Ring lemma In the geometry of circle packings in the Euclidean plane , the ring lemma gives a lower bound on the sizes of adjacent circles in a circle packing .Statement The lemma states: Let be any integer greater than or equal to three. Suppose that the unit circle is surrounded by a ring of interior-disjoint circles, all tangent to it, with consecutive circles in the ring tangent to each other. Then the minimum radius of any circle in the ring is at least the unit fraction where is the th Fibonacci number . The sequence of minimum radii, from, begins and the denominators in this sequence are given as. Generalizations to three-dimensional space are also known.Construction An infinite sequence of circles can be constructed, containing rings for each that exactly meet the bound of the ring lemma, showing that it is tight. The construction allows halfplanes to be considered as degenerate circles with infinite radius, and includes additional tangencies between the circles beyond those required in the statement of the lemma. It begins by sandwiching the unit circle between two parallel halfplanes; in the geometry of circles , these are considered to be tangent to each other at the point at infinity . Each successive circle after these first two is tangent to the central unit circle and to the two most recently added circles; see the illustration for the first six circles constructed in this way. The first circles of this construction form a ring, whose minimum radius can be calculated by Descartes' theorem to be the same as the radius specified in the ring lemma. This construction can be perturbed to a ring of finite circles, without additional tangencies, whose minimum radius is arbitrarily close to this bound.History A version of the ring lemma with a weaker bound was first proven by Burton Rodin and Dennis Sullivan as part of their proof of William Thurston's conjecture that circle packings can be used to approximate conformal maps. Lowell Hansen gave a recurrence relation for the tightest possible lower bound, and Dov Aharonov found a closed-form expression for the same bound.Applications Beyond its original application to conformal mapping , the circle packing theorem and the ring lemma play key roles in a proof by Keszegh et al . that planar graphs of bounded degree can be drawn with bounded slope number .
Popular articles Javier Milei - Argentine libertarian economist, author, radio conductor and public speaker sympathetic to the Austrian School of economic thought. He became widely known for his regular ...Jimmy Carter - American politician, philanthropist, and former farmer who served as the 39th president of the United States from 1977 to 1981. A member of the Democratic Party, he previ...UEFA Euro 2024 - The 2024 UEFA European Football Championship , commonly referred to as UEFA Euro 2024 or simply Euro 2024 , will be the 17th edition of the UEFA European Championship, the quadrennial internationa...Argentina - country located mostly in the southern half of South America. Sharing the bulk of the Southern Cone with Chile to the west, the country is also b...Sam Altman - American entrepreneur, investor, programmer, and blogger. He is the former president of Y Combinator and now the CEO of OpenAI. Early life and education. ...Rosalynn Carter - American who served as First Lady of the United States from 1977 to 1981 as the wife of President Jimmy Carter. For decades, she has been a leading advocate for numerou...Next Argentine presidential election - Next Argentine presidential election - presidential election in Argentina....Popular movies The Hunger Games (film) - 2012 American dystopian action thriller science fiction-adventure film directed by Gary Ross and based on Suzanne Collins’s 2008 novel of the same name. It is the first insta...untitled Captain Marvel sequel - part of Marvel Cinematic Universe....Killers of the Flower Moon (film project) - Killers of the Flower Moon - film project in United States of America. It was presented as drama, detective fiction, thriller. The film project starred Leonardo Dicaprio, Robert De Niro. Director of...Five Nights at Freddy's (film) - Five Nights at Freddy's - film published in 2017 in United States of America. Scenarist of the film - Scott Cawthon....Popular video games Minecraft - sandbox video game developed by Mojang Studios. Created by Markus "Notch" Persson in the Java programming language and released as a public alpha for personal computers in 2...Grand Theft Auto V - 2013 action-adventure game developed by Rockstar North and published by Rockstar Games. It is the first main entry in the Grand Theft Auto series since 2008's Grand Theft ...Roblox - online game platform and game creation system that allows users to program games and play games created by other users. Founded by David Baszucki and Erik Cassel in 2004 and released in...Baldur's Gate III - upcoming role-playing video game developed and published by Larian Studios for Microsoft Windows and the Stadia streaming service. It is the third main game in the Baldur's ...Alan Wake - action-adventure video game developed by Remedy Entertainment and published by Microsoft Studios, released for the Xbox 360 and Microsoft Windows. The story follows best-selling thri...Fortnite - online video game developed by Epic Games and released in 2017. It is available in three distinct game mode versions that otherwise share the same general gameplay and game engine: ...Super Mario RPG - is a role-playing video game developed by Square and published by Nintendo for the Super Nintendo Entertainment System in 1996. It was directed by Yoshihiko Maekawa and Chihiro Fujioka and produced by...Popular books Book of Revelation - The Book of Revelation is the final book of the New Testament, and consequently is also the final book of the Christian Bible. Its title is derived from the first word of the Koine Greek text: apok...Book of Genesis - account of the creation of the world, the early history of humanity, Israel's ancestors and the origins...Gospel of Matthew - The Gospel According to Matthew is the first book of the New Testament and one of the three synoptic gospels. It tells how Israel's Messiah, rejected and executed in Israel, pronounces judgement on ...Michelin Guide - Michelin Guides are a series of guide books published by the French tyre company Michelin for more than a century. The term normally refers to the annually published Michelin Red Guide , the oldest...Psalms - The Book of Psalms , commonly referred to simply as Psalms , the Psalter or "the Psalms", is the first book of the Ketuvim , the third section of the Hebrew Bible, and thus a book of th...Ecclesiastes - Ecclesiastes is one of 24 books of the Tanakh , where it is classified as one of the Ketuvim . Originally written c. 450–200 BCE, it is also among the canonical Wisdom literature of the Old Tes...The 48 Laws of Power - non-fiction book by American author Robert Greene. The book...Popular television series The Crown (TV series) - historical drama web television series about the reign of Queen Elizabeth II, created and principally written by Peter Morgan, and produced by Left Bank Pictures and Sony Pictures Tel...Friends - American sitcom television series, created by David Crane and Marta Kauffman, which aired on NBC from September 22, 1994, to May 6, 2004, lasting ten seasons. With an ensemble cast sta...Young Sheldon - spin-off prequel to The Big Bang Theory and begins with the character Sheldon...Modern Family - American television mockumentary family sitcom created by Christopher Lloyd and Steven Levitan for the American Broadcasting Company. It ran for eleven seasons, from September 23...Loki (TV series) - upcoming American web television miniseries created for Disney+ by Michael Waldron, based on the Marvel Comics character of the same name. It is set in the Marvel Cinematic Universe, shar...Game of Thrones - American fantasy drama television series created by David Benioff and D. B. Weiss for HBO. It...Shameless (American TV series) - American comedy-drama television series developed by John Wells which debuted on Showtime on January 9, 2011. It...
OWIKI.org . Text is available under the Creative Commons Attribution-ShareAlike License.