A box rotated 20 times, and the smallest box containing it
At its defaults it draws a box rotated 20 times, and the smallest box containing it. Two widths against the number of rotations, on a logarithmic vertical axis. The set's width is a flat line, because a rotation is an isometry. The interval enclosure's climbs geometrically — by a factor of 1.4142 a step against a predicted √2 — and after 20 rotations it is 1024 times larger than the thing it encloses.
wrapping-growth is one function in lib/figures/interval.js —
intervals — a bound that is proved rather than measured, and what proving it costs. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
Two widths against the number of rotations, on a logarithmic vertical axis. The set's width is a flat line, because a rotation is an isometry. The interval enclosure's climbs geometrically — by a factor of 1.4142 a step against a predicted √2 — and after 20 rotations it is 1024 times larger than the thing it encloses.
steps: 20
The arguments are the ones Nine steps of pessimism passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two widths against the number of rotations, on a logarithmic vertical axis. The set's width is a flat line, because a rotation is an isometry. The interval enclosure's climbs geometrically — by a factor of 1.4142 a step against a predicted √2 — and after 20 rotations it is 1024 times larger than the thing it encloses.
steps: 8
The arguments are the ones Nine steps of pessimism passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two widths against the number of rotations, on a logarithmic vertical axis. The set's width is a flat line, because a rotation is an isometry. The interval enclosure's climbs geometrically — by a factor of 1.4142 a step against a predicted √2 — and after 8 rotations it is 16 times larger than the thing it encloses.
steps: 12
The arguments are the ones Nine steps of pessimism passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two widths against the number of rotations, on a logarithmic vertical axis. The set's width is a flat line, because a rotation is an isometry. The interval enclosure's climbs geometrically — by a factor of 1.4142 a step against a predicted √2 — and after 12 rotations it is 64 times larger than the thing it encloses.
steps: 20, bits: 11
The arguments are the ones Nine steps of pessimism passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two widths against the number of rotations, on a logarithmic vertical axis. The set's width is a flat line, because a rotation is an isometry. The interval enclosure's climbs geometrically — by a factor of 1.4156 a step against a predicted √2 — and after 20 rotations it is 1044 times larger than the thing it encloses.
steps: 20, bits: 53
The arguments are the ones Nine steps of pessimism passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two widths against the number of rotations, on a logarithmic vertical axis. The set's width is a flat line, because a rotation is an isometry. The interval enclosure's climbs geometrically — by a factor of 1.4142 a step against a predicted √2 — and after 20 rotations it is 1024 times larger than the thing it encloses.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
7 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
the enclosure is (√2)^20 too large after 20 rotations — checked 3 times
a number of rotations the enclosure survives
a precision the arithmetic can round outward at
a rotation does not change the size of the set
the enclosure grows by √2 a rotation at θ = π/4
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
Nine steps of pessimism
A proved bound is 8 to 26 times the error it bounds, at every precision from 16 to 40 significand bits. A carried interval is (√2)ᵐ times too wide after m re-enclosures. The two cross between eight and nine, so the method everybody warns against is the tighter of the two for a short computation.
The arithmetic underneathThree walks and one bound
A left-to-right sum, a chain of three thousand rotations and a conjugate gradient residual recurrence share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against. All three come out at a half — 0.486, 0.554 and 0.507 — and nothing is rescaled.