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''Waterfall'' ( nl, Waterval) is a
lithograph Lithography () is a planographic method of printing originally based on the 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 the German au ...
by the Dutch artist
M. C. Escher Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made Mathematics and art, mathematically inspired woodcuts, lithography, lithographs, and mezzotints. Despite wide popular interest, Escher was for ...
, first printed in October 1961. It shows a
perpetual motion machine Perpetual motion is the motion of bodies that continues forever in an unperturbed system. A perpetual motion machine is a hypothetical machine that can do work infinitely without an external energy source. This kind of machine is impossible, a ...
where water from the base of a waterfall 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 premises, leads to a seemingly self-contradictory or a logically u ...
. The
watercourse A stream is a continuous body of surface water flowing within the bed and banks of a channel. Depending on its location or certain characteristics, a stream may be referred to by a variety of local or regional names. Long large streams are ...
supplying the waterfall (its aqueduct or
leat A leat (; also lete or leet, or millstream) is the name, common in the south and west of England and in Wales, for an artificial watercourse or aqueduct dug into the ground, especially one supplying water to a watermill or its mill pond. Othe ...
) has the structure of two
Penrose triangle The Penrose triangle, also known as the Penrose tribar, the impossible tribar, or the impossible triangle, is a triangular impossible object, an optical illusion consisting of an object which can be depicted in a perspective drawing, but cannot ...
s. A Penrose triangle is an
impossible object An impossible object (also known as an impossible figure or an undecidable figure) is a type of optical illusion that consists of a two-dimensional figure which is instantly and naturally understood as representing a projection of a three-dim ...
designed by
Oscar Reutersvärd Oscar Reutersvärd (29 November 1915 – 2 February 2002) was a Swedish graphic artist, who in 1934 pioneered the art of 3D drawings that may initially appear feasible, yet cannot be physically constructed. He is sometimes described as "the fathe ...
in 1934, and found independently by
Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fell ...
in 1958.


Description

The image depicts a watermill with an elevated aqueduct and
waterwheel A water wheel is a machine for converting the energy of flowing or falling water into useful forms of power, often in a watermill. A water wheel consists of a wheel (usually constructed from wood or metal), with a number of blades or bucke ...
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 Perpetual motion is the motion of bodies that continues forever in an unperturbed system. A perpetual motion machine is a hypothetical machine that can do work infinitely without an external energy source. This kind of machine is impossible, a ...
to compensate for
evaporation Evaporation is a type of vaporization that occurs on the surface of a liquid as it changes into the gas phase. High concentration of the evaporating substance in the surrounding gas significantly slows down evaporation, such as when humidi ...
. The use of the Penrose stairs is paralleled by Escher's ''
Ascending and Descending ''Ascending and Descending'' is a lithograph print by the Dutch artist M. C. Escher first printed in March 1960. The original print measures . The lithograph depicts a large building roofed by a never-ending staircase. Two lines of identically ...
'' (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 (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...
, revealing Escher's interest in mathematics as an artist. The one on the left is a
compound of three cubes In geometry, the compound of three cubes is a uniform polyhedron compound formed from three cubes arranged with octahedral symmetry. It has been depicted in works by Max Brückner and M.C. Escher. History This compound appears in Max Brückner ...
. The one on the right is a
stellation In geometry, stellation is the process of extending a polygon in two dimensions, 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 el ...
of a
rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedr ...
(or a compound of three non-regular
octahedra In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
) 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 flowerless plants in the taxonomic division Bryophyta (, ) ''sensu stricto''. Bryophyta (''sensu lato'', Schimp. 1879) may also refer to the parent group bryophytes, which comprise liverworts, mosses, and hornw ...
and
lichen A lichen ( , ) is a composite organism that arises from algae or cyanobacteria living among filaments of multiple fungi species in a mutualistic relationship.terraced farmland.


See also

*
First stellation of rhombic dodecahedron In geometry, the first stellation of the rhombic dodecahedron is a self-intersecting polyhedron with 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual ...


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