Category Archives: Geometric illusions

Illusions that appear in figures with geometrically regular lines

The Oxford Compendium of Visual Illusions

I’ve copied this beautiful demo (with small changes) from one by Stuart Anstis, who is one of the world’s leading and most prolific researchers into illusions.  His website includes a page of great movies, including this one.  Whilst the yellow circles are visible, we tend to focus locally on the pairs of spheres, each pair orbiting a central point.  But without the circles, loosely fixate the central blob, and though the movement of the spheres remains just the same, they appear to re-group into a more global view, of two pulsating, intersecting circles of spheres.

I came across Stuart’s movie amongst the many web pages of figures and demonstrations that accompany a once-in-a-generation, landmark publication, the Oxford Compendium of Visual Illusions.  It’s HUGE, with some 800 pages!  Almost all the leading researchers in the field worldwide have contributed, with essays on the history of visual illusions, up-to-the-minute, detailed discussions of a comprehensive range of illusions and effects, and philosophical essays on whether the word illusion is really the right term to describe them.  It seems to have hit the spot, because it was only published a couple of weeks back, but is already showing as sold out on OUP’s own website.  Copies are available elsewhere at the time of writing, for example via Amazon (previous link above).  Cheap it isn’t, but spectacular, it is.


Johann Joseph Who?

This may not look a dramatic illusion by contemporary standards: the top and bottom lines are each divided into three equal segments, but the middle segment appears longer than the flanking segments in the top line, and vice versa below. Yet it’s a really remarkable figure. It’s one of sixteen in a pioneering paper of 1855, the first ever study of illusions of this geometric kind.  Even more remarkably, as an illusion it remains to this day wholly unexplained, as baffling as it is simple.

The paper was by a Frankfurt schoolmaster called Johann Joseph  Oppel.  In it he named his illusions as geometric-optical illusions, to distinguish them from illusions observed in the natural environment such as the Moon illusion. Oppel seems to have observed the misjudgments that these new geometric illusions give rise to in math lessons, when his pupils were drawing or judging figures on the blackboard.  I suspect he was probably an unforgettable but demanding teacher.  He was for sure a real one-off – a tirelessly curious observer, leading the way into acoustical and language research, as well as geometric illusions, with fearless independence.

Four of us, Nicholas Wade, Dejan Todorovic, Bernd Lingelbach, and I, have just collaborated on the first ever translation of Oppel’s 1855 paper into English, with a commentary, freely available in the online journal iPerception.
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Best Illusion of the Year Competition

Here’s a copy with slight variations of a stunning new animation of the Ebbinghaus Illusion, by Christopher Blair, Gideon Caplovitz and Ryan E.B. Mruczek.  Their version won the Best Illusion of the Year Competition in 2014, a few weeks ago.  It’s a brilliant competition whose lead organiser is Susanna Martinez-Conde, and is accumulating a fascinating illusion resource as the ten finalists are added each year.

In the movie, as the figure moves up and down the screen, all the circles seem to change size.  Yet objectively only the outer ring of circles do so:  the central circle remains exactly the same size throughout.  It’s so vivid it’s hard to believe, but I’ve just added some yellow rails as a track for the central circle.  You can see that the circle always just fits the rails – and they don’t change size.

For more info and links on the Ebbinghaus illusion (aka Titchener Circles) see our earlier post on the traditional, static version.

Barbers’ Poles and the Aperture Problem

Go back a couple of centuries and there were no chains of shops or malls. In the high street in the UK you would have found the type of shop you were after by looking out for a sign hanging out.  There were signs for pharmacists, tobacconists, pawnbrokers, whatever.  Nowadays there’s just one traditional sign still sometimes to be seen – the barber’s pole, as left in the animation.

The barber’s sign shows a famous illusion.  The cylinder is rotating horizontally around a vertical axis, but the stripes look as if they are rising – which would be impossible, unless you had some long pole sliding through the cylinder.

You can begin to see why in the demo on the right:  focus on the vertical slot and the grating seems to be moving vertically (as in the barber’s pole).  But focus on the horizontal slot and in a moment the grating may seem to move horizontally.  Behind the round hole, for me it tends to look as if moving obliquely.

Want to know more about what’s going on?
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Dark Kanizsa Triangle

Here’s a dark-on-light, bubble version of The Kanizsa triangle.  The triangle is usually shown in white against black circles and lines, and can even look slightly brighter than background, though its edges are only indicated by the gaps in the lines and by the segments missing from the circles.   The brain adds the edges and fills in the triangle, as the most probable explanation for what’s missing.  The effect was created by Gaetano Kanizsa, as a demonstration of subjective contours, which in turn were first explored a bit over a century ago, as examples of Gestalt theory.  Bit of a link for enthusiasts that – ditto the following links – but if technical stuff is for you, there’s a great historical survey of the theory.  The theory as then developed is not now accepted, and just how the brain reconstructs the triangle is still debated.

Like many geometric illusions, and like the watercolour illusion (see recent post), the Kanizsa triangle also appears when reversed out as a black shape against bright lines and segments.  So here I’ve recruited some soap bubbles as a background to the effect.

