Don't believe your eyes

Impossible TriangleWe're all so used to two-dimensional drawings of solids, we don't give them a second glance. But if the artist bends the rules, the result is a boggling double-take

The blueprint seems OK. Three threaded prongs fine, the machine shop can handle that. The metal casting should also be straightforward - two simple arms protruding from a square base. Just get the boss to initial it and send it on to the manufacturing plant.

Blueprint for Devil's Pitchfork

Devil's pitcfork: a blueprint for failure

A classic "impossible figure" from the early 1960s. At first this object seems fine, with each part a recognisable, consistent shape; but it doesn't take long for the brain to clock that there's something amiss. As the eye travels along a line, the line imperceptibly changes function.

But wait a minute. Take a closer look. When you try to get your mind round the picture, things begin to fall apart.
Each end of the drawing makes sense, but when you try to integrate them you're left with rather less than nothing. As for the nuts that are supposed to fit on those prongs - the less said the better.

Welcome to the unsettling world of impossible figures. What you've been looking at is a blueprint for the so-called "devil's pitchfork" - also known as the two-pronged trident or the three- stick cleavis. Impossible artists affectionately call it a "blivet", as in, it asks you to believe it. But whatever you call it, you can't have one. The drawing is not of anything, it simply exploits the way your brain tries to make three-dimensional sense of what it is seeing in only two dimensions.

Impossible figures are a development of the simpler ambiguous figures, which have several possible three-dimensional realities. Look at a set of "baby blocks" carefully (see below left, middle). You'll probably see a pile of boxes stacked in a corner - the shaded diamond-shapes represent their tops. Look again, and with luck you'll experience a mental flip: the boxes turn upside down and now they have dark bottoms.

Toying with the perceptions


Look at the simple Necker cube (above) and watch it flip. Because there's no perspective, either side could be the nearer. At the next level of complexity, the "baby blocks" in  the middle could be a stack of boxes on a floor, or hanging from the ceiling. Try turning the page through 45 degrees.In Roger Penrose's groundbreaking Impossible Tribar from the 1950s (above) each part is correct. But the whole triangle will get you in a twist. The first two are called "ambiguous figures"; the third is an "impossible" or "dimensionally challenged" figure.

Look long and hard and you may see them as simultaneously stacked from the ground and suspended from the ceiling. There are other interpretations too, but don't look too long or you'll give yourself a headache.

Why are we fooled? When we look at any picture the brain instantly processes the visual data and allow us to see a three- dimensional object on a flat surface - although we know it cannot be there. That illusion is common place - and entirely beneficial. The art of painting depends on it for a start, as do all areas of human endeavour where we need to convey a sense of three dimensions on a piece of paper - which is practically everything we do.
Wick's Doghouse

A DIY Nightmare? Prankster Walter is in the doghouse

The American photographer and impossible artist Walter Wick shrugs off his latest handiwork, the Kennel of Confusion. This is the opposite of normal impossible figures: a case of a three-dimensional representation of an impossible three dimensional shape. Got that?

But our brains didn't evolve to appreciate paintings, or blueprints for that matter. They evolved to make life-saving judgments in a world where it was vital to interpret often scant visual information very fast. Danger or dinner? Fight or flee?

Do not adjust your eye sight: evolution has hard-wired these visual glitches into your brain

And it's our brains that sort out the information, not our eyes. Light from the outside world strikes the retina in a pattern of varying brightness and colour, but the pattern means nothing without the power of the brain to process the signals that flash to it along the optic nerve.

That's quite a trick on the part of the brain. Technology has proved very good at producing "artificial retinas" - devices for collecting information from light. There's one in every video camera, and those used in advanced models are vastly more sensitive than the ones in our eyes.

Ernst and Impossible Chess Boards

Board to tears? These impossible chessboards would even make Gary Kasparov weep. Bruno Ernst (left),the world's leading Escher expert,adapted them from Lionel Penrose's Impossible Staircase.

But making sense out of the patterns a retina detects is much harder: programmers have had immense difficulty "teaching" even super-computers to recognise quite simple images.
So it's not surprising that the human brain can sometimes be fooled. Where and how it is fooled is a subject that greatly interests psychologists studying how perception works. But people have been playing around with ambiguous figures for far longer than psychologists have been explaining them. Designs incorporating baby blocks are found on the floors of Romanesque churches dating to the 10th century, and variations can be traced even further back.

