Where the cheap paper is, with a hole and without
holed-sheet is one function. Everything below came out of it during this
build, at arguments 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.
At its defaults
view: "map"
view: "places"
view: "bargain"
What it checked while it drew
Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.
- one round of alignments reaches 212 references with a hole and 9 without ×4
- the deepest of them saves 5.38% against the shallowest's 0.98%, at flaps of 0.12 ×2
- the sheet with a hole is cheaper paper at every one of the 5 flap lengths measured ×2
- a crossing inside the hole is specified by the alignments and is not a reference, because a folder cannot put a finger on it — so the count is of points that land on paper ×1
- a hole is cheaper paper than a notch of the same size at every flap length measured — 1.4% against 0.6% at the shortest, 9.9% against 4.5% at the longest ×1
- every bite here removes the same area, so the comparison is of shape rather than of size ×1
- the hole and the notch remove exactly the same area, so what is compared is where the paper went rather than how much ×1
- the holed sheet is compared with a solid one of the same AREA rather than the same side, so the comparison is about arrangement and not about how much paper there is ×1
- the paper a flap claims at a point is integrated on an equal-area polar grid about that point, so a corner and a hole are priced by one method ×1
- the three places the earlier rung named are priced by the same integration as the two the hole adds, so a whole, a half and a quarter are recovered rather than assumed ×1
Where it is called
Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.
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.
A cut that reaches the edge
A ring of paper with three creases running from its hole to its rim satisfies every condition the subject has — vacuously, because it has no interior vertex at all — and cannot be folded: its panels take no two colours and the two routes to one of them end up 1.75 sheet widths apart. One cut from the hole to the edge, crossing no crease and changing no letter, and it folds exactly. The cut removes an adjacency, which is the one thing neither a fold nor an edge can do.
A hole is an edge
A folder's first fold has to be specified by aligning things that are already there, and what is already there is the sheet's outline. Cut a square hole in the middle and the outline doubles: one round of alignments reaches nine references on a plain square and two hundred and twelve on a holed one — more than the plain square reaches in two rounds.
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.
A notch is not a hole
Remove the same rectangle of paper from the middle of a square and from its edge, and the two sheets are not worth the same. The hole is cheaper paper at every flap length measured — 4.71% cheaper than a plain square of equal area against the notch's 2.58% — because what a cut is worth is rim with paper on both sides of it, and a notch spends one of its four sides on an edge the sheet already had.
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.
The corner premium, with no corners
A square's corners are its most valuable paper: a flap placed there is claimed by a quarter-disc rather than a whole one, so the corner goes four times as far as the middle. A closed sheet has no corners at all, and the accounting that ranks sheet shapes by their corners has nothing left to rank.
The edge was there first
A folder's first fold has nothing to align to but the sheet's own outline, and it shows in where the marks land: four of the five references the first fold adds are on the paper's edge. By the second fold the edge holds forty-eight of five hundred and fifty-six new ones. The rim is where references are cheap and it fills up, because an edge is a line a fold can cross twice while two folds inside the paper cross once each.
Every generator · The designing a base field · The patterns a reader can fold