In mathematics, the Menger sponge is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet, it was first described by Karl Menger in 1926, in his studies of the concept of topological dimension. The construction of a Menger sponge can be described. Divide every face of the cube into 9 squares, like a Rubik's Cube; this will sub-divide the cube into 27 smaller cubes. Remove the smaller cube in the middle of each face, remove the smaller cube in the center of the larger cube, leaving 20 smaller cubes; this is a level-1 Menger sponge. Repeat steps 2 and 3 for each of the remaining smaller cubes, continue to iterate ad infinitum; the second iteration gives a level-2 sponge, the third iteration gives a level-3 sponge, so on. The Menger sponge; the nth stage of the Menger sponge, Mn, is made up of 20n smaller cubes, each with a side length of n. The total volume of Mn is thus n; the total surface area of Mn is given by the expression 2n + 4n.
Therefore the construction's volume approaches zero. Yet any chosen surface in the construction will be punctured as the construction continues, so that the limit is neither a solid nor a surface; each face of the construction becomes a Sierpinski carpet, the intersection of the sponge with any diagonal of the cube or any midline of the faces is a Cantor set. The cross section of the sponge through its centroid and perpendicular to a space diagonal is a regular hexagon punctured with hexagrams arranged in six-fold symmetry; the number of these hexagrams, in descending size, is given by a n = 9 a n − 1 − 12 a n − 2, with a 0 = 1, a 1 = 6. The sponge's Hausdorff dimension is log 20/log 3 ≅ 2.727. The Lebesgue covering dimension of the Menger sponge is the same as any curve. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that every curve is homeomorphic to a subset of the Menger sponge, where a curve means any compact metric space of Lebesgue covering dimension one.
In a similar way, the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar, might be embedded in any number of dimensions; the Menger sponge is a closed set. It has Lebesgue measure 0; because it contains continuous paths, it is an uncountable set. Formally, a Menger sponge can be defined as follows: M:= ⋂ n ∈ N M n where M0 is the unit cube and M n + 1:=. MegaMenger is a project aiming to build the largest fractal model, pioneered by Matt Parker of Queen Mary University of London and Laura Taalman of James Madison University; each small cube is made from 6 interlocking folded business cards, giving a total of 960 000 for a level-four sponge. The outer surfaces are covered with paper or cardboard panels printed with a Sierpinski carpet design to be more aesthetically pleasing. In 2014, twenty level-three Menger sponges were constructed, which
Anton Menger von Wolfensgrün, was an Austrian juridical expert and social theorist who aside from his collegiate works predominantly dedicated himself to propagating socialist literature on juridical grounds. He is the author of "The Right to the Whole Produce of Labor", "The Civil Law and the Poor" among others, he was Austrian economist Carl Menger's brother. Menger was a university professor for the law of civil process in Vienna from 1874 until 1899, where he was the Vice Chancellor from 1895 to 1896. Carl Menger was his brother. Menger's theses and arguments stem from a changed social structure, shaped by the economic crisis and social questions that sought answers from liberal politics in the pursuit of social justice, his juridical interests are assumed to be different than Karl Marx and Friedrich Engels in dealing with all theoretical legal issues. In Menger's juridical theory, he dismissed the basis of positive law from natural law. Rather, he argued that the backbone of law was a gauge of societal power relations.
His name is famous in connection with the collection of original socialist literature in Vienna. Menger collected everything that he could procure and began a book tour through Paris and Berlin, where his special socialist collection was unique in the world for the time. Mengers private library was acquired by the Sozialwissenschaftlichen Studienbibliothek der Kammer für Arbeiter und Angestellte für Wien in the 1920s. Friedrich Engels and Karl Kautsky tried to grapple with Menger's attacks on Marx's "Das Kapital as well as his fundamental project to ground socialism in legal theory in "Juridical Socialism": "The evidence shows that Marx was a plagiarist, this is proof that the concept of surplus value was brought about in another sense before Marx!" The Mengergasse in Floridsdorf was named after him in 1919. Die Zulässigkeit neuen thatsächlichen Vorbringens in den höheren Instanzen. Eine civilprocessualische Abhandlung. 1873. Das Recht auf den vollen Arbeitsertrag in geschichtlicher Darstellung.
