The drawing does not say which corner is nearer

A wireframe cube drawn in isometric flips while it is being looked at. The near face becomes the far one, the solid turns inside out, and after a few seconds it flips back. It is in every book of optical illusions, filed with the impossible triangle and the duck-rabbit, and the filing is wrong. Nothing about the eye is involved. The drawing is a drawing of two cubes and the eye is choosing between them, which is not an error and not an illusion — it is the only honest thing to do with an ambiguous picture.

The claim worth making is stronger than that, and it is arithmetic rather than psychology: there are exactly two, they are both cubes, and they project identically.

Where the second one comes from #

Pohlke’s recovery takes three drawn axes and returns the cube and the projection direction behind them. The last step completes a pair of orthonormal vectors into a frame, and completing a pair into a frame in three dimensions involves one free sign: the third vector may point either way.

Both choices give an orthonormal frame. Both give a cube of the same edge. Both, projected along the recovered direction, reproduce the three drawn vectors exactly. So the count of solutions is two, and it is two for the same reason a square root has two values.

That the second solution is a cube is the part that has to be checked rather than assumed. A recovery that returned a slightly skewed parallelepiped as its second answer would be describing a different phenomenon: the drawing would then determine the solid up to a small deformation, and the flipping would be the eye tolerating an error. The check is on twelve edges and three angles, and the second reading’s edges are equal to each other to arithmetic noise and its angles are 90.000000°.

And they draw the same picture #

The two solids are checked against each other by projecting both and comparing all eight vertices. The disagreement is at the level of the last bit of a double — not small, but the smallest a computer has — at every one of the six drawing systems this site implements, obliques included.

Which makes the ambiguity a fact about parallel projection rather than about cubes, or about wireframes, or about drawings that leave out the hidden lines. Hidden-line removal resolves it — that is what hidden-line removal is for — but it resolves it by adding information the projection destroyed, not by revealing something that was in the geometry.

What perspective does about it #

Move the eye in from infinity and the drawing stops being ambiguous. The interesting question is how fast, and the answer is a clean law.

The reversal still exists as a map, because the rays still exist: send each point to the other point on its own ray at the reciprocal depth, and the picture is unchanged. What changes is the solid. A perspective ray bundle is not parallel, so reversing along the rays does not preserve shape, and the resulting eight points are not the vertices of a box: the edges that were parallel stop being parallel, by an angle that starts at zero and grows as the eye comes in.

Halving the eye’s distance doubles the departure, over the whole sweep, with the drawn size of the box held fixed. That is the signature of a quantity inverse in the distance, and it is asserted as a law rather than as a threshold — the mistake the applied phase had to correct twice was bounding the size of an effect, which holds where the effect is small and therefore exactly where the control lives.

Note what is being held fixed. It is the box’s drawn size, not the focal length. The first version of this measurement swept the field of view with the eye held still and found the departure constant to fifteen digits, which reads as a bug and is a result: a longer lens from the same place makes a larger picture of the same rays, and the reversal is a fact about the rays. What rules out the second reading is being close, not being wide. That is the same distinction stepping closer is not zooming draws in a field two rows away, arrived at from the other end.

The number that matters #

At 128 metres the second reading is 0.33° from being a box. On a 690-pixel drawing that is nothing at all — no drawing can show a third of a degree of edge divergence, and no reader could see it if it did. At 4 metres it is 11.8°, which is obvious.

So there is a distance at which a perspective drawing stops being ambiguous in practice, and it is set by the drawing’s own resolution rather than by geometry. And there is no distance at which it stops being ambiguous in principle: the departure is inverse in the distance, so it approaches zero and never reaches it. A perspective picture of a wireframe box always has a second reading; it is always a solid that is not a box; and at ordinary distances it is a solid so nearly a box that nothing distinguishes it.

What resolves it, and what only appears to #

Four things are commonly said to resolve the ambiguity, and they divide cleanly into two that add information and two that do not.

Hidden-line removal adds information. Deleting the edges a solid would occlude is a statement about which face is in front, and it is a statement the projection had thrown away. The drawing is no longer ambiguous because it is no longer only a projection — an occlusion decision has been printed on it.

Shading adds information, for the same reason and by a different route. A face’s brightness depends on its orientation to the light, so shading encodes the surface normals, and the normals distinguish the two readings.

Perspective adds information, at the rate this essay measures.

And stereo does not resolve it at all in the way it is usually described. Binocular disparity supplies depth directly, so a real wireframe cube held in the hand does not flip. But a drawing of a cube on a flat page is at one depth whichever reading the eye takes, so stereo says the page is flat and says nothing about the cube. That is why the Necker figure keeps flipping when looked at with both eyes, and why the usual account — “the brain lacks depth information” — is right about the drawing and wrong about why.

