When symmetry costs

Almost every origami design is symmetric, and almost every designer will say that this is because the subjects are. A beetle has a mirror plane, so its base does, so its packing does.

That is true and it is not the whole story, because symmetry is also the single most effective way to make an intractable search manageable. Packing the circles is the hard part of design and it has no general algorithm; requiring the arrangement to be symmetric removes most of its parameters at a stroke. Whether the removed parameters were carrying anything is a separate question, and one that can be measured.

What the constraint actually is #

The search here anneals disc centres inside a unit square, maximising the radius at which none of them overlap and none leaves the sheet. That is the standard statement of the packing problem origami design reduces to: each disc is a flap, its radius is the flap’s length, and the answer is how long the limbs can be.

Requiring symmetry means the arrangement has to be unchanged by some motion of the square. Three are used here — reflection in the vertical axis, a half turn, and a quarter turn — and each is imposed by construction rather than by penalty. The search moves a few generator discs and the group produces the rest, so every arrangement it evaluates is symmetric and no annealing schedule can drift out of the set.

That detail matters more than it sounds. A search that could escape the constraint would be answering a different question, and the difference between “constrained search” and “unconstrained search with a preference” is exactly the difference between measuring what a constraint costs and measuring how strongly it was preferred.

Discs left over after the orbits are filled go where the group leaves them no choice: on the mirror line, or at the centre for a rotation. A count that leaves more over than the fixed set can hold has no symmetric arrangement at all under that group, and the search says so rather than quietly returning the free answer.

The case that settles the wording #

Two discs in a square is the smallest interesting case and it is the one that shows what is wrong with the usual claim.

The best packing of two equal discs in a square puts them corner to corner along the diagonal, and that arrangement is symmetric — about the diagonal. It is not symmetric about the vertical axis, and a search restricted to the vertical mirror cannot reach it. The shortfall is fourteen and a half per cent of the radius, which in design terms is a limb one seventh shorter than it needed to be.

So the sentence “the optimum is symmetric” is not false so much as incomplete. Symmetry is not one constraint but a choice among several groups, and choosing the wrong one is as costly as imposing none is expensive. This is the case worth carrying, because it is the case where the intuition and the arithmetic point in opposite directions and the intuition is not exactly wrong.

The fourteen and a half per cent is exact #

That number is reported as a search result and it does not have to be: both packings are small enough to solve in closed form, and doing so turns the essay’s headline into a theorem.

Free. Two discs go corner to corner on the diagonal, so their centres are at $(r, r)$ and $(1-r, 1-r)$ and touching means $\sqrt{2}\,(1 - 2r) = 2r$. That gives

$$r_{\text{free}} = \frac{1}{2 + \sqrt 2} = 0.29289.$$

Mirrored. A vertical mirror forces the two centres to the same height, at $(x, y)$ and $(1-x, y)$. Touching needs $1 - 2x = 2r$, and staying on the sheet needs $x \geq r$; together those give $r \leq 1/4$, attained.

$$r_{\text{mirror}} = \tfrac{1}{4}.$$

So the shortfall is

$$1 - \frac{1/4}{1/(2+\sqrt2)} = \frac{2 - \sqrt 2}{4} = 0.14645,$$

exactly — an irrational number the annealer reproduces to four figures and did not know.

Which makes this case a yardstick as well as an argument #

Two facts follow, and the second is the more useful.

In paper the cost is worse than in length, because area goes as the square: the mirrored pair claims $2\pi r^2 = \pi/8 = 39.3\%$ of the sheet against the free pair’s 53.9%, so the wrong mirror costs 27% of the paper while costing 15% of the limb.

And the mirrored optimum is exactly a quarter, which puts it alongside the four-disc and nine-disc cases as a value the search can be checked against rather than merely compared with. That matters here more than in the other two, because every number in this essay is a difference between two searches — and a difference of two estimates is the quantity most likely to be reporting the estimator. This one is not: one of its two terms is proved, and the search agrees with it.

Where the cost is real #

The mirror constraint is expensive at eight and nine discs — about eleven and ten per cent — and cheap at four, five and seven.