Cotta, Stuttgart 1886. Gutachten über die Vorschläge zur Errichtung einer eidgenössischen Hochschule für Rechts- und Staatswissenschaft. Zürich, J. J. Schabelitz. 1889 Das Bürgerliche Recht und die besitzlosen Volksklassen. Eine Kritik des Entwurfs eines Bürgerlichen Gesetzbuches für das Deutsche Reich. 1890. Die sozialen Aufgaben der Rechtswissenschaft.. 1896 Neue Staatslehre. 1903. Neue Sittenlehre. Jena, G. Fischer, 1905. H. Hörner: "Menger Anton". In: Österreichisches Biographisches Lexikon 1815–1950. Vol. 6, Austrian Academy of Sciences, Vienna 1975, ISBN 3-7001-0128-7, p. 220 f. Gerhard Oberkofler: Anton Menger. In: Bewahren Verbreiten Aufklären. Hrsg. Günter Benser und Michael Schneider. Bonn-Bad Godesberg 2009 ISBN 978-3-86872-105-8, S. 196-201 online Works by or about Anton Menger at Internet Archive
Carl Menger was an Austrian economist and the founder of the Austrian School of economics. Menger contributed to the development of the theory of marginalism, which rejected the cost-of-production theories of value, such as were developed by the classical economists such as Adam Smith and David Ricardo; as a departure from such, he would go on to call his resultant perspective, the “Subjective Theory of Value”. Menger was born in the city of Neu-Sandez in Galicia, Austrian Empire, now Nowy Sącz in Poland, he was the son of a wealthy family of minor nobility. His mother, was the daughter of a wealthy Bohemian merchant, he had two brothers and Max, both prominent as lawyers. His son, Karl Menger, was a mathematician who taught for many years at Illinois Institute of Technology. After attending Gymnasium he studied law at the Universities of Prague and Vienna and received a doctorate in jurisprudence from the Jagiellonian University in Kraków. In the 1860s Menger left school and enjoyed a stint as a journalist reporting and analyzing market news, first at the Lemberger Zeitung in Lemberg, Austrian Galicia and at the Wiener Zeitung in Vienna.
During the course of his newspaper work he noticed a discrepancy between what the classical economics he was taught in school said about price determination and what real world market participants believed. In 1867 Menger began a study of political economy which culminated in 1871 with the publication of his Principles of Economics, thus becoming the father of the Austrian School of economic thought, it was in this work that he challenged classical cost-based theories of value with his theory of marginality – that price is determined at the margin. In 1872 Menger was enrolled into the law faculty at the University of Vienna and spent the next several years teaching finance and political economy both in seminars and lectures to a growing number of students. In 1873 he received the university's chair of economic theory at the young age of 33. In 1876 Menger began tutoring Archduke Rudolf von Habsburg, the Crown Prince of Austria in political economy and statistics. For two years Menger accompanied the prince in his travels, first through continental Europe and later through the British Isles.
He is thought to have assisted the crown prince in the composition of a pamphlet, published anonymously in 1878, critical of the higher Austrian aristocracy. His association with the prince would last until Rudolf's suicide in 1889. In 1878 Rudolf's father, Emperor Franz Josef, appointed Menger to the chair of political economy at Vienna; the title of Hofrat was conferred on him, he was appointed to the Austrian Herrenhaus in 1900. Ensconced in his professorship, he set about refining and defending the positions he took and methods he utilized in Principles, the result of, the 1883 publication of Investigations into the Method of the Social Sciences with Special Reference to Economics; the book caused a firestorm of debate, during which members of the Historical school of economics began to derisively call Menger and his students the "Austrian School" to emphasize their departure from mainstream economic thought in Germany – the term was used in an unfavorable review by Gustav von Schmoller.
In 1884 Menger responded with the pamphlet The Errors of Historicism in German Economics and launched the infamous Methodenstreit, or methodological debate, between the Historical School and the Austrian School. During this time Menger began to attract like-minded disciples who would go on to make their own mark on the field of economics, most notably Eugen von Böhm-Bawerk, Friedrich von Wieser. In the late 1880s Menger was appointed to head a commission to reform the Austrian monetary system. Over the course of the next decade he authored a plethora of articles which would revolutionize monetary theory, including "The Theory of Capital" and "Money". Due to his pessimism about the state of German scholarship, Menger resigned his professorship in 1903 to concentrate on study. Menger used his subjective theory of value to arrive at what he considered one of the most powerful insights in economics: both sides gain from exchange. Unlike William Jevons, Menger did not believe that goods provide units of utility.
