A drawn fold has a phantom

The drawing does not say which corner is nearer is the field’s account of the Necker reversal: a parallel drawing of a box has no depth cue in it, so a corner can be read as the near one or the far one, and the two readings are two different solids drawn identically.

That essay is about a solid. The drawing office draws sheet, and a sheet’s version of a corner is a fold — two plates hinged along a line, at some dihedral angle. It has a reversal too, and the reversal behaves differently.

Where the two readings come from #

Take the hinge along the $x$ axis and the two plate edges perpendicular to it, of known length $L$. A parallel drawing fixes each edge’s component in the picture: two numbers per edge, read straight off the paper.

What it does not fix is the component along the ray. But it is not free — the edge has length $L$, so

$$\text{along the ray} \;=\; \pm\sqrt{L^2 - |\text{drawn}|^2}$$

and there are two choices per edge, so four sign pairs.

An overall reflection — turning the whole plate over, which is the same drawing seen from behind — flips both signs at once, so it identifies $(+,+)$ with $(-,-)$ and $(+,-)$ with $(-,+)$.

Two readings. Exactly two.

And they are not the supplement of each other #

The natural guess is that the phantom is $180° - \phi$, or $360° - \phi$: the fold read the other way, opened out or turned inside out.

Neither. Sweep the true dihedral from forty degrees to a hundred and sixty and the phantom traces a curve that touches neither of those, and by margins of several degrees at their closest.

The reason the guesses are wrong is that they are answers to a different question. The supplement is the dihedral of the fold with one plate reflected in the hinge’s plane, which is an operation on the plate. What the drawing admits is the fold with one plate’s out-of-picture component reversed, which is an operation on the reading — and the two coincide only when the viewing direction is aligned with the fold’s own symmetry.

That distinction is easy to miss because the picture looks the same in both cases: two rays from a point, at some drawn angle. The drawn angle does not determine the dihedral at all, and the essay’s whole content is what does.

The phantom is shallower for a deep fold and deeper for a shallow one #

Read the curve and it has a shape.

For a fold near a hundred and sixty degrees — nearly flat — the phantom is around a hundred and eighteen: much sharper. For a fold near fifty, the phantom is around fifty-six: slightly shallower. So the map compresses: it drags every fold toward a middle.

And the two cross. Somewhere between them is one dihedral that is its own phantom — at sixty-six degrees for the ray this figure uses — and there the ambiguity has no consequence, because both readings are the same plate.

That crossing moves with the viewing direction. Change the ray and it lands somewhere else, which is what says it is a property of the drawing rather than of folds — and it is the reason the essay quotes it with the ray attached rather than as a constant.

The map, written down #

The curve is measured by sweeping, and it has a closed form that explains every feature the sweep shows.

The phantom reverses one edge’s out-of-picture component, which is the same as reflecting that edge in the plane perpendicular to the ray: $\mathbf u_1 \mapsto \mathbf u_1 - 2(\mathbf u_1\!\cdot\!\hat{\mathbf d})\hat{\mathbf d}$. Dotting with the untouched edge,

$$\cos\phi' \;=\; \cos\phi - 2(\mathbf u_1\!\cdot\!\hat{\mathbf d})(\mathbf u_2\!\cdot\!\hat{\mathbf d}).$$

Writing $k = (\hat{\mathbf d}\cdot\hat{\mathbf x})^{2}$ for the squared cosine of the ray’s angle to the hinge, and $\psi$ for where the ray’s transverse part sits relative to the fold’s bisector, the two dot products multiply out and the map becomes

$$\cos\phi' \;=\; k\cos\phi + (1-k)\cos\phi^{*}, \qquad \phi^{*} = 180° - 2\psi.$$

It is affine in the cosine — a contraction toward a fixed point, which is precisely the “drags every fold toward a middle” the sweep shows, now with the middle named and the contraction measured.

Everything the essay observes falls out of those two parameters.

The self-phantom is $\phi^{*} = 180° - 2\psi$, and it depends on the ray’s azimuth about the hinge and on nothing else — not on $k$, not on the fold. That is why it moves when the ray moves and why it is quoted with the ray attached. For this figure’s direction it is 66°, which fixes $\psi$ at 57°.

The contraction is $k$. Taking the essay’s own pair — a 160° fold reading as 118° — gives $k = 0.651$, and the map then predicts a 50° fold reading as 55.9° against the 56 the sweep reports. Two quoted points, one fitted constant, and the second is a prediction rather than a fit.

The two limits are the useful part.

A ray along the hinge gives $k = 1$ and no ambiguity at all: $\phi' = \phi$, the phantom is the truth, and the drawing states the dihedral outright. That is the view a drawing office would choose if it knew this, and it is an ordinary auxiliary view — look along the fold line.

A ray perpendicular to the hinge gives $k = 0$ and total ambiguity: every fold, at every angle, has the same phantom $\phi^{*}$, so the drawing carries no information about the dihedral whatever. A front elevation of a folded sheet whose hinge runs across the page is exactly that view, and it is the one a draughtsman reaches for first.

So the ambiguity is not a fixed hazard of parallel projection; it is a quantity, it is $1-k$, and it is governed by one angle the draughtsman chooses. Turn the sheet so the hinge points at the viewer and the phantom disappears; turn it so the hinge lies across the page and the drawing says nothing. That is a better instruction than any warning, and it is the same shape of answer the true shape of a cut gives for a plane: one direction returns the metric exactly and the rest return a picture with a computable amount of it missing.

Why the drawn angle is not either reading #

There is a number on the paper that a reader will want to be one of the answers, and it is not.

The drawn angle between the two plate edges is what a protractor laid on the drawing measures. For a hundred-degree fold seen from a general direction it comes out at ninety-four — between the two readings and equal to neither.

The reason is that a projection foreshortens the two edges by different amounts. Each edge is drawn at $L$ times the sine of its own angle to the ray, and the two angles differ, so the drawn triangle is not similar to the true one. A protractor is measuring a third thing.

That is the same trap the true shape of a cut reports on a plane section: the area comes off the drawing with one number and the angles do not, because a foreshortening scales one direction and not the other. A fold is that finding on two half-planes instead of one polygon.

So a drawing of a fold carries three angles — the true dihedral, the phantom, and the one on the paper — and the third is the only one anybody can measure directly. It is worth saying plainly, because a reader who has been told the drawing is ambiguous will still reach for the protractor.

The control, which is not the flat plate #

Every claim on this site is paired with a case where it fails, and here the obvious candidate for that case is wrong.

The natural control is a flat plate — a dihedral of a hundred and eighty degrees, the two edges collinear and opposite. Surely a straight drawn line has one reading?

It does not. A straight line is also the drawing of a folded plate seen along the plane that bisects it, and the machinery duly returns two readings that differ by forty degrees. That is not a bug; it is a genuine degeneracy, and a control has to be a case where the ambiguity goes away rather than a case that looks unambiguous.

The case where it goes away is a plate edge lying in the picture plane, drawn at true length. Then the component along the ray is zero, and its two signs are the same number. One reading.

So the control the check uses is a fold viewed so that one plate is edge-on to the ray, and it asserts the two readings agree to six decimal places. Finding that the flat plate was not the control is the kind of correction this site’s habit produces: the assertion was written, it failed, and the failure turned out to be a result rather than a mistake.

What the drawing does fix #

It is worth being explicit about what survives, because “two readings” is easy to hear as “nothing is known”.

The drawn angle is fixed — it is on the paper — and it is neither reading.

The hinge’s direction in the picture is fixed.

And given the two plates’ lengths, the pair of admissible dihedrals is fixed: a finite set with two elements, not a family. That is a much stronger statement than the reversal makes about a general drawn corner, where Pohlke’s theorem says any three drawn segments from a point are the image of a cube’s edges — a one-parameter family of readings rather than two.

The difference is the known lengths. Pohlke’s freedom is exactly the freedom to choose the drawing’s scale and axes; fixing the plates’ actual sizes uses that freedom up, and what is left is the sign.

The dihedral, computed from the edges rather than from the normals #

A small piece of machinery is worth pulling out because the obvious implementation of it is wrong in a way that passes most tests.

The dihedral between two half-planes sharing a hinge is the angle between the two plate edges measured perpendicular to the hinge. Not the angle between the edges themselves — an edge may have a component along the hinge, and that component belongs to neither plane’s inclination.

