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The Banach–Tarski Paradox (book)
''The Banach–Tarski Paradox'' is a book in mathematics on the Banach–Tarski paradox, the fact that a unit ball can be partitioned into a finite number of subsets and reassembled to form two unit balls. It was written by Stan Wagon and published in 1985 by the Cambridge University Press as volume 24 of their Encyclopedia of Mathematics and its Applications book series. A second printing in 1986 added two pages as an addendum, and a 1993 paperback printing added a new preface. In 2016 the Cambridge University Press published a second edition, adding Grzegorz Tomkowicz as a co-author, as volume 163 of the same series. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries. Topics The Banach–Tarski paradox, proved by Stefan Banach and Alfred Tarski in 1924, states that it is possible to partition a three-dimensional unit ball into finitely many pieces and reassemble them into two unit ball ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Tarski Paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their original shape. However, the pieces themselves are not "solids" in the traditional sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces. An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox". The theorem is a veridical paradox: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coincides with the standard measure of length, area, or volume. In general, it is also called '-dimensional volume, '-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by \lambda(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matthew Foreman
Matthew Dean Foreman is an American mathematician at University of California, Irvine. He has made notable contributions in set theory and in ergodic theory. Biography Born in Los Alamos, New Mexico, Foreman earned his Ph.D. from the University of California, Berkeley in 1980 under Robert M. Solovay. His dissertation title was ''Large Cardinals and Strong Model Theoretic Transfer Properties''. In addition to his mathematical work, Foreman is an avid sailor. He and his family sailed their sailboat ''Veritas'' (a built by C&C Yachts) from North America to Europe in 2000. From 2000–2008 they sailed Veritas to the Arctic, the Shetland Islands, Scotland, Ireland, England, France, Spain, North Africa and Italy. Notable high points were Fastnet Rock, Irish and Celtic seas and many passages including the Maelstrom, Stad, Pentland Firth, Loch Ness, the Corryveckan and the Irish Sea. Further south they sailed through the Chenal du Four and Raz de Sein, across the Bay of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randall Dougherty
Randall Dougherty (born 1961) is an American mathematician. Dougherty has made contributions in widely varying areas of mathematics, including set theory, logic, real analysis, discrete mathematics, computational geometry, information theory, and coding theory. Dougherty is a three-time winner of the International Mathematical Olympiad, U.S.A. Mathematical Olympiad (1976, 1977, 1978) and a three-time medalist in the International Mathematical Olympiad. He is also a three-time William Lowell Putnam Mathematical Competition, Putnam Fellow (1978, 1979, 1980). Dougherty earned his Ph.D. in 1985 at University of California, Berkeley under the direction of Jack Silver. With Matthew Foreman he showed that the Banach–Tarski paradox, Banach-Tarski decomposition is possible with pieces with the Baire property, solving a problem of Edward Marczewski, Marczewski that remained unsolved for more than 60 years. With Chris Freiling and Ken Zeger, he showed that linear codes are insufficient t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Set
In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalent definitions of Baire sets, but in the most widely used, the Baire sets of a locally compact Hausdorff space form the smallest σ-algebra such that all compactly supported continuous functions are measurable. Thus, measures defined on this σ-algebra, called Baire measures, are a convenient framework for integration on locally compact Hausdorff spaces. In particular, any compactly supported continuous function on such a space is integrable with respect to any finite Baire measure. Every Baire set is a Borel set. The converse holds in many, but not all, topological spaces. Baire sets avoid some pathological properties of Borel sets on spaces without a countable base for the topology. In practice, the use of Baire measures on Baire sets can often be replaced by the use of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward Marczewski
Edward Marczewski (15 November 1907 – 17 October 1976) was a Polish mathematician. He was born Szpilrajn but changed his name while hiding from Nazi persecution. Marczewski was a member of the Warsaw School of Mathematics. His life and work after the Second World War were connected with Wrocław, where he was among the creators of the Polish scientific centre. He worked at the State Institute of Mathematics, which was incorporated into the Polish Academy of Sciences in 1952. Marczewski's main fields of interest were measure theory, descriptive set theory, general topology, probability theory and universal algebra. He also published papers on real and complex analysis, applied mathematics and mathematical logic. Marczewski proved that the topological dimension, for arbitrary metrisable separable space ''X'', coincides with the Hausdorff dimension under one of the metrics in ''X'' which induce the given topology of ''X'' (while otherwise the Hausdorff dimension is always greater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degree (angle), degrees, or Pi, /2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called square (algebra), squaring. Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art. The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disk (mathematics)
