Three Worlds

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M. C. Escher Dutch graphic artist known for his mathematically-inspired works

Maurits Cornelis Escher was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints. Despite wide popular interest, Escher was for long somewhat neglected in the art world, even in his native Netherlands. He was 70 before a retrospective exhibition was held. In the twenty-first century, he became more widely appreciated, with exhibitions across the world.

Self-reference A sentence, idea or formula that refers to itself

Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding. In philosophy, it also refers to the ability of a subject to speak of or refer to itself, that is, to have the kind of thought expressed by the first person nominative singular pronoun "I" in English.

<i>Gödel, Escher, Bach</i> 1979 book by Douglas Hofstadter

Gödel, Escher, Bach: an Eternal Golden Braid, also known as GEB, is a 1979 book by Douglas Hofstadter. By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher, and composer Johann Sebastian Bach, the book expounds concepts fundamental to mathematics, symmetry, and intelligence. Through illustration and analysis, the book discusses how, through self-reference and formal rules, systems can acquire meaning despite being made of "meaningless" elements. It also discusses what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of "meaning" itself.

Penrose triangle Impossible object

The Penrose triangle, also known as the Penrose tribar or the impossible tribar, is a triangular impossible object, an optical illusion consisting of an object which can be depicted in a perspective drawing, but cannot exist as a solid object. It was first created by the Swedish artist Oscar Reutersvärd in 1934. Independently from Reutersvärd, the triangle was devised and popularized in the 1950s by psychiatrist Lionel Penrose and his son, prominent mathematician Roger Penrose, who described 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.

Penrose stairs Impossible object

The Penrose stairs or Penrose steps, also dubbed the impossible staircase, is an impossible object created by Lionel Penrose and his son Roger Penrose. A variation on the Penrose triangle, it is a two-dimensional depiction of a staircase in which the stairs make four 90-degree turns as they ascend or descend yet form a continuous loop, so that a person could climb them forever and never get any higher. This is clearly impossible in three-dimensional Euclidean geometry.

<i>Still Life with Spherical Mirror</i> print by M. C. Escher

Still Life with Spherical Mirror is a lithography print by the Dutch artist M. C. Escher first printed in November 1934. It depicts a setting with rounded bottle and a metal sculpture of a bird with a human face seated atop a newspaper and a book. The background is dark, but in the bottle can be seen the reflection of Escher's studio and Escher himself sketching the scene.

<i>Regular Division of the Plane</i> drawing series

Regular Division of the Plane is a series of drawings by the Dutch artist M. C. Escher which began in 1936. These images are based on the principle of tessellation, irregular shapes or combinations of shapes that interlock completely to cover a surface or plane.

<i>Reptiles</i> (M. C. Escher) lithograph print by M. C. Escher

Reptiles is a lithograph print by the Dutch artist M. C. Escher first printed in March 1943. It touches on the theme found in much of his work of mathematics in art.

<i>Puddle</i> (M. C. Escher)

Puddle is a woodcut print by the Dutch artist M. C. Escher, first printed in February 1952.

<i>Waterfall</i> (M. C. Escher) Lithograph print by M. C. Escher

Waterfall 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.

<i>Mise en abyme</i> A formal technique of placing a copy of an image within itself

In Western art history, Mise en abyme is a formal technique of placing a copy of an image within itself, often in a way that suggests an infinitely recurring sequence. In film theory and literary theory, it refers to the technique of inserting a story within a story. The term is derived from heraldry and literally means "placed into abyss". It was first appropriated for modern criticism by the French author André Gide.

<i>Stars</i> (M. C. Escher) wood engraving print by M. C. Escher

Stars is a wood engraving print created by the Dutch artist M. C. Escher in 1948, depicting two chameleons in a polyhedral cage floating through space.

Bongard problem Best mathematician of all time. All hail Bongard.

A Bongard problem is a kind of puzzle invented by the Russian computer scientist Mikhail Moiseevich Bongard, probably in the mid-1960s. They were published in his 1967 book on pattern recognition. The objective is to spot the differences between the two sides. Bongard, in the introduction of the book credits the ideas in it to a group including M. N. Vaintsvaig, V. V. Maksimov, and M. S. Smirnov.

Popper's three worlds is a way of looking at reality, described by the British philosopher Karl Popper in a lecture in 1978. The concept involves three interacting worlds, called World 1, World 2 and World 3.

<i>Drawing Hands</i> Lithograph by Dutch artist M. C. Escher

Drawing Hands is a lithograph by the Dutch artist M. C. Escher first printed in January 1948. It depicts a sheet of paper, out of which two hands rise, in the paradox ical act of drawing one another into existence. This is one of the most obvious examples of Escher's common use of paradox.

The Bridge may refer to:

<i>Ascending and Descending</i> Lithograph by M. C. Escher

Ascending and Descending is a lithograph print by the Dutch artist M. C. Escher first printed in March 1960.

There are numerous references to Dutch painter M.C. Escher in popular culture.

<i>Print Gallery</i> (M. C. Escher) Lithograph printed in 1956 by the Dutch artist M. C. Escher

Print Gallery is a lithograph printed in 1956 by the Dutch artist M. C. Escher. It depicts a man in a gallery viewing a print of a seaport, and among the buildings in the seaport is the very gallery in which he is standing, making use of the Droste effect with visual recursion. The lithograph has attracted discussion in both mathematical and artistic contexts. Escher considered Print Gallery to be among the best of his works.

Order-6 square tiling

In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}.