The reflection, named #

The two readings differ by a reflection, and it is worth saying which reflection, because “mirror image” covers several different maps and only one of them is right here.

It is not a reflection in the picture plane — that would leave the drawing unchanged only for an orthographic projection along the picture normal, and the obliques have the ambiguity too. It is not a reflection in the plane through the object’s centre perpendicular to the page, which is a different plane again.

It is a reflection in the plane the recovery’s two orthonormal rows span — which is to say, the plane perpendicular to the projection’s own null direction. That is the plane every ray meets at right angles once the ray direction is taken as the axis, and reflecting in it exchanges each point with the other point of its ray at equal distance on the far side.

For an orthographic system that plane is parallel to the picture plane and the reflection is the familiar one. For an oblique system it is tilted, by exactly the obliquity, so cavalier’s two cubes are reflections in a plane 45° from the paper. Both are still cubes, because a reflection in any plane takes a cube to a cube.

That precision matters for one reason: it says the reversal is a property of the projection rather than of the page. Two solids that a given parallel projection cannot distinguish are related by that projection’s own reflection, and a different projection of the same scene pairs different solids.

The count, in the other fields #

A recovery on this site always returns something, and the question that separates a measurement from a story is how many things could have produced the picture. The answers form a short and useful list.

One, for the camera from three orthogonal vanishing points. The orthocentre relation fixes the focal length and the principal point, and there is nothing left over.

Two, here.

Four, for the relative pose of two cameras from an essential matrix — resolved to one by requiring the reconstructed points to be in front of both cameras, which is a condition from outside the algebra rather than a further equation.

A one-parameter family, for the scale of a scene from a single picture, resolved by nothing available in the picture.

And a seven-parameter family, for a bundle adjustment’s gauge, resolved by convention rather than by measurement.

Reading down that list, the parallel drawing’s two is unusual in one respect: it is the only entry resolved by nothing at all. The four-fold pose ambiguity has a physical condition to appeal to; the scale ambiguity has an external length; the gauge has a convention everybody agrees on. The depth reversal has none of those. Both cubes are in front of the projection, both are the same size, and no convention prefers either.

What it means for a drawing that is meant to be read #

A practical consequence, and the reason the count matters outside the geometry.

An axonometric drawing intended to be understood — an assembly diagram, an exploded view, a map of a building — is ambiguous unless something else in it breaks the tie. Hidden lines and shading are the usual devices and they are not decoration; they are the carriers of the depth order, and a drawing that omits both has genuinely not said which way round the object is.

The failure is quiet because most objects are not cubes. A drawing of a chair has a seat and legs and the reader’s knowledge of chairs settles it in a fraction of a second. The Necker cube flips because it is a shape with no natural orientation, which makes it a demonstration rather than an anomaly: it shows what every parallel drawing is doing and what every other drawing is being rescued from by its subject matter.

Why the illusion story survives #

Two reasons, and both are worth naming because they are the reason the geometry is not the usual account.

The eye does not flip at random. It flips because it is doing constraint satisfaction on an underdetermined problem and there are two solutions with nothing to choose between; and the timing of the flip, the fact that it happens after a few seconds and can be influenced by attention, genuinely is psychology. So there is a real perceptual phenomenon here, and describing it is not a mistake. Calling the ambiguity itself perceptual is.

And the demonstration is nearly always drawn in isometric, which makes it look like a property of that system. The examples in the books are drawn in isometric because isometric is easy to draw by hand on graph paper, not because the effect needs it.

The general shape #

Every recovery on this site returns something, and the question that separates a measurement from a story is always the same: how many things could have produced this picture, and is the answer one?

For the camera from three vanishing points, the answer is one, and the round trip is worth what it is worth because of that. For a scene from two views, the answer is four poses of which one puts the points in front of both cameras — a count, resolved by a condition from outside the algebra. For scale from one view, the answer is a one-parameter family and no condition resolves it.

Here the answer is two, resolved by nothing, and the correct report of a two-fold ambiguity is both answers rather than the more plausible one.

There is one more consequence, and it is the reason a phase about the thinnest rows of this subject put an essay here. The Necker cube’s ambiguity is normally presented as a curiosity at the edge of the subject. It is not at the edge: it is what “parallel projection destroys depth” means once it is made precise. Depth is not merely compressed or approximated by a parallel projection. Along the rays it is not there at all, and the drawing is exactly as consistent with the solid turned inside out as with the solid.