In geometry, a disk (Spelling of disc, also spelled disc) is the region in a plane (geometry), plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius r, an open disk is usually denoted as D_r, and a closed disk is \overline. However in the field of topology the closed disk is usually denoted as D^2, while the open disk is \operatorname D^2. Formulas In Cartesian coordinates, the ''open disk'' with center (a, b) and radius ''R'' is given by the formula D = \, while the ''closed disk'' with the same center and radius is given by \overline = \. The area (geometry), area of a closed or open disk of radius ''R'' is π''R''2 (see area of a disk). Properties The disk has circular symmetry. The open disk and the closed disk are not topologically equivalent (that is, they are not homeomorphism, homeomorphic), as they have different topological properties from each other. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski's Circle-squaring Problem
Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. It is possible, using pieces that are Borel sets, but not with pieces cut by Jordan curves. Solutions Tarski's circle-squaring problem was proven to be solvable by Miklós Laczkovich in 1990. The decomposition makes heavy use of the axiom of choice and is therefore non-constructive. Laczkovich estimated the number of pieces in his decomposition at roughly 1050. The pieces used in his decomposition are non-measurable subsets of the plane. Laczkovich actually proved the reassembly can be done ''using translations only''; rotations are not required. Along the way, he also proved that any simple polygon in the plane can be decomposed into finitely many pieces and reassembled using translations only to form a square of equal area. It follows from a result of that it is pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miklós Laczkovich
Miklós Laczkovich (born 21 February 1948) is a Hungarian mathematician mainly noted for his work on real analysis and geometric measure theory. His most famous result is the solution of Tarski's circle-squaring problem in 1989.Ruthen, R. (1989) ''Squaring the Circle'', Scientific American 261(1), 22-24. Career Laczkovich received his degree in mathematics in 1971 at Eötvös Loránd University, where he has been teaching ever since, currently leading the Department of Analysis. He was also a professor at University College London, where he is now a professor emeritus ''Emeritus/Emerita'' () is an honorary title granted to someone who retirement, retires from a position of distinction, most commonly an academic faculty position, but is allowed to continue using the previous title, as in "professor emeritus". .... He became corresponding member (1993), then member (1998) of the Hungarian Academy of Sciences. He has held several guest professor positions in the UK, Canada, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Measure
Jordan, officially the Hashemite Kingdom of Jordan, is a country in the Southern Levant region of West Asia. Jordan is bordered by Syria to the north, Iraq to the east, Saudi Arabia to the south, and Israel and the occupied Palestinian territories to the west. The Jordan River, flowing into the Dead Sea, is located along the country's western border within the Jordan Rift Valley. Jordan has a small coastline along the Red Sea in its southwest, separated by the Gulf of Aqaba from Egypt. Amman is the country's capital and largest city, as well as the most populous city in the Levant. Inhabited by humans since the Paleolithic period, three kingdoms developed in Transjordan during the Iron Age: Ammon, Moab and Edom. In the third century BC, the Arab Nabataeans established their kingdom centered in Petra. The Greco-Roman period saw the establishment of several cities in Transjordan that comprised the Decapolis. Later, after the end of Byzantine rule, the region beca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension ''n'' of the space, and is commonly denoted E(''n'') or ISO(''n''), for ''inhomogeneous special orthogonal'' group. The Euclidean group E(''n'') comprises all translations, rotations, and reflections of \mathbb^n; and arbitrary finite combinations of them. The Euclidean group can be seen as the symmetry group of the space itself, and contains the group of symmetries of any figure (subset) of that space. A Euclidean isometry can be ''direct'' or ''indirect'', depending on whether it preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the special Euclidean group, often denoted SE(''n'') and E+(''n''), whose elements are called rigid motions or Euclidean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
