A current transformer is a type of transformer, used to measure alternating current. It produces a current in its secondary, proportional to the current in its primary. Current transformers, along with potential transformers, are instrument transformers. Instrument transformers scale the large values of voltage or current to small, standardized values that are easy to handle for measuring instruments and protective relays; the instrument transformers isolate measurement or protection circuits from the high voltage of the primary system. A current transformer provides a secondary current, proportional to the current flowing in its primary; the current transformer presents a negligible load to the primary circuit. Current transformers are the current-sensing units of the power system and are used at generating stations, electrical substations, in industrial and commercial electric power distribution. Like any transformer, a current transformer has a primary winding, a core and a secondary winding, although some transformers, including current transformers, use an air core.
In principle, the only difference between a current transformer and a voltage transformer is that the former is fed with a'constant' current while the latter is fed with a'constant' voltage, where'constant' has the strict circuit theory meaning. The alternating current in the primary produces an alternating magnetic field in the core, which induces an alternating current in the secondary; the primary circuit is unaffected by the insertion of the CT. Accurate current transformers need close coupling between the primary and secondary to ensure that the secondary current is proportional to the primary current over a wide current range; the current in the secondary is the current in the primary divided by the number of turns of the secondary. In the illustration on the right,'I' is the current in the primary,'B' is the magnetic field,'N' is the number of turns on the secondary, and'A' is an AC ammeter. Current transformers consist of a silicon steel ring core wound with many turns of copper wire as shown in the illustration to the right.
The conductor carrying the primary current is passed through the ring. The CT's primary, consists of a single'turn'; the primary'winding' may be a permanent part of the current transformer, i.e. a heavy copper bar to carry current through the core. Window-type current transformers are common, which can have circuit cables run through the middle of an opening in the core to provide a single-turn primary winding. To assist accuracy, the primary conductor should be centered in the aperture. CTs are specified by their current ratio from primary to secondary; the rated secondary current is standardized at 1 or 5 amperes. For example, a 4000:5 CT secondary winding will supply an output current of 5 amperes when the primary winding current is 4000 amperes; this ratio can be used to find the impedance or voltage on one side of the transformer, given the appropriate value at the other side. For the 4000:5 CT, the secondary impedance can be found as ZS = NZP = 800ZP, the secondary voltage can be found as VS = NVP = 800VP.
In some cases, the secondary impedance is referred to the primary side, is found as ZS′ = N2ZP. Referring the impedance is done by multiplying initial secondary impedance value by the current ratio; the secondary winding of a CT can have taps to provide a range of five taps being common. Current transformer shapes and sizes vary depending on the end switch gear manufacturer. Low-voltage single ratio metering current transformers are either a ring type or plastic molded case. Split-core current transformers either have a core with a removable section; this allows the transformer to be placed around a conductor without having to disconnect it first. Split-core current transformers are used in low current measuring instruments portable, battery-operated, hand-held. Current transformers are used extensively for measuring current and monitoring the operation of the power grid. Along with voltage leads, revenue-grade CTs drive the electrical utility's watt-hour meter on many larger commercial and industrial supplies.
High-voltage current transformers are mounted on porcelain or polymer insulators to isolate them from ground. Some CT configurations slip around the bushing of a high-voltage transformer or circuit breaker, which automatically centers the conductor inside the CT window. Current transformers can be mounted on high voltage leads of a power transformer. Sometimes a section of a bus bar can be removed to replace a current transformer. Multiple CTs are installed as a "stack" for various uses. For example, protection devices and revenue metering may use separate CTs to provide isolation between metering and protection circuits and allows current transformers with different characteristics to be used for the devices; the burden impedance should not exceed the specified maximum value to avoid the secondary voltage exceeding the limits for the current transformer. The primary current rating of a current transformer should not be exceeded or the core may enter its non linear region and saturate; this would occur near the end of the first half of each half of the AC sine wave in the primary and would compromise the accuracy.
