Every flap along one line
uniaxial-base is one function. Everything below was drawn by it when the site was
last published, at settings taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and when the generator changes, this
page changes with it.
With nothing specified
view: "metric", show: "spend", tree: "bird"
view: "metric", show: "marginal", tree: "bird", delta: 0.2
view: "metric", show: "spend-curve", tree: "bird"
What it checked while it drew
Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.
- 5 units: the nearest rounding 2 5 3 3 4 gives up 9.79% at 23.3% error; the largest, 1 5 2 2 4, is 0.287410 at 30.4%; on the frontier ×7
- every flap is at most the sheet's own diagonal, which is what a packing on it can produce — the longest here is 0.78 ×7
- 3 units: the nearest rounding 1 3 2 2 2 gives up 17.24% at 31.6% error; the largest, 1 3 2 1 3, is 0.280778 at 37.0% ×5
- and over the range drawn it costs 27.6 per cent of the scale ×4
- 3 of the 10 pairs are at the limit in the best arrangement found, and the rest have room ×3
- 4 of 5 leaves cannot be moved at all without lowering the scale, so the arrangement is mostly decided ×2
- 6 of the 7 leaves are pinned by the pairs that are tight and the walls they touch; 1 can be moved without costing the arrangement anything ×2
- along the frontier size rises from 0.2651 at 0.0% error to 0.2874 at 30.4%, 0.28 per cent of size per point of error on average ×2
- every rounding with no error at all is the tree as drawn, at its size 0.2651 ×2
- every step moves length from the edge the arrangement prices highest to the one it prices lowest, so the subject's total length stays 7.50 throughout ×2
- fewer than half the pairs are at their limit on any of these trees — 5 of 15, 3 of 10, 8 of 21 — so most of the condition is slack at the best arrangement ×2
- lengthening the flap lowers the scale at every step — 0.2848, 0.2651, 0.2378, 0.2231, 0.2207 — and never raises it, which it cannot, because every pair through it asks for more and none asks for less ×2
- on every grid tried, rounding the dearer edges down and the cheaper ones up gives the largest model of all the roundings — 4.9% larger than the nearest rounding on 4 units, 2.2% larger than the nearest rounding on 6 units, 5.7% larger than the nearest rounding on 8 units ×2
- spending the same total length where it is cheapest raises the scale from 0.2651 to 0.2817 by step 4, and the gap between the dearest and cheapest price narrows from 0.092 to 0.027 on the way ×2
- a tree with no internal edge gives the same scale both ways, which is the case the circle picture was drawn for ×1
- and at least one tree leaves a leaf free to move: 1 of 5 on a body with four legs and a tail, 1 of 7 on a body of three segments ×1
- and at least one tree leaves a leaf free to move: 1 of 6 on a body with wings, legs, a head and a tail, 1 of 5 on a body with four legs and a tail, 1 of 7 on a body of three segments ×1
- and every tree with a body is overstated by the circles — 16.0%, 29.7% ×1
- and they are priced very unevenly — the dearest, back–tail, costs 0.0451 a unit against a middle edge's 0.0169, while 2 of them cost nothing at all ×1
- and they are priced very unevenly — the dearest, chest–head, costs 0.0786 a unit against a middle edge's 0.0361 ×1
- and they are priced very unevenly — the dearest, head–ant2, costs 0.0547 a unit against a middle edge's 0.0248 ×1
- at 12 flaps there are 66 requirements and 12 circles, so what is drawn and what is required have parted company ×1
- lengthening the body lowers the scale at every step — 0.2418, 0.2225, 0.2112, 0.2008, 0.1895 — and never raises it, which it cannot, because every pair through it asks for more and none asks for less ×1
- lengthening the body lowers the scale at every step — 0.3228, 0.2982, 0.2783, 0.2621, 0.2489, 0.2336 — and never raises it, which it cannot, because every pair through it asks for more and none asks for less ×1
- no edge of a body of three segments can be lengthened at a profit — every one of the 9 costs the scale something or nothing ×1
- no edge of a body with four legs and a tail can be lengthened at a profit — every one of the 6 costs the scale something or nothing ×1
- no edge of a body with wings, legs, a head and a tail can be lengthened at a profit — every one of the 7 costs the scale something or nothing ×1
- on every grid tried, rounding the dearer edges down and the cheaper ones up gives the largest model of all the roundings — 5.7% larger than the nearest rounding on 8 units ×1
- the base is drawn from 5 flaps, each with a length worth drawing ×1
- the frontier over 121 roundings on 12 grids holds 9, from grids 5, 6, 7, 9, 10, 14, 16; grids 3, 4, 8, 11, 12 contribute none, and the nearest rounding is on it only on 6 and 16 ×1
- the frontier over 82 roundings on 12 grids holds 13, from grids 4, 5, 7, 8, 9, 11, 12, 16; grids 3, 6, 10, 14 contribute none, and the nearest rounding is on it only on 7 and 9 and 16 ×1
Where it is called
Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.
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.
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.
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.
Getting close instead of getting it right
When the best answer is out of reach the question stops being what it is and becomes how much is lost. For packing discs into a square the loss is measurable: a seeded search in this repository comes within a fifth of a percent of the best radius anybody has proved, and proves nothing.
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.
The grid belongs to the subject
Rounding a subject's limbs to whole grid units trades shape for size, and on any one grid the cheap-direction rounding is the largest. Put every rounding on every grid from three to sixteen units side by side and only a handful are worth having. For the bird, nine of 121 roundings are beaten by nothing on both counts; they come from seven grids, starting with the exact spelling on sixteen and buying size at about a quarter of a per cent for each point of worst-limb error. Eight, twelve, eleven, four and three units contribute nothing — every rounding on them is beaten by one on another grid. For the lizard, eight and twelve are on the frontier and six, ten and fourteen are not. No grid is good in general; a grid is good for a subject.
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.
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.
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