Origami & Folding
what one sheet can do
Series

Uniaxial bases — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. the treeleg0.62leg0.62body0.34arm0.50head0.78the basethe axis0.620.620.340.500.78the flaps are the tree's edges, at the tree's lengths, all square to one lineso the base's shadow along the axis is the tree, and nothing else can be designed this waywhich is the restriction the circle argument quietly depends on

    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.

    part 1 · design
  2. a flap of 0.12 of the sheet's side, on two sheets of the same areadarker is cheaper: less of the flap's disc is paper that has to be paid forwith a holesolid, same areamean claim 0.8646mean claim 0.8959

    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.

    part 2 · design
  3. a body with four legs and a tail — one internal edge of 0.8circles of radius m·ℓ, and every pair's requirement drawn — the ones through the body ask for more than the two circles do

    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.

    part 3 · design
  4. a body of three segments — two internal edges, 0.6 and 0.9the segments are the pairs at their limit — 4 of 8 of them measured through the body rather than around it

    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.

    part 4 · design
  5. what lengthening each edge of a body with wings, legs, a head and a tail coststhe scale falls from 0.2651 by this much per unit of extra length, measured by re-solving the arrangementchest–head0.0786a flap, 5.9% of the scale per 0.2rump–ll0.0471a flap, 3.6% of the scale per 0.2chest–rump0.0429the body, 3.2% of the scale per 0.2chest–wl0.0361a flap, 2.7% of the scale per 0.2chest–wr0.0264a flap, 2.0% of the scale per 0.2rump–tail0.0123a flap, 0.9% of the scale per 0.2rump–lr0.0120a flap, 0.9% of the scale per 0.2an edge no tight pair passes through is an edge the design can spend freely, and the condition says which

    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.

    part 5 · design
  6. spending the bird's length where the arrangement prices it lowesta price is the scale lost per unit of length added to every edge of a group, re-solved at every stepstepscaledearestpricecheapestpricetight pairs00.2651head0.092tail-0.000510.2753head0.102wings0.034520.2748legs0.081wings0.012530.2781body0.072wings0.037640.2817head0.081wings0.054350.2778tail0.074head0.000560.2780head0.076tail0.0026a price holds for as long as the dearest and the cheapest group stay the same groups

    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.

    part 6 · design
  7. the bird rounded to a grid two waysunits in the order body, wings, head, legs, tail; the drawn tree unrounded has size 0.2651gridnearesterrorsizecheap wayerrorsize4 units1 4 2 2 316%0.26751 4 2 2 423%0.28076 units2 6 3 3 511%0.27241 6 3 3 545%0.27848 units3 8 5 4 715%0.26312 8 4 4 720%0.2782size is the scale times the sheet length one unit of the subject's own length receives; error is the worst limb's

    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.

    part 7 · design

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