Impossible Triangle Explained: Henry Segerman's Real-Life Model

VideoHenry SegermanGeometry  ·  Illusion

The first thing Henry Segerman does in this video is hold up a white impossible triangle and push a screwdriver through the hole in the middle. The screwdriver goes in, comes out the other side, and nothing about the shot looks wrong. The object sits there in the light being a thing that cannot exist.

The figure is usually drawn flat, and it usually carries the Penrose name after Roger Penrose and his father Lionel, who published it in the British Journal of Psychology in 1958 and called it impossibility in its purest form. As tends to happen, someone else got there first. The Swedish artist Oscar Reutersvärd drew it in 1934 at the age of eighteen, out of a row of cubes, and spent the rest of his life producing thousands more impossible figures while the credit went elsewhere.

What makes it impossible is worth stating carefully, because the argument is cleaner than the picture. You see three apparently flat faces belonging to one three-dimensional object. Follow the figure around. This face turns a corner and covers part of that one, so it must be nearer to you. Keep going and the next face covers the one after it, so it too must be nearer. Go all the way around and you arrive back where you started, having moved nearer at every step. The part you began with has to be in front of itself.

Occlusion is supposed to be an ordering. If A hides B then B does not hide A, and the relation cannot loop. The impossible triangle is a three-cycle in a relation that is not allowed to have cycles, which is a more precise way of saying that your visual system has been handed a set of local facts with no global solution. Every corner is fine. The tour around them is not. The figure’s surface can in fact be embedded without stretching in five-dimensional space, which is a polite way of saying that three dimensions are simply not enough room.

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So a physical one has to cheat, and the interesting part of Segerman’s video is the audit of exactly how. He walks through the options like someone eliminating suspects.

You can curve the faces. A model he shows by a designer called tato_713 does this, letting the faces bulge so they can pass in front of one another at the corners and then bend back to arrive in the right place at the next one. The geometry works. The lighting does not. Curved surfaces shade continuously, and the gradient across each face announces that nothing here is flat.

You can keep the faces perfectly flat by drawing the whole thing on a sheet of paper, which is both a complete solution and a worthless one. Now there is no object, and every face lies in the same plane, so no face can be lit differently from its neighbor. You have to draw a line where the shading should have been.

You can keep flat faces pointing in genuinely different directions and give up on having them meet properly at the edges. Then the faces tilt past each other, and the shadows shift where they do. You can see it.

Every version of this object is a claim about where the viewer is standing.

What is left is Segerman’s answer. Keep the faces flat, keep them meeting properly almost everywhere, and concentrate the entire lie into one place that you then decline to point at. In his model the faces really do get closer to the camera as you go around, except that one pair of apparently neighboring faces are not neighbors at all. They only look adjacent from one position.

Three real rectangular bars, drawn here as actual solids. Their free ends are separated by three units along the line of sight, which is why the figure closes from exactly one direction and only from that one. Turn the object and the loose end lifts away, marked here in amber.

The figure above is the honest version of the trick. Three bars, built as real boxes, arranged so the gap between the two loose ends runs precisely along the direction you are looking. A displacement parallel to your line of sight projects to nothing. The gap is there in the object and absent from the image, and the whole illusion is that one fact.

Which leaves lighting as the last line of defense, and this is where the video earns its keep. With a soft light from overhead and the camera in its one correct position, the model holds up. Segerman then switches on a hard light from the left and kills the overhead, and an edge along the side lights up that has no business lighting up. The light diffuses through the side face and illuminates a region that ought to be sitting in the shadow of the piece behind it, except that the piece behind it is not behind it. The shadow that should be there is missing, and the missing shadow is the tell.

I like that the whole thing runs on shadows rather than on shape. You cannot check the geometry of the object by looking at it, because looking at it is precisely the operation the object is built to defeat. What you can check is whether the light behaves. Light does not care where the camera is, so it goes on falling honestly on faces that are pretending to be somewhere else, and eventually a shadow fails to show up where the illusion has promised one.

Segerman signs off by calling it his fake impossible triangle, which is the right description and also a slightly unusual thing for the builder of an illusion to do on camera. He spends the video finding the one lighting setup that destroys his own model, and then shows you.

Where to go next

Segerman teaches mathematics at Oklahoma State University and works on three-dimensional geometry and topology, which is the day job behind the workshop. His book on visualizing mathematics with 3D printing has a companion site where the models can be turned around and downloaded, and it is the closest thing to holding one of these arguments in your hand.

★★★★★

In short

A physical impossible triangle always cheats somewhere. Segerman finds the one lighting angle that makes the cheat visible, then does it on camera.

The fake impossible triangle ·  Henry Segerman  ·  Geometry & Illusion  ·  abakcus.com