The pattern behind that is not mysterious. Four discs pack into the corners, which is mirror-symmetric already; the constraint costs nothing because it was not a constraint. Nine pack into a three-by-three grid, which is also symmetric, and yet the measured cost is ten per cent: the free search at this effort does not reliably find the grid, and the symmetric search does not find it either, for a different reason. What the number is reporting there is the difficulty of the search rather than the price of the symmetry, and the figure’s caption says so.

Where the constraint helps instead #

The more interesting half of the measurement is where the numbers go negative.

A constrained search sometimes wins. Under a half turn it wins at eight discs and at two; under a quarter turn it wins at eight and five. Not by much — fractions of a per cent — but the sign is what matters.

Nothing paradoxical is happening. The constrained search has a quarter or a half as many parameters, so the same number of restarts covers its space far more thoroughly. It is not finding better packings than exist; it is finding better packings than the free search found.

That is precisely why symmetry is a good heuristic and precisely why it is not a theorem. Its value is in the search, and a value in the search is a statement about the algorithm rather than about the geometry.

A count that suits its group is worth seeing beside one that does not.

Three groups, three different answers #

Running the same comparison under three symmetries makes a point that one of them alone would not.

The mirror is the expensive one. It is a strong constraint — it halves the free parameters and it fixes an axis — and it is the one that costs double figures at several counts. It is also the one designers actually use, because it is the one animals have.

The half turn is much cheaper and occasionally free. It halves the parameters too, but it fixes no axis: an arrangement can be rotated to suit it. The only count where it is badly wrong is three, where one disc has to sit at the centre and the other two are locked opposite each other, which is a poor arrangement for three discs by any measure.

The quarter turn is the most restrictive of the three and, at the counts where it is possible at all, the least costly. That looks contradictory and is not: a quarter turn is only possible when the disc count is a multiple of four or one more, and those are exactly the counts whose good packings are grid-like and already have it. The constraint is nearly free because it is nearly always inherited.

The pattern across all three is worth stating as a rule of thumb: a symmetry costs what it forbids, and what it forbids depends on whether the good arrangements had it anyway. For a design that means the useful question is not “should this be symmetric” but “does the tree already have this symmetry”, and the answer is usually available before any packing is attempted.

What the shortfall means in a limb #

The percentages are radii, and a radius is a flap length, so the translation into design terms is direct and unflattering.

A ten per cent shortfall in radius is a limb ten per cent shorter from the same sheet. Put the other way round, reaching the intended limb length from a constrained packing needs a sheet about eleven per cent larger on a side, which is twenty-three per cent more paper. For a model already at the limit of what a folder can handle, that is the difference between a workable design and one that needs a larger square than the shop sells.

It compounds with everything else the sheet is being spent on. A colour change costs twice what it shows; the rivers between groups of flaps cost their own width; and the packing itself is never as efficient as the hexagonal ideal. A symmetry imposed carelessly is one more claim on the same sheet, and it is the only one on that list that buys nothing when it is wrong.

Which theorem was checked, and how #

The word cost is avoided in the figures, and avoiding it is the check.

Neither number is a proved optimum. Both come out of an annealing search, and the honest description of the gap is the difference between two searches rather than the price of a constraint. The generator reports it that way, its axis is labelled shortfall of the symmetric search, and its closing line says in as many words that the comparison is between two searches.

What can be checked is that neither search is cheating. The free search is the same one used elsewhere on this site, and it is asserted against packings somebody else proved optimal: it may fall short of a published value and it may never exceed one, because exceeding one would be either a discovery or a bug and a figure is the wrong place to find out which.

Both searches are also given the same restarts and the same steps, which the generator sets explicitly rather than inheriting, because the two functions’ defaults differ and a comparison at unequal effort would measure the defaults.

The reason symmetry usually arrives for free is visible one step further down the pipeline.

What this is not about #

Symmetry as a mathematical subject — the classification of what symmetries a figure can have, the group structure, which arrangements of the plane are possible — is somebody else’s ground and this essay stays off it. No group is composed here, no orbit is classified, and the three symmetries used are named as motions of a square and nothing more.