Getting out of the DIY doghouse

The Wick's doghouse is of course an optical illusion - not mind bending reality.
 The pieces that make up the frame contain judiciously sawn - out gaps. Photographed from just the right angle they are filled in by those behind - which appear to be in front.
 To further confuse the eye clever shaping means that some of the pieces of wood are in places,and so youbelieve them to be closer.
the seemingly horizontal strut on which a puppy seems to be climbing in the original trick shot is actually a separate piece of wood on the ground behind the kennel.

Doghouse Solution

Two-dimensional space has been manipulated since the earliest cave paintings to give an approximation of 3-D space, but it wasn't until half-way through the present century that artists thought of taking it as their subject matter. The Surrealists, in the 1930s and '40s, delighted in using ambiguous and impossible figures to draw attention to the absurdity of the "modern condition".

The first person to write about ambiguous figures was L A Necker , a mineralogy professor from Geneva, who described the "Necker cube" in 1832. This is simply a wire-frame cube, drawn from an angle and without perspective, and it has the same effect on our perception as the blocks. The "front" corner flips to become the "back" corner and vice versa. In other words, there are at least two solutions to the perceptual question "what is it?".

As 19th-century interest in perception grew, more ambiguous figures were developed. One major if elementary discovery was the effect of shadow a picture of a sun-lit mound appears to be a hollow when turned over - and the crucial role light and shade play in art. In effect, visual clues are interpreted by the brain in a programmed way - which can be tricked remarkably easily.

Concave and Convex


Convex and Concave (above left) employs the same principle as the blivet: the two sides are correct and it's the joining up that creates the inconsistency.

Where the blivet spreads its connection over some distance (so the eye has trouble seeing the two ends simultaneously), Escher placed the two parts beside each other, skilfully using shade to fudge the join. Note the baby blocks on the hanging banner.

Waterfall (above),three tribars joined together is a perplexing contribution to the perpetual motion debate.

MC Escher - founding father of a special school of art

Maurits Escher

After World War ll the Dutch artist Maurits Cornelis Escher (1898-1972) began drawing his "impossible figures" , which have become some of the best - known images of our time.

It is a small step from the Necker cube to the truly impossible "tribar" , though that step took more than a century. The figure seems to have been invented twice: once, in 1934, by the Swedish art historian Oscar Reutersväard (with a bubble) ; and again, quite independently in 1958 by Roger Penrose, now Sir Roger, an Oxford mathematics professor and author of  The Emperor's New Mind.

Figuring the impossible


The first impossible figure was by the Swedish artist Oscar Reutersvärd in 1934. Impossible artists who have followed in his strange footsteps regard him highly for his formal economy

Visual Illusions

A knotty problem A impossible fourbar is more fiendish when it is intersected by a possible cross
Up the impossible staircase to nowhere
The eye takes its cue from the most obvious fact,here that the steps must be coming up out of the hole in the ground.The first storey cannot be flat
Another one that doesn't quite square
The individual corners of a cuboid can be combined into impossible forms

A mad spiralling thing
 In this spiral the illusion that all the lines are straight gets you into trouble.A variety on the impossible cuboid theme.

More subtle than the blivet, the Tribar has three linked sections (some variants have even more), and any two of the links make perfect sense. It is only when the eye follows the loop all the way round to the third link that the inconsistency leaps out. The Tribar has become a familiar image in commercial logos and other graphics helped by the fact that Penrose, not foreseeing how popular it would become never copyrighted his design.
Put the Necker cube together with the Penrose tribar, add some skilful photography, and you get the apparently three-dimensional dog kennel on the previous page. Created by American prankster William Wick back in the early 1980s, it too is impossible. The clues the brain uses to reconstruct a solid image from a flat picture have been cunningly manipulated, and added to real three dimensional red herrings, until the mind boggles at the contradictions it is being asked to accept.

Are impossible figures an art form, or a research tool for psychologists? Both, of course. But as the master of creating art from optical paradox, M C Escher shows us, there's no need to get tied up in knots about them.

David Singmaster