The symmetry the letters cannot keep

Symmetry is the first thing anybody says about a crease pattern and it is nearly always said about the drawing. The preliminary base is both diagonals and both midlines of a square, so all eight symmetries of the square carry it to itself. A twist unit has the same eight. A Miura has a mirror. Those are facts about lines on paper.

A folded object is not lines on paper. It is lines on paper plus a lettering, and a symmetry of the drawing survives into the object only if it also carries the lettering to itself — a mountain onto a mountain, a valley onto a valley, everywhere. So there are two groups, the second sits inside the first, and this site has quoted the first throughout without once measuring the second.

When symmetry costs and the shapes the optimum has both measured symmetry in the packing — a search over where the circles go. This is symmetry in the pattern, and the answer is not the same kind of answer.

What a symmetry of a pattern is here #

One of the eight maps of the square — four rotations and four reflections — about the pattern’s own centre, required to carry the vertex set to the vertex set and the edge set to the edge set, with the boundary going to the boundary.

That last clause matters. A map that took a crease onto a raw edge would be carrying the sheet somewhere else, and there is no sense in which the result is the same pattern. So the assignment is checked to the extent of is this a crease or is this the edge of the paper, and no further — whether a mountain goes to a mountain is precisely the question being asked afterwards and must not be built into the test.

The eight rather than the whole plane group, because every pattern printed here sits on a square or a rectangle and the interesting maps are the ones that keep the sheet. A hexagon twist has a six-fold symmetry the square’s group cannot see, so this test reports the subgroup it can find and says so — which understates rather than overstates, and is the right direction for a test to be wrong in.

The base keeps its mirrors and loses every turn #

The preliminary base admits 112 letterings that satisfy every condition at its single interior vertex. Sorted by which symmetry each keeps:

No rotation survives at all. Not one lettering out of 112 is carried to itself by turning the pattern through a quarter, a half or three quarters of a turn — and the half turn is the one that would be most expected to survive, since a half turn does not exchange the two sides of the paper.

The mirrors do survive, and in a tenth of the letterings each. So the base’s folded objects have a mirror available to them and no rotational symmetry available to them at all, and that is a fact nobody could read off the drawing, which has both.

The reason is Maekawa’s condition meeting the pattern’s own structure. The eight creases at the vertex must split five and three; a quarter turn permutes them in two four-cycles, so a lettering it fixes must be constant on each cycle, giving four and four or eight and none — and neither of those is five and three. The rotations are excluded by arithmetic, not by accident.

Every row of that table can be derived #

The rotations are explained above by an arithmetic argument and the mirrors are left as a measurement. The same argument gives all eight entries exactly, and it is worth running because it turns the table from a report into a consequence.

A symmetry permutes the eight crease-rays at the base’s centre. A lettering fixed by it must give the same letter to every ray in a cycle of that permutation, so the letterings it fixes number two to the number of cycles — and among those, the ones that survive are the ones Maekawa admits, which at a degree-eight vertex means five mountains and three valleys or the reverse.

The quarter turns. Rotating the eight rays by ninety degrees is two four-cycles. Two cycles gives four fixed letterings, whose mountain counts are nought, four, four and eight. None is five or three, so the answer is nought.

The half turn. Rotating by a hundred and eighty is four transpositions. Sixteen fixed letterings, and every one of them has an even mountain count, because the rays come in pairs that must agree. Five and three are odd, so again nought.

The mirrors. A mirror through a midline sends one ray to itself, its opposite to itself, and pairs up the other six — two fixed rays and three transpositions, so five cycles and thirty-two fixed letterings. Writing $a$ for the transpositions lettered mountain and $b$ for the fixed rays lettered mountain, the mountain count is $2a + b$ with $a$ at most three and $b$ at most two. That equals five for $a = 2$, $b = 1$, in three times two ways, and three for $a = 1$, $b = 1$, in another six. Twelve, which is the measured entry.

A diagonal mirror has the same cycle structure — one ray fixed, its opposite fixed, three pairs — so it gives twelve as well, and all four mirrors agree.