What is claimed is narrow and belongs to design: imposing an invariance on a search changes what the search finds, in both directions, and the size of the change can be measured on the one problem origami design actually reduces to.

The idealisation should be named as well. Discs of equal radius mean flaps of equal length, and real designs have flaps of many lengths, plus rivers between groups of them, plus a tree that decides which flaps are adjacent. A symmetric subject produces a symmetric tree, and a symmetric tree makes the symmetric packing the natural one — which is the good case, where the symmetry is inherited rather than imposed.

The same inheritance applies to everything else a tree specifies.

Why real designs are symmetric anyway #

None of the above argues against symmetric design, and it is worth saying why not.

When the subject has a mirror plane, the tree has one, and the packing that follows has one for free. The constraint is not being imposed on the search — it is a property of the problem, and the search is smaller because the problem was smaller. Nothing has been given up.

The cost measured here appears when a designer imposes a symmetry the problem does not have, or the wrong one, or imposes it on a sub-problem that was not symmetric even though the whole was. That is a real failure mode and the two-disc case is its cleanest instance: an arrangement that is beautifully symmetric about the wrong axis, and one seventh worse than it needed to be.

The other reason is duller and is about people. A symmetric crease pattern is easier to fold accurately, because a folder can work both halves the same way and check one against the other. Errors that accumulate along a strip accumulate in mirror image on a symmetric design, so they show up as asymmetry, which is the one kind of error a person notices immediately.

A third reason is about diagrams rather than paper. A folding sequence for a symmetric model is half as long to write and half as long to read, because every step that applies to one side applies to the other and the instruction says so once. That is not a small saving in a discipline whose principal publishing format was, for most of its history, a numbered sequence of drawings, and it biased the repertoire toward symmetric models for reasons that had nothing to do with either geometry or zoology.

None of the three is an argument that the symmetric packing is optimal. They are arguments that it is cheaper to find, easier to fold and shorter to publish, and that combination is more than enough to explain why nearly everything in the field is symmetric without any claim about optima being true.

There is one design style where the question does not arise at all.

Who found what, and when #

The circle-packing formulation of origami design is from the 1990s and is the reason a design problem could be handed to a search at all. That the search is intractable in general, and that no algorithm for optimal packings exists, is stated here alongside the efficiency figures rather than glossed.

Symmetry as a search restriction is much older than origami design and belongs to optimisation practice generally: it is one of the standard ways of cutting a space down when the space is too large to explore. The specific measurements above are this repository’s, they are searches rather than proofs, and every one of them would be superseded by an actual optimum.

The known optima for small disc counts, which the free search is checked against, come from a long line of case-by-case proofs — several of them computer-assisted, none of them following from any general theory. That the answers are known for a handful of small counts and unknown for almost every other is the honest state of the subject, and the reason a design tool’s output is a good arrangement rather than the best one.

What a search cannot tell anybody #

A closing caution, because everything above is a search and searches have a characteristic way of misleading.

An annealing search reports the best arrangement it found, and the honest name for that quantity is the best arrangement it found. It is a lower bound on the optimum and nothing else. Two such numbers can be compared, and the comparison inherits both of their uncertainties.

That is why the figures here label their axis a shortfall rather than a cost, why both searches are given identical effort, and why the free search is checked against packings somebody else proved optimal. Those three precautions are not enough to turn a search into a theorem and they are enough to keep the comparison honest.

The place where it would matter most is precisely where the numbers are smallest. A gap of a fifth of a per cent between the free and the symmetric searches is well inside what a different random seed would produce, and no conclusion should be drawn from its sign. A gap of fourteen per cent, as at two discs, is far outside it and is real — and in that case there is a proof available anyway, because the optimal two-disc packing is known and it is on the diagonal.

The findings worth keeping from this essay are the large ones: that a mirror imposed about the wrong axis is expensive, that the expense reaches double figures at several counts, and that a constrained search can beat an unconstrained one at equal effort. The small ones are noise wearing a number.

Where the ladder goes next #

The obvious neighbouring constraint is the one nobody thinks to question at all. Every packing above is into a square, because origami paper is square. Whether it should be is measurable in exactly the same way, and for some numbers of flaps the answer is no.