The other natural route is through the two planes’ normals: build $\hat n_a = \hat x \times \hat u_a$ and $\hat n_b = \hat x \times \hat u_b$, take the angle between them, and take the supplement. That works and it carries a sign convention that has to be got right, and getting it wrong produces an answer that is correct for half the range and reflected for the other half — which looks like a plausible curve and is two functions glued together.

foldReadings takes the first route. It projects each edge perpendicular to the hinge, asserts that neither projection has collapsed to nothing, and takes the angle between what is left. The assertion is the part that matters: an edge drawn along the hinge has no perpendicular component and no dihedral to contribute, and a routine that returned an angle for it would be returning the arc-cosine of a zero over a zero.

That is the refusal the gate exercises, and it fires on a fold viewed along its own hinge — the one direction from which a fold draws as a single line and says nothing whatever.

A workshop reading of it #

The practical form is short and is worth having, because a fold is a thing somebody makes.

A folded part on a drawing carries its bend angle as a number, never as something to be read off the drawn angle. That is universal practice and this is why: the drawn angle is not the bend angle, and even the pair of dihedrals the drawing admits needs the projection direction to be known before it can be worked out.

And a part’s bend direction — up or down, toward the reader or away — is marked with a symbol or a section rather than left to the drawing. That is the phantom being ruled out by an annotation.

Both conventions exist because a fold’s drawing is ambiguous, and neither of them says by how much. The map above says by how much: a hundred-degree fold reads as eighty-seven as well, which is thirteen degrees of ambiguity on a dimension that a bracket’s fit depends on.

The check that disagreed with its author #

The control above is the second thing in this essay that came out of an assertion failing, and both are worth recording because the pattern is the site’s own.

assertADrawnFoldHasAPhantom was written with three parts. Every true dihedral has to be one of its own two readings — otherwise the reading machinery is wrong rather than the drawing ambiguous. The phantom has to differ from both the supplement and the reflex by a real margin. And a flat plate has to have one reading.

The third failed, and the reason turned out to be the finding above: a straight drawn line is also the drawing of a fold seen along its bisecting plane. The assertion was replaced by the edge-on control and the flat case was kept as a positive assertion in the other direction — that the two readings differ by more than a degree — so the degeneracy is on the record rather than quietly dropped.

That is the third time in two phases on this site that a check disagreeing with its author has produced the result rather than a repair. The habit that makes it work is writing the assertion before knowing the answer, which is uncomfortable and is the whole point.

Where this sits among the field’s ambiguities #

The parallel field has now measured four different things a parallel drawing leaves free, and they are four different kinds of freedom.

The reversal leaves a binary choice about which corner is nearer, on a solid.

Pohlke leaves a continuum: any three drawn axes are some cube’s.

One oblique drawing leaves a one-parameter family of solids, sheared along the projection’s own kernel.

And a drawn fold leaves a pair, of which one is the plate and the other is a specific different plate whose dihedral is computable.

Putting them side by side is worth doing because the fold’s case is the one with the most structure. A binary choice is a choice; a continuum is a shrug; a pair whose second member is a computable other object is something a reader can go and look for.

What a second view does #

The obvious repair, and it works.

A second parallel view from a different direction fixes each plate edge’s component along the first ray, because that component is in the second view’s picture. So two views determine the fold — one reading, no phantom — which is the same statement as two parallel views collapsing the reversal and leaving only a relief.

What the second view does not fix is the relief: the whole arrangement can still be stretched along the common viewing direction, and a fold stretched that way has a different dihedral. So two views give the dihedral only if the two directions are known relative to each other, which on a drawing they are — the standard three views are perpendicular by construction, and that is what makes the convention work.

What this does not settle #

It does not treat a fold with plates of unknown length. Then the drawing admits a family rather than a pair, and the pair here is bought entirely by knowing how big the plates are.

It does not treat a curved fold — a plate bent through a radius rather than along a crease. That is the bent plate and its ambiguity is a different question, because a bend has a whole family of generators rather than one hinge.

And it does not say a reader ever mistakes one for the other in practice. Shading, hidden lines, an annotation and knowing what the part is for all remove the ambiguity long before geometry does — which is true and is a fact about drawings rather than about projection.

An ambiguity worth reporting is one whose second answer is a nameable object. “The drawing does not determine it” is a shrug; “the drawing is equally consistent with this other plate, whose dihedral is eighty-seven degrees” is something to check.