M.C. ESCHER AND IMPOSSIBLE SHAPES
An impossible object (also known as an impossible figure or an
undecidable figure) is a type of optical illusion consisting of a twodimensional figure which is instantly and subconsciously interpreted
by the visual system as representing a projection of a threedimensional object although it is not geometrically possible for such
an object to exist (at least not in the form interpreted by the visual
system).
The Penrose triangle, also known as the Penrose tribar, is an
impossible object. It was first created by the Swedish artist Oscar
Reutersvärd in 1934. The mathematician Roger Penrose
independently devised and popularized it in the 1950s, describing
it as "impossibility in its purest form". It is featured prominently
in the works of artist M. C. Escher, whose earlier depictions of
impossible objects partly inspired it.
The tribar appears to be a solid object, made of three straight beams of square cross-section
which meet pairwise at right angles at the vertices of the triangle they form. The beams may be
broken, forming cubes or cuboids; the logo of National Westminster Bank, comprising three
chevrons that form a broken hexagon, has sometimes been depicted with the chevrons as two
sides of cubes forming the corners of a Penrose triangle.
Impossible triangle sculpture as an optical illusion, East Perth, Western Australia
Impossible Triangle sculpture,
Gotschuchen, Austria
M.C. Escher's lithograph Waterfall depicts a
watercourse that flows in a zigzag along the long sides of
two elongated Penrose triangles, so that it ends up two
stories higher than it began. The resulting waterfall,
forming the short sides of both triangles, drives a water
wheel. Escher helpfully points out that in order to keep
the wheel turning some water must occasionally be
added to compensate for evaporation.
The Necker cube is an ambiguous line drawing.
On the left is one possible interpretation of the
Necker cube, often claimed to be the most
common interpretation. On the right is another
possible interpretation
The effect is interesting because each part of the picture is ambiguous by itself, yet the human
visual system picks an interpretation of each part that makes the whole consistent. The Necker
cube is sometimes used to test computer models of the human visual system to see whether they
can arrive at consistent interpretations of the image the same way humans do. With the cube on
the left, most people see the lower-left face as being in front most of the time. This is possibly
because people view objects from above, with the top side visible, far more often than from
below, with the bottom visible, so the brain "prefers" the interpretation that the cube is viewed
from above. Another reason behind this may be due to the brain's natural preference of viewing
things from left to right, therefore seeing the leftmost square as being in front. There is evidence
that by focusing on different parts of the figure one can force a more stable perception of the
cube. The intersection of the two faces that are parallel to the observer forms a rectangle, and the
lines that converge on the square form a "y-junction" at the two diagonally opposite sides. If an
observer focuses on the upper "y-junction" the lower left face will appear to be in front. The
upper right face will appear to be in front if the eyes focus on the lower junction. Blinking while
being on the second perception will probably cause you to switch to the first one.
It is possible to cause the switch
to occur by focusing on different
parts of the cube. If one sees the
first interpretation on the right it
is possible to cause a switch to
the second by focusing on the
base of the cube until the switch
This is an example of two identical Necker cubes, the one on
occurs to the second
the left showing an intermediate object (blue bar) going in
interpretation. Similarly, if one
"down from the top" while the one on the right shows the
is viewing the second
object going in "up from the bottom" which shows how the
interpretation, focusing on the
image can change its perspective simply by changing which
left side of the cube may cause a
face (front or back) appears behind the intervening object.
switch to the first.
Necker cube on the left,
impossible cube on the right.
This cube appears to defy the
laws of geometry
Humans do not usually see an inconsistent interpretation of the cube. A cube whose edges cross
in an inconsistent way is an example of an impossible object, specifically an impossible cube.
Belvedere is a lithograph print by the Dutch artist M. C. Escher,
first printed in May 1958. It shows a plausible-looking building
that is actually an impossible object. In this print, Escher uses
two-dimensional images to depict objects free of the confines of
the three-dimensional world. The image is of a rectangular
three-story building. The upper two floors are open at the sides
with the top floor and roof supported by pillars. From the
viewer's perspective, all the pillars on the middle floor are the
same size at both the front and back, but the pillars at the back
are set higher. The viewer also sees by the corners of the top
floor that it is at a different angle than the rest of the structure.
All these elements make it possible for all the pillars on the middle floor to stand at right angles,
yet the pillars at the front support the back side of the top floor while the pillars at the back
support the front side. This paradox also allows a ladder to extend from the inside of the middle
floor to the outside of the top floor.
There is a man seated at the foot of the building holding an
impossible cube. He appears to be constructing it from a
diagram of a Necker cube at his feet with the intersecting lines circled. The window next to him
is closed with an iron grille that is geometrically valid but practically impossible to assemble.
Belvedere built with Lego pieces!