Rather, he wrote, goods are valuable. Menger came up with an explanation of how money develops, still accepted by some schools of thought today. 1871 – Principles of Economics 1883 – Investigations into the Method of the Social Sciences with Special Reference to Economics 1884 – The Errors of Historicism in German Economics 1888 – The Theory of Capital 1892 – On the Origins of Money Methodenstreit History of macroeconomic thought Menger space White, Lawrence H.. "Menger, Carl". In Hamowy, Ronald; the Encyclopedia of Libertarianism. Thousand Oaks, CA: SAGE. Pp. 325–26. Doi:10.4135/9781412965811.n130. ISBN 978-1-4129-6580-4. LCCN 2008009151. OCLC 750831024. Golland and Karl Sigmund "Exact Thought in a Demented Time: Karl Menger and his Viennese Mathematical Colloquium" The Mathematical Intelligencer 2000, Vol 22,1 The Epistemological Import of Carl Menger's Theory of the Origin of Money Ludwig von Mises in Human Action on Menger's Theory of the Origins of Money Profile on Carl Menger at the History of Economic Thought Website Principles of Economics, online version provided by the Ludwig von Mises Institute.
Grundsätze der Volkswirtschaftsleh
Andreas Menger is a former German football player and goalkeeping coach of VfB Stuttgart. The netminder played in German professional football between 1997 until 2005, but in his first season at the Cologne side he assured a regular spot. After the Billy Goats relegated, he had only eight appearances. In winter of the 1998–99 season, he was transferred to MSV Duisburg. While playing twelve times with the Zebras, he never played in one game when he was under contract for Frankfurt; the executives were convinced of his abilities and from 2005 to 2011 he was the goalkeeping coach of the eagles, after splitting his duties between being a stand-by goalkeeper and coaching the keepers. In July 2011, Menger became new goalkeeping coach of VfB Stuttgart. On 8 January 2013, Menger extended his contract with VfB Stuttgart until June 2016
A given name is a part of a person's personal name. It identifies a person, differentiates that person from the other members of a group who have a common surname; the term given name refers to the fact that the name is bestowed upon a person to a child by their parents at or close to the time of birth. A Christian name, a first name, given at baptism, is now typically given by the parents at birth. In informal situations, given names are used in a familiar and friendly manner. In more formal situations, a person's surname is more used—unless a distinction needs to be made between people with the same surname; the idioms "on a first-name basis" and "being on first-name terms" refer to the familiarity inherent in addressing someone by their given name. By contrast, a surname, inherited, is shared with other members of one's immediate family. Regnal names and religious or monastic names are special given names bestowed upon someone receiving a crown or entering a religious order; such a person typically becomes known chiefly by that name.
The order given name – family name known as the Western order, is used throughout most European countries and in countries that have cultures predominantly influenced by European culture, including North and South America. The order family name – given name known as the Eastern order, is used in East Asia, as well as in Southern and North-Eastern parts of India, in Hungary; this order is common in Austria and Bavaria, in France, Belgium and Italy because of the influence of bureaucracy, which puts the family name before the given name. In China and Korea, part of the given name may be shared among all members of a given generation within a family and extended family or families, in order to differentiate those generations from other generations; the order given name – father's family name – mother's family name is used in Spanish-speaking countries to acknowledge the families of both parents. Today the order can be changed in Spain and Uruguay using given name – mother's family name – father's family name.
The order given name – mother's family name – father's family name is used in Portuguese-speaking countries to acknowledge the families of both parents. In many Western cultures, people have more than one given name. One of those, not the first in succession might be used as the name which that person goes by, such as in the cases of John Edgar Hoover and Mary Barbara Hamilton Cartland. A child's given name or names are chosen by the parents soon after birth. If a name is not assigned at birth, one may be given at a naming ceremony, with family and friends in attendance. In most jurisdictions, a child's name at birth is a matter of public record, inscribed on a birth certificate, or its equivalent. In western cultures, people retain the same given name throughout their lives. However, in some cases these names may be changed by repute. People may change their names when immigrating from one country to another with different naming conventions. In certain jurisdictions, a government-appointed registrar of births may refuse to register a name that may cause a child harm, considered offensive or which are deemed impractical.
