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.

Read on for more on size-constancy.

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An Ambiguous Skull-Mask Illusion

Is this a picture of a mask looking at a skull it’s holding up for inspection, or vice versa?

I got the idea from a print by Picasso, Young man with mask of a bull, faun and profile of a woman. There’s a copy in the Art Gallery of New South Wales in Sydney, and you can see it by calling up,

search for Picasso, scroll down the results and you’ll find it!

Classic illusion books


Here’s another rotating head illusion, just to introduce a list of my favourite illusion books. It’s not one you’ll find in the books, because I only just drew it.  Still, that leaves plenty of old illusions, and there are stacks of fun books on the subject, many of them excellent. Here are some I think are real classics:

Richard GREGORY, Eye and Brain: the psychology of seeing (5th Ed.) OUP 1998

The standard introduction to vision and illusions, say, sixth form to first year college level, authoritative but great fun.


J.O.ROBINSON, The Psychology of Visual Illusion, New Ed., Dover Pubs, 1998

A bit more technical, but with a comprehensive selection of geometric illusions.


Jacques NINIO, The Science of Illusions, Cornell Uni Press 2001

Another fascinating general introductory text, by an eminent researcher


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Tessellations with figures

A tessellation is a pattern made up of elements that repeat with no gaps and no overlaps.  The elements may be abstract shapes, or may be recognisable objects or creatures, like the ones in the tessellations of M.C.Escher.  When I begun playing around with tessellations, I thought understanding the procedures needed to make patterns that tessellate would be the hard part.  I thought it would be fairly easy to find creatures in whatever shapes I ended up with.  Not so.  The procedures aren’t so hard.  But fitting creatures into them I found really difficult.  Here’s one of my first attempts.  I started with one of the most complicated recipes for a tessellation.  


For the details of the procedures, which give shapes that tessellate, see my tessellation tutorial. Essentially, the boundary of every shape that will tessellate is made up of pairs of lines. Within each pair, an identical line repeats, either by rotation, reflection, or just by shifting over. In the example above, there are four pairs of lines, two of them with rotations and two with reflections.

But what creature could I discover in this shape? Here’s what I came up with, a cross between an elephant and a rhino, with a little man on its back.

And it does tessellate!  It gives a pattern in which the elenoceros repeats four times, right way up facing both ways, and then upside down facing both ways.

Ambiguous patterns

I reckon some images look beautiful because they bamboozle the brain processes we normally depend on to make sense of the world.  I don’t know why  that can help make patterns and pictures look beautiful. Nor do I think perceptual puzzlement is the essence of art, or anything like that.  But just from a practical point of view, if you are an artist (or a composer, poet or architect), a motif that’s puzzling can seem to offer a stepping off point for aesthetic effects.

Here are two beautiful examples from architectural decoration, both just about 500 years old.  The first is the dome of the Mausoleum of Sultan Qaitbay in Cairo.

What’s puzzling about this is that a single line segment can be part of the edge of an object, such as a star, and at the same time part of a line that meanders over the whole surface.  Edges don’t behave like that in everyday vision.  Here’s the dome with added lines, left below, to show what I mean. Look at the segment that is labeled with both blue and yellow lines.

Then note that you can do just the same with the lines that outline the octagons on the ceiling in the picture to the right – every edge is also part of a fan of lines.  That ceiling is in Christchurch Cathedral in Oxford, and we even know who designed it – William Orchard, the Master Mason.  Now we’d call him the architect.  Here’s a picture showing a bit more of the ceiling.

I don’t think it’s just the puzzling features that make these patterns so beautiful.  Interlace patterns like these look like small segments of patterns that go on for ever, and in both Christianity and Islam were a metaphor for perfection and heaven.  And that’s how I reckon artists turn perceptually puzzling effects into something more than amusing images – they choose motifs that are also metaphors for some deeper meaning.

You may be more familiar with interlace as a technology for managing video images.  For interlace as a pattern motif in christian art, see the Wikipedia article on Celtic Knots.  Or try here to find out about interlace patterning in the context of islamic faith.

Revolving Heads

Heads that present one character one way up and another when rotated have been favorite illusions for over a century. Here are two heads from a cartoon story I devised about a boy who gets stuck in a weird hotel. The receptionist and chef, (Mr. and Mrs. Turner …. ) seem OK at first, but then transform into two sinister old men when their heads rotate.

For an animation of Mr. and Mrs. Turner see below:

There are loads of great rotating heads at: Rotating Heads

Soap Bubbles

Soap bubbles aren’t illusions, but I am fascinated by them, and have a special technique for photographing them, (with a little help from Photoshop).

Here’s a picture of a bubble moon over London:

I’ll return to the subject of bubble pictures. Meanwhile, the most beautiful images of soap films are made by Karl Deckart.  Jason Tozer is another photographer who has recently made some stunning new photos of soap films and planet like segments.

Optical Illusion Cartoon Story

For years I’ve wished someone would make an animated cartoon in which the events depend on the kind of visual transformations we see in many illusion pictures. It won’t be easy. Salvador Dali loved effects of these kinds, and helped sketch out a scheme for a Disney movie (though not one with a real storyline) in 1945/6. It’s called Destino. It didn’t get made, until Disney’s nephew Roy Disney made a version in about 2000. I don’t think it was so successful, but it was a fascinating chance to see what works, and what is less successful when animated. Take a look at a trailer and decide,

I reckon Goo-Shun Wang’s wonderful, recent animation of a character trapped on an Escher-style, never-ending staircase is far more successful:

To explore the kind of effects I think might work in a narrative, I devised a couple of still-picture cartoon stories. Here’s a pair of frames from one you can view on the www, in which the characters are almost trapped on another never-ending staircase, when a spiral stair suddenly transforms:

Check out the whole thing at:

it also includes loads of hints on drawing illusion pictures.

Illusions and visual special effects – explanations and tutorials