And why every entry is even #

The essay notes that nothing was arranged to make the counts even, and there is a reason rather than a coincidence.

Reversing every letter carries a folding to a folding, and it commutes with any permutation of the creases: reflecting and then reversing gives the same lettering as reversing and then reflecting. So the set of letterings a symmetry fixes is closed under reversal — and reversal fixes nothing, since a lettering equal to its own reverse would give some crease both letters.

A set closed under a fixed-point-free involution pairs off, so every entry in the table is even, necessarily. Twelve, thirty-two, sixteen, four and nought all are, and a count coming back odd would be a fault in the enumeration rather than a finding about a pattern. That is the cheapest possible check on the whole measurement and it costs nothing to apply.

The twist keeps its turns and loses half its mirrors #

Now the same test on a pattern drawn with exactly the same eight symmetries, and it loses a different half.

The square twist admits 256 letterings. Of those, four keep a quarter turn, sixteen keep the half turn, thirty-two keep each diagonal mirror — and not one keeps either of the mirrors through the sheet’s own edges.

Two patterns, both carried to themselves by all eight symmetries of the square, and they lose opposite halves. The base keeps the four mirrors and no rotation; the twist keeps the rotations and only two of the mirrors.

Which part of a drawing’s group survives is not readable off the group. It is a fact about the pattern’s structure, and it has to be computed.

Why a twist loses those two mirrors and not the others #

The twist’s answer has a name and this site has already reached it from another direction.

A square twist’s central square rotates as the sheet closes, and which way it turns is a property of the lettering. The four creases forming that square are buried — each has an interior vertex at both ends — so their letters are fixed when the pattern is drawn and cannot be changed by hand. That is the handedness.

A mirror through one of the sheet’s edges reverses the sense of the central rotation, so it exchanges a left twist with a right twist. No lettering can be carried to itself by it, and none is.

A diagonal mirror also reverses the sense — every reflection does — and yet thirty-two letterings keep each of them. Which means the diagonal mirrors are not acting on the pattern the way the edge mirrors are: the diagonal reflection permutes the twist’s four creases with an odd permutation that composes with the reversal of sense to give back the same lettering, and the edge reflection does not.

That is the honest account: the effect is not chirality alone, it is chirality composed with how each reflection permutes the creases, and the two mirrors of a square twist differ in that permutation. The square that turns is where the turning itself was established. Which is a very good reason to compute the table rather than to reason about the picture.

Every symmetry a tessellation has, gone #

The patterns above are small enough to enumerate. The tessellations are not — a Miura patch has thirty-eight free creases and a Yoshimura eighty-six — so they are sampled, and the sampling can only ever show that a symmetry is sometimes kept.

It shows the opposite, uniformly.

Two hundred independent letterings of the Miura keep its single mirror none of the time. Two hundred of the Yoshimura keep its mirror none of the time. Two hundred of the tapered corrugation, the same. And the waterbomb tessellation, which is drawn with four symmetries — the identity, a half turn and two mirrors — keeps none of the three in any of two hundred draws.

A sample of two hundred that finds nothing is not a proof that nothing exists, and the essay says so. What it is, on four separate patterns, is a strong indication that the effect gets worse with size — which is what the arithmetic suggests it should. A symmetry has to be kept on every crease at once, so the chance of a lettering keeping one falls off with the crease count, and a tessellation has a great many creases — and a bigger patch has proportionally more of them buried.

The one that has no symmetry at all #

The control is the fold-and-cut triangle, and it earns its place by being dull.

It is drawn on a square, and none of the eight maps carries it to itself except the identity. The test reports one symmetry, the one is the identity, and every one of its thirty letterings keeps it.

Without that row the whole table would be open to the reading that the test finds symmetries because it is looking for them. It finds exactly one where there is exactly one, and it reports the trivial answer as trivial rather than as a result.

How much symmetry is lost, counted rather than described #

The tables above are per-symmetry. Read the other way — per lettering — they say something a designer can act on.

