Editing Impossible cube

{{Short description|2D drawing of impossible 3D object}}
[[Image:Impossible cube illusion angle.svg|thumb|right|Viewed from a certain angle, this cube appears to defy the laws of geometry.]]
The '''impossible cube''' or '''irrational cube''' is an [[impossible object]] invented by [[M.C. Escher]] for his print ''[[Belvedere (M. C. Escher)|Belvedere]]''. It is a two-dimensional figure that superficially resembles a [[perspective drawing]] of a three-dimensional [[cube]], with its features drawn inconsistently from the way they would appear in an actual cube.

==Usage in art==
In Escher's ''Belvedere'' a boy seated at the foot of a building holds an impossible cube. A drawing of the related [[Necker cube]] (with its crossings circled) lies at his feet, while the building itself shares some of the same impossible features as the cube.<ref>{{citation
 | author = Bruno Ernst (Hans de Rijk)
 | editor1-last = Schattschneider | editor1-first = D. | editor1-link = Doris Schattschneider
 | editor2-last = Emmer | editor2-first = M.
 | contribution = Selection is Distortion
 | pages = 5–16
 | publisher = Springer
 | title = M. C. Escher's Legacy: A Centennial Celebration
 | year = 2003}}</ref><ref>{{citation|title= Impossibility: The Limits of Science and the Science of Limits|author=John D. Barrow|author-link=John D. Barrow|publisher=Oxford University Press|year=1999|isbn=9780195130829|page=14|url=https://books.google.com/books?id=0jRa1a4pD5IC&pg=PA14}}</ref>

Other artists than Escher, including [[Jos De Mey]], have also made artworks featuring the impossible cube.<ref>{{citation
 |author=Jos De Mey
 |author-link=Jos De Mey| editor1-last = Schattschneider | editor1-first = D.
 | editor2-last = Emmer | editor2-first = M.
 | contribution = Painting After M. C. Escher
 | pages = 130–141
 | publisher = Springer
 | title = M. C. Escher's Legacy: A Centennial Celebration
 | year = 2003}}</ref>
A doctored photograph purporting to be of an impossible cube was published in the June 1966 issue of ''[[Scientific American]]'', where it was called a "Freemish crate".<ref name="new angle"/><ref>{{citation|author=C. F. Cochran|year=1966|title=Letters|journal=[[Scientific American]]|volume=214|issue=5 |page=8|doi=10.1038/scientificamerican0566-8 }}</ref> An impossible cube has also been featured on an [[Austria]]n [[postage stamp]].<ref>{{citation|title=Stamping Through Mathematics|author=Robin J. Wilson|publisher=Springer|year=2001|isbn=9780387989495|page=102|url=https://books.google.com/books?id=vT1J3KOYkI0C&pg=PA102}}</ref>

==Explanation==
[[Image:Impossible cube different angle.png|thumb|A possible non-cube object that, viewed from appropriate angle, looks like an impossible cube.]]
[[File:Rotterdam kunstwerk De onmogelijke kuboïde.jpg|thumb|Impossible cube with forced perspective in [[Rotterdam]], by [[Koos Verhoeff]]]]
The impossible cube draws upon the ambiguity present in a Necker cube illustration, in which a cube is drawn with its edges as line segments, and can be interpreted as being in either of two different three-dimensional orientations.

An impossible cube is usually rendered as a Necker cube in which the line segments representing the edges have been replaced by what are apparently solid beams.
In Escher's print, the top four [[Woodworking joints|joints]] of the cube, and the upper of the two crossings between its beams, match one of the two interpretations of the Necker cube, while the bottom four joints and the bottom crossing match the other interpretation. Other variations of the impossible cube combine these features in different ways; for instance, the one shown in Escher's painting draws all eight joints according to one interpretation of the Necker cube and both crossings according to the other interpretation.

The apparent solidity of the beams gives the impossible cube greater visual ambiguity than the Necker cube, which is less likely to be perceived as an impossible object. The illusion plays on the human eye's interpretation of two-dimensional pictures as three-dimensional objects. It is possible for three-dimensional objects to have the visual appearance of the impossible cube when seen from certain angles, either by making carefully placed cuts in the supposedly solid beams or by using [[forced perspective]], but human experience with right-angled objects makes the impossible appearance seem more likely than the reality.<ref name="new angle">{{citation|journal=Perception|year=1984|volume=13|pages=153–154|title=A new angle on the freemish crate|author=Nancy E Smith|issue=2 |pmid=6504675|doi=10.1068/p130153}}</ref>

== See also ==
* [[Penrose triangle]]
* [[Blivet]]

==References==
{{reflist}}

[[Category:Optical illusions]]
[[Category:Impossible objects]]
[[Category:Cubes]]
[[Category:M. C. Escher]]