The crease, its rulings, and the line where they cross
ruling-envelope 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: "turning", show: "sweep"
view: "turning", show: "picture"
view: "turning", show: "picture", rate: 1.4
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.
- on a circular crease the rulings meet at exactly R·sin β, which is 0.2350 at this angle for a radius of 0.3 ×3
- and only where the crease is momentarily straight is neither surface bounded, which happens at 0 of the 201 points sampled ×2
- and turning it costs reach before it costs the sign: the bounded surface comes in from 0.2956 to 0.0000 as the rate rises, falling at every step ×2
- at a turning rate of 1.25 both of the fold's surfaces are bounded, on a crease whose curvature never changes sign — which a constant ruling angle can never produce ×2
- a crease of one sign of curvature bounding both of its surfaces, because the ruling angle turns faster than it bends ×1
- a crease that never changes the way it bends bounds one surface only while the ruling angle turns slower than the crease does — every rate at or below one leaves the outer surface unbounded along its whole length, and every rate above one bounds part of it ×1
- a curved fold reaches sin β of its own tightest radius of curvature, on every crease curve tried ×1
- and a crease that never changes the way it bends has one surface that never runs out at all, anywhere along it — which is true of the circle, the ellipse and the parabola at every angle tried, and of the wave at none ×1
- at every point of every crease measured, at most one of the fold's two surfaces has its rulings converging — the two sides never both run out at the same place, on 16 crease-and-angle pairs ×1
- at no point of this crease do both surfaces have converging rulings — the two denominators are exact negatives, so a point where one is positive is a point where the other is not ×1
- four different crease curves reach the same share of their own tightest radius at every angle, to 2e-16 ×1
- on 16 crease-and-angle pairs the two surfaces' converging regions never overlap, and 12 of them have a surface that never runs out ×1
- one crease bounding both of its surfaces, over 100 and 100 of the sampled points ×1
- the one line the four curves lie on is sin β, so a curved fold's reach is set by the angle its rulings leave at and by nothing else about the curve ×1
- the surface that runs out changes at a turning rate of one, where the ruling angle turns exactly as fast as the crease bends ×1
- the two surfaces' envelope denominators are exact negatives, so their bounded regions partition the crease rather than overlapping ×1
- this crease bounds both of its surfaces, each over the stretch where the crease bends toward it — 100 of the sampled points on one side and 100 on the other ×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.
An angle that turns faster than the crease
Which of a curved fold's two surfaces runs out is decided by a sum of two rates — how fast the crease bends and how fast the ruling angle turns — and every measurement so far has set the second to zero. Let it turn and it carries the sign on its own: past a rate of exactly one, a crease of unchanging curvature bounds both of its surfaces, which no constant angle on that crease can do. Below that rate the turning costs reach without changing anything else.
Only one side can run out
A curved fold has two surfaces and every reach ever computed here has been one of them. The closed form's denominator is the crease's curvature plus the rate the ruling angle turns at, and crossing to the other surface negates both — so at any point of any crease at most one of the two surfaces can have its rulings converge. A crease that never changes the way it bends therefore has a surface with no such boundary at all, anywhere along it.
The gap between two curves
The rulings leaving a curved crease are not parallel, so they cross, and the surface exists only as far as the first crossing. That bound is usually read as a limit on how far a design extends outward. It is not: the paper between two curved creases has to be reachable from both, so the bound bites hardest where the circles are smallest, and a concentric pleat has a hole in the middle that no sheet size removes.
Where the rulings run out
A curved fold's surface is made of straight lines leaving the crease, and the lines are not parallel, so they cross. Past the first crossing there is no surface: two points of the paper have been sent to one point of space. The boundary is a curve nobody drew, no crease pattern shows it, and it sits at the sine of the ruling angle times the crease's own tightest radius — on every curve tried.
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