In an impossible figure, seemingly real objects—or parts of objects—form geometric relations that physically cannot happen. Dutch artist M. C. Escher, for instance, depicted reversible staircases and perpetually flowing streams. Mathematical physicist Roger Penrose drew his famously impossible triangle, and visual scientist Dejan Todorovi of the University of Belgrade in Serbia created a golden arch that won him third prize in the 2005 Best Illusion of the Year Contest. These effects challenge our hard-earned perception that the world around us follows certain, inviolable rules. They also reveal that our brains construct the feeling of a global percept—an overall picture of a particular item—by sewing together multiple local percepts. As long as the local relation between surfaces and objects follows the rules of nature, our brains don't seem to mind that the global percept is impossible.

Several contemporary sculptors recently have taken up the challenge of creating impossible art. That is, they are interested in shaping real-world 3-D objects that nonetheless appear to be impossible. Unlike classic monuments—such as the Lincoln Memorial in Washington, D.C.—which can be perceived by either sight or touch, impossible sculptures can be interpreted (or misinterpreted, as the case may be) only by the visual mind.

PENROSE TRIANGLE
The impossible triangle (also called the Penrose triangle or the tribar) was first created in 1934 by Oscar Reutersvärd. Penrose attended a lecture by Escher in 1954 and was inspired to rediscover the impossible triangle. Penrose (who at the time was unfamiliar with the work of Reutersvärd, Piranesi and other previous discoverers of the impossible triangle) drew the illusion in its now most familiar form (left) and published his observations in the British Journal of Psychology in 1958, in an article co-authored with his father, Lionel. In 1961 the Penroses sent a copy of the article to Escher, who incorporated the effect into Waterfall, one of his most famous lithographs (right).

IMPOSSIBLE ARCH
Elusive Arch, by Todorovi, shows a new impossible figure. The left-hand part of the figure appears as three shiny oval tubes. The right-hand part looks corrugated, with three alternating pairs of shallow matte ridges and grooves. The bright streaks on the figure's surface are seen either as highlights at the peaks and troughs of the tubes or as inflections between grooves. Determining the direction of the apparent illumination falling on the figure is difficult: it depends on whether we interpret the light as falling on a receding or an expanding surface. Further, determining the exact position and shape of the transition region near the center of the arch is maddening, because the local 3-D interpretations defy the laws of illumination. For more about the arch, see http://illusioncontest.neuralcorrelate.com/2005/elusive-arch.

HOMAGE TO ESCHER
Escher's Belvedere (left) showcases columns that switch walls between their bases and capitals, a straight ladder whose base rests inside the building yet nonetheless enters the building from the outside at its top, and a sitting man holding an impossible cube. Mathieu Hamaekers, a Belgian mathematician and sculptor, created an homage to Belvedere that features a real-life impossible cube. This photograph (below) shows the artist holding the sculpture Upside Down, built in 1985.

IMPOSSIBLE BOX
Hans Schepker has built outstanding sculptures of impossible objects, such as this Crazy Crate made from glass (above, left). Other views of the crazy crate show the method behind the madness (above, center and right). Notice that the illusion works only from a specific vantage point. At any other angle, the illusion fails. Scientists refer to this as the accidental view, but there is nothing accidental about it. To perceive the illusion, the view must be carefully staged and choreographed, or else the audience will fail to see the “impossible” sculpture.

BACKYARD MAGIC
The late magician Jerry Andrus created this crazy crate, shown here from two different angles, in his backyard. The photograph on the right reveals the magic.

Industrial-Size Triangle
Artist Brian McKay created a giant version of the impossible triangle (below, left) in Perth, Australia, in collaboration with architect Ahmad Abas. How did they do that? A photograph taken from another angle (below, right) reveals the trick.

A Twist on the Triangle
Unity, an impossible triangle created by Hamaekers in 1995, is now installed in Ophoven, Belgium. Again the viewer's location relative to the object is critical. But in this case, Hamaekers used a different physical method to achieve the illusion.

A Closed Triangle
Unlike most 3-D Penrose triangles, the sculptures by French artist and magician Francis Tabary are neither twisted nor open. They look impossible from a relatively large range of vantage points, although they do fail when seen from some viewpoints. The Tabary sculpture shown here is a four-cube-sided Penrose triangle.

Making Escher 3-D
Andrew Lipson, a self-described “professional nerd” with no official connection to the Lego Group, and his friend Daniel Shiu have rendered five Escher works in Lego blocks, including this model of Escher's Ascending and Descending (left). The original work by Escher, a 1960 lithograph, shows a large building with an endless staircase on its roof (bottom right). Some of the people are ascending the staircase, while others are descending.

Lipson and Shiu spent considerable time studying Escher's work before beginning construction. In their photograph of the finished sculpture, it looks as though the staircase is continuous. But in this picture taken from another angle (top right), you can see that the edges of the staircase do not meet. The Lego illusion works only if the photograph is taken from exactly the right viewing angle.

WHICH WAY IS UP?
The Terrace, a 1998 work by British artist David MacDonald, is an example of impossible perspective. Are we looking at this scene from above or below the checkerboard? MacDonald produces impossible perspectives akin to those created by Escher, but photographically. He made this image by creating a computer wireframe matrix and filling it in with digitally photographed textures and objects.

IT'S ALL RELATIVE
Lipson and Shiu also worked together on a Lego rendition of Escher's Relativity (top). The original version, a popular lithograph first printed by Escher in 1953, depicts a surreal architectural structure in which there seem to be three separate sources of gravity (bottom left). The stairways are double-sided, and each stair is double-treaded.

A photograph taken from a slightly different angle and farther away (bottom right) shows how the sculpture is made. Lipson and Shiu used lots of scaffolding to hold it up. This was their fourth Escher picture rendered in Lego blocks.

ONE-MAN BAND
Encore, by the late Japanese artist Shigeo Fukuda, uses similar principles to represent a pianist and violinist in the same sculpture when viewed from two vantage points. You can see only half of the duet at once, and neither is visible unless the sculpture is viewed from the side.

MIRROR IMAGE
Another work by Fukuda, Underground Piano, looks like a pile of piano parts unless you stand in the right place and view the “reassembled” piano in the mirror.

SHADOW PLAY
Fukuda welded together 848 forks, knives and spoons to make Lunch with a Helmet On. Here he cleverly resolves the illusion by placing a light at the critical vantage point, making the motorcycle obvious only in the shadow cast by the utensil pile.

IMELDA'S DREAM COME TRUE
Imelda Marcos, widow of the former Philippines dictator Ferdinand Marcos, was infamous for her shoe collection but also for quotes such as this one: “People say I'm extravagant because I want to be surrounded by beauty. But tell me, who wants to be surrounded by garbage?” Well, Imelda, now you can be surrounded by both, courtesy of artists Tim Noble and Sue Webster, who create eye-catching artwork from rubbish.

In 1998 Noble and Webster created this sculpture, Dirty White Trash (with Gulls), using six months' worth of their own garbage. Like Fukuda, they used a strategically placed light source to cast their own shadows on the wall. The sculpture appeared in a 2003 exhibition at the P.S.1 Contemporary Art Center in Long Island City, N.Y.

AND THE WINNER IS...
For several years, Italian sculptor Guido Moretti has donated copies of his Three-Bar Cube and other impossible sculptures as trophies for the Best Illusion of the Year Contest. Depending on your vantage point, Three-Bar Cube can appear to be a cube, a solid structure or an impossible triangle. For more information, see http://illusioncontest.neuralcorrelate.com/trophies.