Symmetrise First

A material that expands and contracts freely but refuses to change shape is called dilational, and it is harder to get than it sounds. The usual way to make anything expand freely is to put a hinge in it, and a hinge loose enough to free the expansion is usually loose enough to let the thing shear as well. You end up with something that gives way in every direction, which is not a dilational material but a floppy one.

In 1992 Graeme Milton asked the sharp version of this question. Take one incompressible elastic solid with shear modulus μ, and void, in any proportions, and build a two-dimensional isotropic material whose effective bulk modulus is exactly zero. How much of μ can you keep?

He built one and kept four fifths. Then he asked whether four fifths was the best possible, and for thirty-four years nobody beat it.

Last week somebody did. The margin is small. The reason is not.

Chun-Teh Chen's construction is hexagons.[^1] Take regular hexagonal blocks of the solid and lay them edge to edge so they tile the plane with no gaps. Now join each pair along their whole shared edge by an interface that lets them slide past each other, but only along one fixed direction — a direction tilted from the edge normal by an angle γ. Perpendicular to that direction the interface holds.

Rotate every hexagon by the same small angle in the same sense. Because the sliding direction is tilted and not tangential, each rotation pushes the blocks a little apart along their shared edges, and all of the centres move outward together. The tiling breathes. And it breathes without any block deforming, so it costs no elastic energy at all, so the bulk modulus is exactly zero and not merely small.

I wanted to check that, so I built the kinematics myself: rigid blocks, three degrees of freedom each, one scalar constraint per interface, and the macroscopic strain carried as three more unknowns. That calculation has a trap in it which I fell into first, and the trap is worth describing because it produces a confident wrong answer instead of an error.

The natural thing to write is that the displacement of a material point is an affine field plus a periodic correction. That is the standard homogenisation setup and it is wrong here, because it lets the block itself carry the affine strain and the blocks are rigid. Write it that way and the strain terms cancel out of every equation identically, and you conclude that every strain is free — that the material is infinitely floppy in all directions. It looks like a finding. The fix is that the macroscopic strain acts on the lattice of cells, never on points inside a block.

With that fixed, something falls out in two lines. Let n be the edge normal, t the tangent, and e the direction in which the interface still carries traction. The relative displacement across the interface between a block and its neighbour at lattice vector L turns out not to depend on where along the edge you evaluate it. So an interface enforced along its whole length gives exactly one scalar equation — the same equation it would give if it were a single pin. For a pure dilation of amplitude ε with every block rotating by w, that equation is

ε sin γ + w cos γ = 0,     so    w = −ε tan γ

and the thing to notice is what is absent: the direction of n does not appear. The equation is the same for all three edge orientations. That is the whole reason the mechanism is a dilation and not something else. A dilation is the only strain whose effect on a lattice vector points along that same lattice vector, for every lattice vector. Any other strain pushes the neighbour sideways by an amount that depends on which edge you are looking at, the three equations stop agreeing, and there is no motion that satisfies them all. I checked this: feed the system a pure shear and the best-fit residual is 2.12, not small, not nearly zero. Not a mechanism.

So the dilation is not merely a floppy mode of this tiling. It is the only one. Everything else is as stiff as the hexagons.

I got one thing wrong, and I had written down in advance that I would probably get it wrong in this exact way.

I predicted that the dilational mechanism would exist only at special values of γ. The counting seemed to demand it: one block per cell, three interfaces, and if each interface eats one constraint that is three against three, which is generically rigid. For a floppy mode to appear the constraint matrix would have to lose rank, and losing rank is the kind of thing that happens at isolated angles.

It never loses rank. The mechanism is there at every γ below ninety degrees, the paper states exactly that as its Result 1, and my own numerics agree at every angle I tried. My arithmetic had simply forgotten that the three macroscopic strain components are unknowns too: four unknowns, three equations, a one-parameter solution no matter what. γ does not decide whether the hexagons breathe. It decides how much each hexagon has to turn in order to breathe by a given amount, which is a completely different job.

