Category Archives: Geometric illusions

Illusions that appear in figures with geometrically regular lines

The vertical/horizontal illusion


This aimiable looking old guy was Austin Crothers, governor of Maryland USA in the first years of the last century, and a notable scourge of deception and corruption.  His top hat however presents one mysterious deception that even he couldn’t unravel.  It looks to me about as wide as it is high. But now look at the measuring rod, first when vertical, and running the full height of the hat.  When horizontal, at the foot of the picture, we can see that the same line stretches only a touch over two thirds of the way across the width of the hat. The hat is MUCH wider than it is high.  It’s an example of the horizontal/vertical illusion – we tend to overestimate height. Check out pictures of the St. Louis Arch, seen from the front, for example.  It’s just as wide as it’s high, but looks higher.

There’s no agreement on why.  There are lots of speculations, for example that the effect arises from some adjustment to allow for the inequality between the width and height of the visual field in normal binocular viewing.

The discovery of the illusion is attributed to J.J.Oppel in 1855.  It’s usually seen in this simplified version.

It works the other way up too, and is sometimes called the T illusion.  It’s one of many illusions for which you’ll find a brilliant interactive demo on Michael Bach’s site.

It’s amazing that we’ve made so little decisive progress with simple illusions like this one, after more than a century.  I can’t think of another area of science in which progress has been quite so hard, except of course some areas of maths.  But with these illusions, the explanations proposed in papers from over a century ago are sometimes much the same as those we are still discussing today.

The photo of Crothers is from the Grantham Bain collection in the Library of Congress and can I believe be used without copyright restrictions. 

Competing illusions


Here’s a rather subtle effect. It’s a competition underway, when the Zollner illusion is seen embedded in a staircase. In the staircase lower left, where two of the long lines are either side of the outside edge of a step (in other words like lines a and b here, on the sides of a convex step), the lines seem to get further apart with distance, as they would in a normal presentation of the Zollner illusion. But wherever on that lower left stair the lines are like b and c here, either side of the inner edge of a step, (so on a concave step), they tend to look much more parallel. In a normal version of the illusion, as below, the equivalent long lines appear to get closer together to the right.

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The Poggendorff Illusion and depth processing

One of the most obstinately puzzling illusions is Poggendorff’s, in which a slanting line interrupted by a gap no longer looks aligned. For over a century specialists have been unable even to agree whether it arises from 2D properties of the image, or as a result of attempts by the brain to interpret the configuration as 3D. Papers written a hundred years ago treat the problem in very much the same terms as we do today. I’m betting on 2D (I argue for that on another, website devoted to the Poggendorff illusion). It’s not likely my speculations are spot on, and they may well not even be in the right direction. But read on here if you’d like to see demonstrations that show why I don’t think depth processing can be the answer.

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Paradoxical Size Constancy

Size constancy is the term for our tendency to see distant objects as larger than they are. So the far end of a shape with parallel sides looks wider than the near end. (See the earlier post on The Wonky Window). It seems to be such a basic feature of vision that it can give rise to amazing effects.  In the photo, first note note that the “sculpture” is impossible! All four blocks are receding from us, so they could only connect up in real space as a bendy snake. Instead they join up in an impossible, ever-receding, endless loop.  (See the earlier post on M.C.Escher’s Waterfall for how that kind of impossible figure works). Here the endless loop leads to a paradox, thanks to size constancy. The distant end of each block seems wider than the near end, and yet at the same time seems to be exactly the same size as the apparently smaller, near end of the next block. Measure the sides of the blocks and you’ll find them parallel. It’s one of many demonstrations that perceptual space is not always geometrically consistent, (or it can be non-Euclidean, as the specialists put it).


I located my impossible sculpture in a deeply receding space because that makes the effect just a bit stronger.

Update January 2010: How could I have overlooked this?  The stripes I’ve added to these blocks will be enhancing the effect of divergence by adding the chevron illusion to the size-constancy effect.  The chevron illusion was first reported 500 years ago, by French writer Montaigne, as related in Jaques Ninio’s book on illusions, page 15.  The chevron effect is a special case of the illusion later re-discovered a bit over a century ago as the Zollner illusion.  Some specialists would say both effects depend on the brain’s attempts to make sense of figures as shapes in space.  I suspect that’s true of the size-constancy effect, but that the chevron effect is 2D, pattern driven.  That seems supported by the observation that whilst in the picture above the chevron and size-constancy effects are acting in consort, they can also oppose one another, reducing the effect of divergence.

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(Not so) Geometric Illusions

Many of the illusions in popular books are geometric ones, in which lines that are really parallel look wonky, or lines that are aligned seem not to be. Most of these figures were discovered by German researchers, a hundred to a hundred and fifty years ago. But how geometric do they have to be? With graphics packages it’s easy and fun to explore. Here are versions of two famous illusions, one showing apparent divergence where the other presents convergence, against the same “zebra skin” background.

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