model

A dynamics model tracks forces, displacements and energies across any network of bits and gates. It maps where signals propagate (the design regions) and how efficient a gate can be — the 63% best case and its narrowing tolerance.

See the real devices move: the article’s Supplementary Videos S1–S5 show the micro AND gate, the 10-bit chain, macro bit actuation, a printed micro chain, and the macro AND gate — open the article at Materials Horizons.

What this brings closer

Dominoes need a hand to stand back up. These circuits do not: push the input and the signal goes one way; pull it back and the circuit resets itself, through memory and through logic. The authors present this as “a robust foundation for the next generation of reversible, mechanical computing,” extending “toward practical, scalable applications in computation, robotics, and adaptive systems.”

Picture a sensor carved from one piece of material that registers two events, combines them and remembers the answer with no battery — the kind of embedded, powerless device the authors have in mind. Their next steps: more logic operators (OR, NOR, NAND, XOR), longer and microscale circuits, and a closer study of how the pulse stiffness ratio shapes propagation.

Push. Pull. No hand reset.

Source & credits

Hilary A. Johnson, Robert M. Panas, Amin Farzaneh, Frederick Sun, Logan Bekker, John Cortes, Melika Ahmadi, Julie Mancini, Andrew J. Pascall and Jonathan B. Hopkins. “Signal propagation in reversible digital mechanics.” Materials Horizons 12, 10623–10629 (2025). doi:10.1039/d5mh00509d

Open access under a Creative Commons BY-NC 4.0 licence. The paper’s figures and videos are not reproduced here: mechanisms are redrawn in simplified form after its Figs. 1–3, and the efficiency boundary is redrawn approximately from Fig. 3h. Quoted numbers (63% efficiency, the 0.5–3.5 design region, ~200 mm modules, 200 µm scale bars, ~0.3 s crossing time) come from the paper; the crossing time is read approximately from a plot.

All interactive simulations are simplified teaching models written for this page, not the authors’ model; they reproduce the mechanisms qualitatively. The opening circuit is drawn and simulated after the paper’s Fig. 1a–c (its two-layer, four-input AND design), with the same simplified bit and gate models used in chapters 02–04.