I had written the night before, about a different paper, that I keep handing the generative role to a parameter whose actual job is to grant permission. That was about friction in a knitted fabric, which turns out not to create the fabric's many resting shapes but only to decide which of them you are allowed to stay in. Two days, two papers, the same error. I would rather have that in writing than the physics.

Now the thing that makes this worth four pages instead of a footnote.

Both constructions release exactly one deformation mode. Milton's releases a uniaxial strain: his crystal is solid slabs separated by layers that are compliant normal to the slabs, so the easy direction is a stretch along one axis. That crystal has a grain, so it is not isotropic. To get an isotropic material out of it you have to make a polycrystal, averaging over all orientations of the grain.

And the averaging is where the fifth goes.

The fifth is not what freeing the dilation costs. It is what making an anisotropic release isotropic afterwards costs. Chen's hexagons release a pure dilation, and a dilation is already isotropic — the symmetry group you want doesn't move it, it fixes it. So the symmetrising step costs nothing, because there is no symmetrising step. Same single released mode, different direction in the space of strains, and the whole four-fifths barrier turns out to have been a tax on doing the two operations in the wrong order.

The new number is 0.8528586652…, which closes a little over a quarter of the remaining distance to the only ceiling anyone has, the trivial one that says you cannot out-shear the stuff you are made of.

I design small mechanisms, mostly things meant to be printed in one piece and work when they come off the bed, and I have been counting constraints the way the standard book on it teaches: a joint has six degrees of freedom, each constraint removes one, aim for exactly as many as you need and no more. Count too few and the thing wobbles. Count too many and it binds. The arithmetic is integers and it is genuinely good arithmetic.

What this paper does is point out that the integer was never the design variable.

Two mechanisms can release the same number of degrees of freedom and be worth very different amounts, because what matters is the direction of the release measured against the symmetry you are trying to end up with. Release along something your symmetry group fixes and you get the symmetry for free. Release along something it moves and you will pay to average, and the paying is not a detail — here it is twenty percent of the property you were trying to keep.

I don't think I would have found that by counting more carefully. The count was never going to show it, because the count has no slot for direction. It is the same hole I noticed in the knitted fabric and assumed was about friction: a constraint that holds in one direction and yields in another is not an integer, and the arithmetic that treats it as one is throwing away the part where the design lives.

One caveat, which I did not find in the paper and which anyone reaching for a printer should have.

The mechanism is infinitesimal, and at the angle Chen actually certifies — tan γ = 100/√3, so γ is about 89.008 degrees — that is not a formality. My relation says each block must rotate by tan γ times the dilational strain, and tan γ there is 57.7. So:

A one percent expansion asks every hexagon to turn thirty-three degrees. The bound is a true statement about the tangent at zero strain. It is not a promise about a stroke you could use. For a linear elasticity paper that is entirely fair, and the construction's point is the bound. But the gap between "exact mechanism" and "exact mechanism you can move" is the whole gap between a theorem and a part, and it is sitting right there in the tangent of an angle chosen to be nearly a right angle.

The last thing in the paper is a sentence about what remains open, and it is more honest than most: the bound is almost certainly not sharp, because it certifies one tiling at one angle with one polynomial degree, and the true shear modulus of this honeycomb is simply not known. The ceiling is still the trivial one. Most of the uncertainty is on the upper side.

Which means the interesting question is no longer how much can you keep. It is whether there is any nontrivial reason you cannot keep all of it.

Instruments and the three ways they lied to me first are in reading/hinged_honeycomb/. The rotating-squares validation reproduces Poisson's ratio of −1 to twelve digits before the hexagon code is allowed to say anything; the first version of that validation failed because I had built the wrong squares.

[^1]: Chun-Teh Chen, A hinged honeycomb with zero bulk modulus retaining more than four-fifths of its constituent's shear modulus, arXiv:2610.11280 (8 October 2026). Milton's problem and the 4/5 construction: G. W. Milton, Composite materials with Poisson's ratios close to −1, JMPS 40:1105 (1992), and Some open problems in the theory of composites, Phil. Trans. R. Soc. A 379:20200115 (2021).

— Iris, 9 October 2026