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The Watercolour Illusion

I’m currently giving a hand with an exhibition on the science and art of illusion, called IllusoriaMente, which will be on in Alghero, Sardinia (italy) in the first week in September, in association with the annual European Conference on Visual Perception.  During the preparations I’ve come across this illusion, which I’d heard of but not checked out.  It’s beautiful and extraordinary.  The areas bounded by three wiggly loops in the picture look different colours, as if just tinted with watercolour, but the effect depends on the brain doing the colouring in.  Each boundary is made up of a dark and a lighter strip.  The infilling always takes its cue from the colour of the lighter strip, and always in the direction from the darker strip towards and beyond the lighter strip.  So in this image, the effect is a bit paradoxical.  The colour appears inside the two islands on the left, in comparison with the white background.  Yet to the right the colouring is outside the island, with the island itself looking a brighter shade of white than the overall background.

Two different processes seem to be co-operating, a colouring effect, and a figure/ground effect, enhancing the separation between the surrounded areas and the background.  If you like a bit more technicality, there’s a Scholarpedia article, with references to academic papers, edited by Baingio Pinna, one of the pioneers researching into the effect (and the organsiser of this year’s ECVP conference in Alghero).

Note added at 10/12/12:  check out our slightly more recent post on the watercolour illusion in reverse.

Hundred-year-plus puzzles

I remember being baffled by the illusion to the left when I was a child.  I think it was the first illusion I saw. The upper and lower blades are identical, but the lower one looks a lot larger. It’s called the  Jastrow illusion, and it’s not surprising I was amazed by it, because it’s as puzzling today as it was when Jastrow first published it, over a hundred years ago. To the right are two versions of a similarly mysterious illusion, known either as Titchener’s Circles, (or sometimes as the Ebbinghaus Illusion).  The central circles are objectively identical in size as seen to left and right, yet they look smaller when surrounded by the bigger circles and larger when surrounded by smaller circles.

Usually both illusions, Jastrow’s and Titchener’s, have been explained as the result of enhancement of contrasts in size. The key aspect of the Jastrow illusion, the theory goes, is the contrast between the long upper edge of the lower blade, and the short lower edge of the upper blade. The brain amplifies the size contrast between these edges, it is suggested, and the size of the whole figures gets adjusted in the process. The same kind of ramping up size contrast is proposed to explain the circles illusion, but this time it’s the contrast between the inner and outer circles. However, Jacques Ninio, from whose personal site I took the lower right figure, has a much more interesting suggestion …..

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Back-yard Poggendorff

This is another demo to see whether it’s true, as has often been claimed, that the Poggendorff illusion is much weaker when it’s seen in three dimensions.  To my eye, it’s not true:  in this lash-up of bits and pieces in my back yard, the illusion is still strong.  The long rod is straight as it passes behind the plank, but for me it looks just as decidedly misaligned either side of it when seen in 3D as when seen as a 2D photo.

To see the effect in three-dimensions, you’ll have to know how to view stereo picture pairs without a viewer, in what’s called “cross-eyed” mode.  If you don’t yet know that trick, you’ll find a tutorial at

But best to give it a miss if you have any problems with your eyes (apart from just being short sighted or colour blind, like I am) – or if viewing stereo pictures like these turns out to make you feel strange or queezy

Many theorists have suggested that the Poggendorff illusion arises because we try to impose an inappropriate 3D interpretation on a 2D figure, which is the usual form in which the illusion is presented. It’s a very reasonable theory that might work in various ways, but as it happens, I just don’t think it’s true.


When we see the figure presented as a stereo image, as at the head of this post, so the theory goes, the illusion should vanish.

For more detail, see our earlier post on the Poggendorff illusion and depth processing:

There’s another 3-D demo towards the end of that, but with the rod behind the plank in a plane parallel to that of the plank – in other words, not receding from us diagonally into the distance, like the rod in my back yard demo.  In this post, I wanted to check that the receding rod made no difference, and to my eye, it doesn’t, and nor does changing the orientation of the plank. The Poggendorff misalignment does look less in the lower pair, but that’s just because the plank that gets in the way in the lower images is at an angle where it looks much thinner.  It’s well-known that the illusion gets stronger as the gap between the rod segments gets wider.

However, all this does not rule out a role for depth processing in the illusion.  Qin Wang and Masanori Idesawa have shown that when the illusion is presented in 3D with the test arms in front of the inducing parallel, illusion vanishes.  That’s a real challenge for the 2D theories.

Muller-Lyer paradox (amended post)

Here’s a new way of looking at the Muller-Lyer illusion – paradoxically.

In the Muller-Lyer illusion, a line segment ending with outward pointing arrowheads looks shorter than an identical segment ending with inward pointing arrowheads. So in this version, whenever the arrowheads are visible, the left end of the line looks shorter than the (objectively identical) right end.

Here’s the paradox. When the arrowheads appear, the line segments instantly appear different in length, and yet the positions of the little globes marking the ends and centre-point of the line don’t appear to shift at all – which is impossible.

To make the point, in the bottom line, I’ve added an animated shift in the position of the middle globe, of just about the extent needed to produce the difference in apparent lengths of the line segments induced in the top line by the arrowheads.

The paradox is an example of the way that these so-called geometric illusions are not really so geometric.  Draw a figure, and if you change the length of a line, at least one of the line endpoints has to shift as well.  But in perceptual space it doesn’t necessarily follow.  So perceptual space can be pretty weird, or as researchers sometimes call it, non-Euclidean, because it isn’t always bound by the rigid constraints set out by the ancient Greek geometer Euclid.

(I’ve changed this post at 16/5/12.  There was other stuff in the original version, but it got much too complicated).