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Harshbarger, Eric (2010-08-19). "The Never-Ending Stories: Inception's Penrose Staircase". Wired. ISSN 1059-1028 . Retrieved 2020-06-05. Each shape represents a matrix, and tensor multiplication is done horizontally, and matrix multiplication is done vertically. In the language of tensor algebra, a particular tensor is associated with a particular shape with many lines projecting upwards and downwards, corresponding to abstract upper and lower indices of tensors respectively. Connecting lines between two shapes corresponds to contraction of indices. One advantage of this notation is that one does not have to invent new letters for new indices. This notation is also explicitly basis-independent. [3] Matrices [ edit ] In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. [1] A diagram in the notation consists of several shapes linked together by lines. In the language of multilinear algebra, each shape represents a multilinear function. The lines attached to shapes represent the inputs or outputs of a function, and attaching shapes together in some way is essentially the composition of functions.

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While the Penroses credited Escher in their article, Escher noted in a letter to his son in January 1960 that he was: The "continuous staircase" was first presented in an article that the Penroses wrote in 1959, based on the so-called "triangle of Penrose" published by Roger Penrose in the British Journal of Psychology in 1958. [5] M.C. Escher then discovered the Penrose stairs in the following year and made his now famous lithograph Klimmen en dalen ( Ascending and Descending) in March 1960. Penrose and Escher were informed of each other's work that same year. [7] Escher developed the theme further in his print Waterval ( Waterfall), which appeared in 1961. Graphical notation for multilinear algebra calculations Penrose graphical notation (tensor diagram notation) of a matrix product state of five particles The Penrose stairs or Penrose steps, also dubbed the impossible staircase, is an impossible object created by Oscar Reutersvärd in 1937 [1] [2] [3] [4] and later independently discovered and made popular by Lionel Penrose and his son Roger Penrose. [5] 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 but possible in some non-Euclidean geometry like in nil geometry. [6]Torre, Matteo. "Impossible Pictures: When Art Helps Math Education" (PDF). Impossible Pictures: When Art Helps Math Education . Retrieved 9 October 2020. Mr Penrose recommends further work to strengthen and speed up enforcement of consumer and competition law. In nature and on our bathroom walls, we typically see tile patterns that repeat in “a very predictable, regular way”, says Dr Craig Kaplan, an associate professor of computer science at the University of Waterloo in Ontario. What mathematicians were interested in were shapes that “guaranteed non-periodicity” – in other words, there was no way to tile them so that the overall pattern created a repeating grid.

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John Penrose MP has today (16 February) published proposals to update the UK’s competition and consumer regime. In November 2020, the government announced the formation of a new Digital Markets Unit to oversee a pro-competition regime for platforms including those funded by digital advertising, such as Google and Facebook. Biography working on the design of a new picture, which featured a flight of stairs which only ever ascended or descended, depending on how you saw it. [The stairs] form a closed, circular construction, rather like a snake biting its own tail. And yet they can be drawn in correct perspective: each step higher (or lower) than the previous one. [...] I discovered the principle in an article which was sent to me, and in which I myself was named as the maker of various 'impossible objects'. But I was not familiar with the continuous steps of which the author had included a clear, if perfunctory, sketch, although I was employing some of his other examples. [10]I want to thank John Penrose for his hard work on this independent report, which considers how the UK’s competition regime can promote productivity, reward and encourage innovation and, most importantly, get consumers a better deal.

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a b … n ε a b … n {\displaystyle \varepsilon _{ab\ldots n}\,\varepsilon There’s been a thread of beautiful mathematics over the last 60 years or so searching for ever smaller sets of shapes that do this,” Kaplan says. “The first example of an aperiodic set of shapes had over 20,000 shapes in it. And of course, mathematicians worked to get that number down over time. And the furthest we got was in the 1970s,” when the Nobel-prize winning physicist Roger Penrose found pairs of shapes that fit the bill.Mr Penrose brings a unique perspective from his experience in business, an understanding of everyday consumer issues from 15 years as a constituency MP, and long-running interest in the subject. At an Escher conference in Rome in 1985, Roger Penrose said that he had been greatly inspired by Escher's work when he and his father discovered both the Penrose tribar structure (that is, the Penrose triangle) and the continuous steps. The Penrose stairs appeared twice in the movie Inception. This paradoxical illusion can only be realized in the dream worlds of the film. In the film, the hero descends the stairs fleeing from a guard. In the real world, the hero should always be in front of the villain throughout this chase. However, in the case of the Penrose stairs the hero descends another flight of stairs to catch up to the antagonist and catch him unawares. [15] John Penrose was appointed the Prime Minister’s Anti-Corruption Champion in December 2017 and was reappointed in July 2019. He was previously a Minister of State in the Northern Ireland Office from November 2018 to July 2019. John was first elected as MP for Weston, Worle and the Villages in 2005. Escher was captivated by the endless stairs and subsequently wrote a letter to the Penroses in April 1960: