''Waterfall'' () is a

lithograph Lithography () is a planographic method of printing originally based on the miscibility, immiscibility of oil and water. The printing is from a stone (lithographic limestone) or a metal plate with a smooth surface. It was invented in 1796 by ...

by the Dutch artist

M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...

, first printed in October 1961. It shows a perpetual motion machine where water from the base of a

waterfall A waterfall is any point in a river or stream where water flows over a vertical drop or a series of steep drops. Waterfalls also occur where meltwater drops over the edge of a tabular iceberg or ice shelf. Waterfalls can be formed in seve ...

appears to run downhill along the water path before reaching the top of the waterfall. While most two-dimensional artists use relative proportions to create an illusion of depth, Escher here and elsewhere uses conflicting proportions to create a visual

paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictor ...

. The

watercourse A stream is a continuous body of water, body of surface water Current (stream), flowing within the stream bed, bed and bank (geography), banks of a channel (geography), channel. Depending on its location or certain characteristics, a strea ...

supplying the waterfall (its aqueduct or leat) has the structure of three Penrose triangles. A Penrose triangle is an impossible object designed by Oscar Reutersvärd in 1934, and found independently by

Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...

in 1958.

Description

The image depicts a watermill with an elevated aqueduct and

waterwheel A water wheel is a machine for converting the kinetic energy of flowing or falling water into useful forms of power, often in a watermill. A water wheel consists of a large wheel (usually constructed from wood or metal), with numerous blade ...

as the main feature. The aqueduct begins at the waterwheel and flows behind it. The walls of the aqueduct step downward, suggesting that it slopes downhill. The aqueduct turns sharply three times, first to the left, then to the right, and finally to the left again. The viewer looks down at the scene diagonally, which means that from the viewer's perspective the aqueduct appears to be slanted upward. The viewer is also looking across the scene diagonally from the lower right, which means that from the viewer's perspective the two left-hand turns are directly in line with each other, while the waterwheel, the forward turn and the end of the aqueduct are all in line. The second left-hand turn is supported by pillars from the first, while the other two corners are supported by a tower of pillars that begins at the waterwheel. The water falls off the edge of the aqueduct and over the waterwheel in an impossible infinite cycle; in his notes on the picture, Escher points out that some water must be periodically added to this perpetual motion machine to compensate for

evaporation Evaporation is a type of vaporization that occurs on the Interface (chemistry), surface of a liquid as it changes into the gas phase. A high concentration of the evaporating substance in the surrounding gas significantly slows down evapora ...

. The use of the Penrose stairs is paralleled by Escher's '' Ascending and Descending'' (1960), where instead of the flow of water, two lines of monks endlessly march uphill or downhill around the four flights of stairs. The two support towers continue above the aqueduct and are topped by two compound

polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

, revealing Escher's interest in mathematics as an artist. The one on the left is a compound of three cubes. The one on the right is a

stellation In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific ...

of a

rhombic dodecahedron In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ...

(or a compound of three non-regular octahedra) and is known as Escher's solid. Below the mill is a garden of bizarre, giant plants. This is a magnified view of a cluster of

moss Mosses are small, non-vascular plant, non-vascular flowerless plants in the taxonomic phylum, division Bryophyta (, ) ''sensu stricto''. Bryophyta (''sensu lato'', Wilhelm Philippe Schimper, Schimp. 1879) may also refer to the parent group bryo ...

and

lichen A lichen ( , ) is a hybrid colony (biology), colony of algae or cyanobacteria living symbiotically among hypha, filaments of multiple fungus species, along with yeasts and bacteria embedded in the cortex or "skin", in a mutualism (biology), m ...

that Escher drew in ink as a study in 1942. The background seems to be a climbing expanse of terraced farmland.

References

External links

Escher's Solid—from Wolfram MathWorld
Includes a great deal of metric data

from George W. Hart {{Optical illusions Mathematical artworks Works by M. C. Escher 1961 paintings Impossible objects Water in art

Description

See also

References

External links