In France, the agency can refer the case to a local judge. Some jurisdictions, such as Sweden, restrict the spelling of names. Parents may choose a name because of its meaning; this may be a personal or familial meaning, such as giving a child the name of an admired person, or it may be an example of nominative determinism, in which the parents give the child a name that they believe will be lucky or favourable for the child. Given names most derive from the following categories: Aspirational personal traits. For example, the name Clement means "merciful". English examples include Faith and August. Occupations, for example George means "earth-worker", i.e. "farmer". Circumstances of birth, for example Thomas meaning "twin" or the Latin name Quintus, traditionally given to the fifth male child. Objects, for example Peter means "rock" and Edgar means "rich spear". Physical characteristics, for example Calvin means "bald". Variations on another name to change the sex of the name or to translate from another language.
Surnames, for example Winston and Ross. Such names can honour other branches of a family, where the surname would not otherwise be passed down. Places, for example Brittany and Lorraine. Time of birth, for example day of the week, as in Kofi Annan, whose given name means "born on Friday", or the holiday on which one was born, for example, the name Natalie meaning "born on Christmas day" in Latin. Tuesday, May, or June. Combination of the above, for example the Armenian name Sirvart means "love rose". In many cultures, given names are reused to commemorate ancestors or those who are admired, resulting in a limited repertoire of names that sometimes vary by orthography; the most familiar example of this, to Western readers, is the use of Biblical and saints' names in most of the Christian countries (with Ethiopia, in which names were ideals or abstractions
A manger, or feeding trough, is a structure or feeder used to hold food for animals. The word comes from Latin mandere; the feeding trough is a rectangular boxlike receptacle made of either wood, or stone, or metal, sometimes of plastic. Mangers are used in livestock raising and found at stables and farmhouses, they are used to feed wild animals, e.g. in nature reserves. A similar trough providing drinking water for domestic or non-domestic animals is a watering trough and may be part of a larger watering structure called abreuvoir. A manger is a Christian symbol, associated with nativity scenes where Mary and Joseph, forced by necessity to stay in a room for animals instead of a guest room, used a manger as a makeshift crib for the Baby Jesus.. Away in a Manger, a Christmas carol The Dog in the Manger, a metaphor Media related to Feeding troughs at Wikimedia Commons The dictionary definition of manger at Wiktionary
Karl Menger was an Austrian-American mathematician. He was the son of the economist Carl Menger, he is credited with Menger's theorem. He worked on mathematics of algebras, algebra of geometries and dimension theory, etc. Moreover, he contributed to social sciences. Karl Menger was a student of Hans Hahn and received his PhD from the University of Vienna in 1924. L. E. J. Brouwer invited Menger in 1925 to teach at the University of Amsterdam. In 1927, he returned to Vienna to accept a professorship there. In 1930 and 1931 he was visiting lecturer at The Rice Institute. From 1937 to 1946 he was a professor at the University of Notre Dame. From 1946 to 1971, he was a professor at Illinois Institute of Technology in Chicago. In 1983, IIT awarded Menger a Doctor of Humane Sciences degree, his most famous popular contribution was the Menger sponge, a three-dimensional version of Sierpinski's carpet. It is related to the Cantor set. With Arthur Cayley, Menger is considered one of the founders of distance geometry.
The characteristic mathematical expressions appearing in those definitions are Cayley–Menger determinants. He was an active participant of the Vienna Circle which had discussions in the 1920s on social science and philosophy. During that time, he proved an important result on the St. Petersburg paradox with interesting applications to the utility theory in economics, he contributed to the development of game theory with Oskar Morgenstern. Menger's longest and last academic post was at the Illinois Institute of Technology, which hosts an annual IIT Karl Menger Lecture and offers the IIT Karl Menger Student Award to an exceptional student for scholarship each year. Crilly, Tony, 2005, "Paul Urysohn and Karl Menger: papers on dimension theory" in Grattan-Guinness, I. ed. Landmark Writings in Western Mathematics. Elsevier: 844–55. Golland and Karl Sigmund "Exact Thought in a Demented Time: Karl Menger and his Viennese Mathematical Colloquium" The Mathematical Intelligencer 2000, Vol 22,1, 34-45 O'Connor, John J..
Karl Menger at the Mathematics Genealogy Project