The plane, and the two vectors inside it, at a gap of 10⁻⁶
At its defaults it draws the plane, and the two vectors inside it, at a gap of 10⁻⁶. A circle representing the invariant plane, with two perpendicular radii for the eigenvectors and two more for the perturbed ones, rotated within the same circle.
plane-and-vectors is one function in lib/figures/subspace.js —
subspaces — the gap, the eigenvector, and the plane that survives both. 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.
A circle representing the invariant plane, with two perpendicular radii for the eigenvectors and two more for the perturbed ones, rotated within the same circle.
gap: 0.000001
The arguments are the ones The plane survives what its vectors do not passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A circle representing the invariant plane, with two perpendicular radii for the eigenvectors and two more for the perturbed ones, rotated within the same circle.
gap: 1e-8
The arguments are the ones The plane survives what its vectors do not passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A circle representing the invariant plane, with two perpendicular radii for the eigenvectors and two more for the perturbed ones, rotated within the same circle.
gap: 0.1
The arguments are the ones The plane survives what its vectors do not passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A circle representing the invariant plane, with two perpendicular radii for the eigenvectors and two more for the perturbed ones, rotated within the same circle.
gap: 0.001
The arguments are the ones The plane survives what its vectors do not passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A circle representing the invariant plane, with two perpendicular radii for the eigenvectors and two more for the perturbed ones, rotated within the same circle.
gap: 0.00001
The arguments are the ones The plane survives what its vectors do not passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A circle representing the invariant plane, with two perpendicular radii for the eigenvectors and two more for the perturbed ones, rotated within the same circle.
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.
6 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.
a gap inside the range drawn
and what leaves the plane is the sine of the angle it turned through
Jacobi needs a symmetric matrix
matmul shapes agree
the plane has barely moved
the two unperturbed vectors are drawn at a right angle because they are at one
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.
The plane survives what its vectors do not
At a gap of 10⁻⁹ a perturbation of 10⁻⁶ turns the two eigenvectors through half a radian and turns the plane they span through 7.6·10⁻⁸ — a ratio of six million. Ask for the subspace instead of the vectors and a hopeless computation becomes a well-conditioned one, with no change to the arithmetic.
Least squares, and the road not to takeThe projection and the right angle
The least-squares solution is the one whose residual is perpendicular to everything the columns can reach. That is not a mnemonic — it is an equation, Aᵀr = 0, and the computed answer satisfies it to 10⁻¹⁶.