Symmetry — where it appears
Named by 25 essays across 7 fields — each of them below, with the objects they name alongside it.
The star that was cut before it was proved
Fold a sheet into ten wedges, make one straight cut, and a regular five-pointed star falls out. The trick is at least two centuries old and the theorem that any straight-line drawing can be released by one cut is of 1998 — because the traditional method is not the theorem, and works only on shapes with the symmetry the folding imposes.
When symmetry costs
Design software and designers both reach for symmetry, and for a good reason: it makes the search enormously easier. It is a heuristic and not a theorem, and how much it gives away can be measured — including the case where the optimum is symmetric about an axis nobody imposed.
The biggest one that can also be folded
Which regular polygon uses a square sheet best, and which of them a fold can actually construct, are two questions with completely different pedigrees. Answered side by side over sixteen polygons, they turn out to agree — and the reason is that both are questions about the arithmetic of the same number.
Thickness has a sign
Swap every mountain for a valley and back again. Kawasaki does not notice, Maekawa gets the same condition the other way round, the lemma still asks the two creases to differ, and the layers come out mirrored. Every theorem on this site is blind to which side of the paper it is looking at — and a hinge in a panel with depth is not. The fold closes one way and jams at nothing at all the other.
The Miura folds two ways
One vertex repeated is what makes the Miura buildable: identical panels, identical creases, one degree of freedom. It is also what makes it ambiguous. At one fold angle on one crease the sheet has two folded states, differing in three letters and in half its width, and both of them close exactly — while a mesh with no two vertices alike has one.
The shapes the optimum has
Requiring a circle packing to be its own mirror image halves the number of coordinates a search has to find, so the same effort covers a much smaller space. Whether that helps depends on something the search cannot know in advance — whether the best packing was symmetric — and measured flap count by flap count the answer alternates without a pattern anybody could use.
Thirty-two rules, one object
Five hundred and twelve repeating rules for the waterbomb tessellation, fifty-six that pass on a small patch, thirty-two that pass on one containing every kind of vertex. Fold all thirty-two and compare their panels: the same panels, in the same places, with the same areas, every time. The rules are thirty-two labels on one object, and a count of them has counted the labels.
The sheet decides which points exist
Every measurement of what folding can locate has been made on a square, because origami paper is sold square. Hold the area fixed and change the proportion: one fold reaches nine marks on a square and twenty-nine on the A-series rectangle, and two folds reach 565 against 45,705. The square is the worst of five proportions at both depths, and the reason is its own symmetry.
The symmetry the letters cannot keep
Every pattern in this subject is drawn symmetric and the symmetry is always quoted of the drawing. A folded object is a drawing and a lettering together, so a symmetry survives only if the letters keep it — and the preliminary base loses every rotation while the square twist, drawn with the same eight, loses the other half.
The count counts labels
One, two, six, sixteen, fifty, a hundred and forty-four: the oldest sequence in the subject counts foldings of a strip of numbered stamps. A folded strip of blank paper has no first stamp and no top side, and neither of those operations ever leaves a folding alone — so the count of objects is 1, 2, 5, 14, 38, 120, and it is not the count over four.
The order that is its own mirror
Trying a mountain first and trying a valley first are two different searches, and on a hundred and forty-two crease patterns they cost the same number of steps — not on average, not nearly, but identically, pattern for pattern. The reason is a symmetry of every condition the subject has, and it is four lines long.
A sheet with no edge
A twist tessellation repeats, so a rectangle of it is a description of the whole plane rather than a piece of paper. Joining the rectangle's opposite sides makes that explicit and produces an object every gate in this collection can read: twenty-five drawn panels become sixteen, forty crease pieces become thirty-two, sixteen vertices are all interior, and the three counts add to nothing.
Folding it flat is one similarity
Where a cell of paper goes when a twist tessellation collapses is a scale and a turn: multiply the plane's lattice by 0.410373441 and rotate it by 36.62°. That is the answer on the square grid, the triangular grid, the honeycomb, the elongated triangular tiling and the rhombille alike, agreeing to eight figures — while the collection's other answer to how much smaller it gets gives those five tilings five different numbers.
The turn a column costs
The Yoshimura's drawing repeats every column. Folded flat, it does not: the fold carries one column onto the next by a turn of two hundred and forty degrees, so the folded state repeats every third column and not before. A pattern has two periods and only one of them has ever been written down.
The period nobody measured
Every repeating pattern in this collection has its drawn period recorded, because a drawing cannot be generated without one. Its folded period is recorded nowhere, and on one of the families measured the two differ by a factor of three — which means the number that has always been quoted is the wrong one for anything about the folded object.
The seam that is not a symmetry
Gluing a cell's edges looks like a symmetry of the drawing and is not. It is an instruction about which points of the paper are the same point, the drawing has to agree with it along the whole of a glued edge, and a rectangle that is not a period of the pattern does not glue at all — which turns out to be the only real restriction on which cylinders exist.
