Uniaxial base — where it appears
Named by 16 essays across one field — each of them below, with the objects they name alongside it.
A flap costs a circle
A flap of a given length uses up every point of the sheet within that distance of it. Two flaps whose circles overlap are asking for the same paper twice — and that one observation turned origami design from an art into an algorithm.
What joins the flaps
Circles are the rule for flaps that all hang from the same point. As soon as two groups of flaps hang from different places, the paper between them has to be paid for too — and the payment is a strip whose width is the distance between them.
Every flap on one axis
The tree method does not design a shape. It designs a base whose flaps all lie along a single line — and that restriction, which is almost never stated aloud, is what makes the circle argument true.
From a packing to a crease pattern
The circles say where the flaps are. They do not say where to fold, and the step in between is a construction rather than a search — two families of crease, both determined by the packing, neither of them visible in the picture of the discs.
Paying in paper
A feature added to a finished design costs exactly the paper inserted for it. Cut the crease pattern along a line, slide in a strip, and every existing crease continues across it unchanged — so the bill is the strip's width times the length of the cut, and there is no second term.
The corner is worth four times the middle
A flap consumes every point of paper within its own length of its tip — but only the paper that is actually there. On an edge of the sheet that is half a disc, and in a corner a quarter, so the same flap costs four different prices depending on where it stands and the boundary is the cheapest paper on the sheet.
Two packings, one radius
A packing search reports a number, and the number is not the design. What a crease pattern is built from is the graph of which discs touch which — and at five and six flaps, runs of the same search that agree about the best radius to four decimal places come back with contact graphs that are provably not the same graph. The answer an optimiser gives has not determined the pattern it is supposed to have found.
A hole is cheap paper
A flap claims every point of paper within its own length of its tip, but only the paper that is actually there — so an edge is half price and a corner a quarter. A hole in the middle of the sheet is more edge, and holding the area fixed, a square with a hole in it is cheaper paper than a solid square at every flap length measured.
Spending the cheap paper
A hole makes a sheet cheaper by the square inch, because a flap against its rim claims only half a disc. Whether a design can spend that was left open, because no packing search here could express a region that is not paper. It can now, and the mechanism turns out not to be the discount at all: across a hole, two flaps may overlap, and two tips three fifths of a sheet apart carry flaps of 0.500 rather than 0.300.
A tree cannot argue
A molecule fills a polygon with creases taken from its straight skeleton, and a straight skeleton is a tree. So a molecule's panels have almost no closed chains for its letters to contradict themselves round — one to three, against thirty-six on the smallest tessellation patch. Two hundred and eighty independent letterings across seven outlines, including an L and a five-pointed star, and not one of them disagrees with itself.
A base needs an edge to point at
Every flap of a uniaxial base ends in a point, and every point is made of paper that came from the sheet's boundary. A sheet with no boundary has nowhere for a point to come from, and a sheet with half a boundary can only have points at the half it has left.
Every pair, not every circle
A uniaxial base is designed by packing a circle for each flap, and the circles are not the condition. The condition is that every pair of the subject's extremities be separated on the sheet by the distance between them through the tree — which for two flaps meeting at one point is the sum of their lengths, and for two flaps across a body is more. Circles are the case with no body in it, so a design read off circles alone is promised a base sixteen to thirty per cent larger than the sheet can give.
What the condition does not decide
A tree of seven leaves imposes twenty-one separations and eight of them bind. The rest are slack, the eight pin six of the seven leaves against the sheet's own edges, and the seventh can be moved half a per cent of the sheet for nothing. The requirement that looks quadratic is doing linear work, and what it leaves undecided is the part a designer is actually choosing.
The price of a limb is not its length
Lengthening an edge of a subject's tree costs the design some of its scale, and the amount can be measured by re-solving the arrangement. It is not proportional to the edge, and it is not the body that is dearest. On a bird whose wings are twice its legs, the head — nine tenths of a unit against the wings' one and six — costs three times what a wing costs, and two edges of a lizard cost nothing at all.
A price holds until the arrangement moves
Every edge of a subject's tree has a price — the scale lost per unit of extra length — and the obvious use of a price list is to spend a fixed total of limb where it is cheapest. Done a tenth of a unit at a time, re-pricing at every step, it works and then stops: the bird's scale rises 6.3 per cent in four steps and no further. But the prices do not hold while it happens. The bird's free tail stops being free after the first tenth, and its legs nearly treble in price without being touched. The lizard's prices hold for four steps, because its arrangement keeps the same three pairs at their limit for four steps. A price is a statement about which pairs are at or near their limit, and it lasts as long as they stay there.
Rounding in the cheap direction
A tree spelled on a grid has every limb rounded to a whole number of units, and the rounding is chosen to keep the subject's proportions. Each rounding is also a small move of length between edges, and the edges have prices. Rounding the bird's dearer edges down and its cheaper ones up gives the largest model of every rounding tried, on grids of four, six and eight units — 2 to 6 per cent larger than rounding to the nearest unit, and larger than the unrounded bird itself on all three. The proportions pay for it, by five points of error on eight units and by thirty-four on six, which is the trade the grid had been making silently in whichever direction the arithmetic happened to fall.
Named alongside it
The objects these essays reach for when they reach for this one.
Circle packingTree methodFlapDesignSheet shapeTrade-offConservationConstraintHinge creaseRidge creaseBoundaryDesign technique