Impossible objectAn 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-dimensional object but cannot exist as a solid object. Impossible objects are of interest to psychologists, mathematicians and artists without falling entirely into any one discipline.
TriangleShape with three sidesTriangle, a polygon with three corners (vertices) and three lines (sides)A triangle or trigon is a polygon consisting of three sides connected at three corners. It is one of the basic shapes in geometry and the simplest of polygons. The corners, also called vertices, are zero-dimensionalpoints while the sides connecting them, also called edges, are one-dimensional line segments.
Euclidean spaceEuclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimensionn, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes.
Waterfall (M. C. Escher)Waterfall (Dutch: Waterval) is a lithograph by the Dutch artist M. C. Escher, first printed in October 1961. It shows a perpetual motion machine 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.
Roger PenroseEnglish mathematician, mathematical physicist (born 1931)Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, and philosopher of science. He is EmeritusRouse Ball Professor of Mathematics at the University of Oxford, an emeritus fellow of Wadham College, Oxford, and an honorary fellow of St John's College, Cambridge, and University College London.
3-manifoldIn mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe to a small enough observer. This is made more precise in the definition below.
Lionel PenroseLionel Sharples PenroseFRS (11 June 1898 – 12 May 1972) was an English psychiatrist, medical geneticist, paediatrician, mathematician and chess theorist, who carried out pioneering work on the genetics of intellectual disability. Penrose was initially the Galton professor of eugenics (1945–1963) at University College London, before having his title changed to professor of human genetics (1963–1965) at his request. He was later emeritus professor.
SolidSolid is a state of matter in which atoms are closely packed and are difficult to move past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree of resistance being dependent upon the specific material under consideration.
Oscar ReutersvärdOscar Reutersvärd (Swedish pronunciation:; 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 father of the impossible figure", although there are much older examples, e.g. Hogarth's Satire on False Perspective.
Optical illusionIn visual perception, an optical illusion (also called a visual illusion) is an illusion caused by the visual system and characterized by a visual percept that arguably appears to differ from reality. Illusions come in a wide variety; their categorization is difficult because the underlying cause is often not clear but a classification proposed by Richard Gregory is useful as an orientation.
M. C. EscherMaurits Cornelis Escher (; Dutch:; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithographs, and mezzotints, many of which were inspired by mathematics. Despite wide popular interest, for most of his life Escher was neglected in the art world, even in his native Netherlands. He was 70 before a retrospective exhibition was held.
Intrinsic metricIn the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to (or very close to) that distance.*As an Amazon Associate I earn from qualifying purchases.