Of the preliminary base’s 112 letterings, how many keep any non-trivial symmetry at all? The four mirrors are kept twelve times each and no lettering keeps two of them at once, because two mirrors compose to a rotation and no rotation is kept. So forty-eight of the 112 letterings keep exactly one symmetry, and sixty-four keep none.

Fifty-seven per cent of the base’s folded candidates are completely asymmetric objects made from a pattern with eight symmetries.

The square twist is the same shape of answer with different numbers. Its 256 letterings include sixteen keeping the half turn, four keeping each quarter turn, and thirty-two keeping each diagonal mirror; the four that keep a quarter turn keep the half turn too, since a quarter turn squared is a half turn. Counting carefully, sixty-four of the 256 keep something and 192 keep nothing.

So on both patterns, the typical folded object has no symmetry at all. Symmetry in a folded model is a minority outcome of a symmetric drawing, and it is a minority that has to be selected for.

The one place symmetry is guaranteed #

There is a symmetry every lettering keeps and it is not in the table, because it is not one of the eight.

Turning the model over — swapping every mountain for every valley — carries a folding to a folding, always, on every pattern. It is not a map of the plane at all: it changes no vertex position and permutes no edge. It is a relabelling, and it is the one operation whose survival needs no computation.

That is worth stating beside the rest because it is the reason a folded model’s symmetry is a slippery thing to talk about. A model and the same model seen from behind are the same object with every letter reversed, so any statement about which symmetries survive is a statement about letterings up to that reversal — and the tables above count both members of each reversed pair, which is why every number in them is even.

Both the preliminary base’s twelve and the square twist’s thirty-two are even, and so is every other entry. Nothing was arranged to make that true.

What a designer should take from it #

Three things, and the first is the one that will save somebody an evening.

Drawing a symmetric pattern does not produce a symmetric model. If the model is meant to have a mirror, the lettering has to have that mirror, and on some patterns no lettering does. A designer who wants a mirror-symmetric twist has to build it out of two twists of opposite hand rather than out of one twist drawn symmetrically.

Symmetry of the drawing is still worth having, for the reason the packing rungs gave: it halves the number of coordinates a search has to find, and it makes a pattern possible to check by hand. Those benefits are about the design process and survive intact.

And the two groups can be measured, cheaply, on any pattern small enough to enumerate. The test is a permutation of the edge list and a comparison of strings. It costs less than drawing the pattern.

Against the packing, one more time #

This anchor’s first two rungs were about symmetry in a circle packing, and it is worth being explicit about how differently the two questions behave, because they use one word.

A packing’s symmetry is a property of a search’s answer. Requiring the packing to be its own mirror image halves the coordinates the search has to find, so the same effort covers a smaller space — and whether that helps depends on something the search cannot know, namely whether the best packing happened to be symmetric. Measured flap count by flap count, the answer alternates without a pattern anybody could use.

A pattern’s symmetry is not a property of an answer. It is a property of a drawing, and what this essay measures is what happens to it downstream. There is no search, no optimum, and no question of whether symmetry helps: the drawing has the symmetry or it does not, and the letters keep it or they do not.

The two do interact once, and the interaction runs the wrong way for a designer. A symmetric packing produces a symmetric arrangement of molecules, which produces a symmetric crease pattern — and the crease pattern’s symmetry is then subject to everything above. So the search’s symmetry is inherited by the drawing and lost again by the letters, and a design built symmetric all the way from the packing can still fold into an object with nothing left of it.

What is not claimed #

The group here is the square’s, so a pattern with a finer symmetry has that symmetry undercounted — the hexagon twist reports two where it visibly has more, and the essay’s numbers for it are about the subgroup this test can see.

And the letterings counted are the locally admissible ones: satisfying developability, Kawasaki, Maekawa and the big-little-big lemma at every interior vertex. Whether any of them folds globally is NP-hard, so a symmetry kept by twelve letterings is a symmetry kept by twelve candidates, and how many of those are objects is not decided here.

Neither caveat moves the finding. A symmetry kept by zero candidates is kept by zero objects, and the three rotations of the preliminary base and the two edge mirrors of the square twist are all zero.