The symmetry a gluing adds
A patch of a tessellation has whatever symmetry its outline allows — a few reflections, a rotation or two. Glue its edges and it acquires translations, and a lettering of the glued sheet has to be invariant under them. That is a much stronger requirement than a lettering of the patch, and it is why one answer covers every patch at once.
The square is in the answer
The largest regular polygon a square sheet holds is not increasing in the number of sides, and the octagon's win is the striking part: it uses 82.8% of the paper against the twelve-gon's 80.4% and the hexagon's 69.6%. Run the same census over rectangles and the octagon's advantage is gone — on every proportion tried the hexagon leads, and the order among the even-sided polygons reverses outright.
How many wedges the paper allows
A k-pointed folk star is folded into 2k equal wedges and cut once, so the scissors go through 2k thicknesses of the sheet. The geometry is indifferent to k and the paper is not: at a stack a pair of scissors will shear cleanly in one pass, ordinary copier paper takes a three-pointed star and nothing more, and the five-pointed one everybody knows needs washi or thinner.
Fourteen states are one pile
A machine that folds every layer at once reaches every folded state of a strip of six equal stamps and misses fourteen piles at seven. The fourteen are not fourteen things. Taking a pile's bottom stamp and putting it on top maps foldings to foldings, so the 462 piles of seven stamps fall into 33 classes of exactly fourteen, and the missed piles are one whole class: the pile 0 6 1 2 3 4 5 — an accordion of five stamps with the last stamp wrapped round it and slid into the fold that holds the first — seen from each of its seven stamps. At eight stamps the machine misses 64 piles, and they are exactly the piles that leave that one when an end stamp is removed.
Nine stamps have piles of their own
A machine that folds every layer at once reaches every folding of six equal stamps, misses one class of fourteen piles at seven, and at eight misses exactly the piles that shrink to that class when an end stamp is taken off. The prediction was that nine would do the same. Counted directly, nine equal stamps have 4,536 foldings and the machine reaches 4,052. Of the 484 it misses, 324 are inherited from eight and 160 are new — among them the nine-stamp tucked pile. And the symmetry that made the misses whole classes breaks: at nine stamps, turning a pile's bottom stamp to the top takes 52 piles the machine reaches to piles it cannot.
A sheet that turns with the twist
Holding a different panel of a folded twist can give a different pair of shrink factors: on a square sheet the three- and six-sided twists gave two pairs each and the five-sided twist five. Cut each twist from a regular sheet of its own number of sides and every panel gives the same pair. The reason is a count of rotations. A K-sided twist's panels hold its folded state in frames that are multiples of 180⁄K or 360⁄K degrees; a sheet that shares g of the twist's rotations makes the widths agree in every frame those rotations reach, with the half-turn a width cannot see thrown in. On twenty-two pairs of twist and regular sheet, that count gives the exact number of answers every time — a hexagon twist on a triangle gives one, a pentagon twist on a square five.
Equal effort was counted in steps
A circle-packing search required to be symmetric was found to beat a free search of the same effort, and the reason given was that a smaller space is easier to search. Past eleven discs the test gets sharp: the free search falls behind the best packings known, 0.13 per cent at ten discs and 6.07 at twenty, and a symmetric search leads it by as much as 4.18. But tighten the free search's packings to the arrangement they were heading for and most of that shortfall disappears — at twenty discs, all of it. Give the free search twice the steps and the lead reverses. The effort was equal in steps and unequal in what a step does: a symmetric search moves half as many coordinates, so each is moved twice as often.
No placement cleans every drawing
Five of a hundred and twenty clipped twist patches carry fragments — creases a fraction of a millimetre long, left where the sheet's edge cut through the pattern — and the proposed repair was to nudge the sheet. A fragment turns out to be a corner of a twist polygon lying within a seventh of a millimetre of the rim, and moving the pattern 0.127 mm clears any one drawing. No single movement clears them all. Laid at random, a sheet puts a fragment on one drawing in nine; of twenty-four random placements the cleanest still leaves three drawings with one, and the centred placement every drawing was made at leaves five. A fragment is not a fault of the construction. It is the threshold meeting a placement, and the only clean set of drawings is one placed drawing by drawing.
Clamped, a stack counts in pairs
A Kresling tube of four courses, its ends free to turn, can be pressed and pulled into all sixteen of its words. Clamp the ends so they cannot turn and the courses stop being independent switches, because a course that clicks shut turns by a fixed angle — eighty degrees here, whatever height it was made at — and the others must give it back. With every course of one hand the stack does not click at all, at more than twice the force that shuts any course free. With two courses of each hand it clicks in pairs, one of each hand together, so that their turns cancel: every word it reaches has as many right-handed courses shut as left-handed, and it reaches exactly four. The clamp turns a sixteen-word register into a two-switch one whose every state leaves the ends where they were.
Named alongside it
The objects these essays reach for when they reach for this one.
PeriodicityTessellationOptimalityCrease patternGluingShrinkageUnit cellBoundaryCircle packingThe folded periodFolded stateMap folding