Current transformers are used to monitor high currents or currents at high voltages. Technical standards and design practices are used to ensure the safety of installations using current transformers; the secondary of a current transformer should not be disconnected from its burden while current is
International Business Machines Corporation is an American multinational information technology company headquartered in Armonk, New York, with operations in over 170 countries. The company began in 1911, founded in Endicott, New York, as the Computing-Tabulating-Recording Company and was renamed "International Business Machines" in 1924. IBM produces and sells computer hardware and software, provides hosting and consulting services in areas ranging from mainframe computers to nanotechnology. IBM is a major research organization, holding the record for most U. S. patents generated by a business for 26 consecutive years. Inventions by IBM include the automated teller machine, the floppy disk, the hard disk drive, the magnetic stripe card, the relational database, the SQL programming language, the UPC barcode, dynamic random-access memory; the IBM mainframe, exemplified by the System/360, was the dominant computing platform during the 1960s and 1970s. IBM has continually shifted business operations by focusing on higher-value, more profitable markets.
This includes spinning off printer manufacturer Lexmark in 1991 and the sale of personal computer and x86-based server businesses to Lenovo, acquiring companies such as PwC Consulting, SPSS, The Weather Company, Red Hat. In 2014, IBM announced that it would go "fabless", continuing to design semiconductors, but offloading manufacturing to GlobalFoundries. Nicknamed Big Blue, IBM is one of 30 companies included in the Dow Jones Industrial Average and one of the world's largest employers, with over 380,000 employees, known as "IBMers". At least 70% of IBMers are based outside the United States, the country with the largest number of IBMers is India. IBM employees have been awarded five Nobel Prizes, six Turing Awards, ten National Medals of Technology and five National Medals of Science. In the 1880s, technologies emerged that would form the core of International Business Machines. Julius E. Pitrap patented the computing scale in 1885. On June 16, 1911, their four companies were amalgamated in New York State by Charles Ranlett Flint forming a fifth company, the Computing-Tabulating-Recording Company based in Endicott, New York.
The five companies had offices and plants in Endicott and Binghamton, New York. C.. They manufactured machinery for sale and lease, ranging from commercial scales and industrial time recorders and cheese slicers, to tabulators and punched cards. Thomas J. Watson, Sr. fired from the National Cash Register Company by John Henry Patterson, called on Flint and, in 1914, was offered a position at CTR. Watson joined CTR as General Manager 11 months was made President when court cases relating to his time at NCR were resolved. Having learned Patterson's pioneering business practices, Watson proceeded to put the stamp of NCR onto CTR's companies, he implemented sales conventions, "generous sales incentives, a focus on customer service, an insistence on well-groomed, dark-suited salesmen and had an evangelical fervor for instilling company pride and loyalty in every worker". His favorite slogan, "THINK", became a mantra for each company's employees. During Watson's first four years, revenues reached $9 million and the company's operations expanded to Europe, South America and Australia.
Watson never liked the clumsy hyphenated name "Computing-Tabulating-Recording Company" and on February 14, 1924 chose to replace it with the more expansive title "International Business Machines". By 1933 most of the subsidiaries had been merged into one company, IBM. In 1937, IBM's tabulating equipment enabled organizations to process unprecedented amounts of data, its clients including the U. S. Government, during its first effort to maintain the employment records for 26 million people pursuant to the Social Security Act, the tracking of persecuted groups by Hitler's Third Reich through the German subsidiary Dehomag. In 1949, Thomas Watson, Sr. created IBM World Trade Corporation, a subsidiary of IBM focused on foreign operations. In 1952, he stepped down after 40 years at the company helm, his son Thomas Watson, Jr. was named president. In 1956, the company demonstrated the first practical example of artificial intelligence when Arthur L. Samuel of IBM's Poughkeepsie, New York, laboratory programmed an IBM 704 not to play checkers but "learn" from its own experience.
In 1957, the FORTRAN scientific programming language was developed. In 1961, IBM developed the SABRE reservation system for American Airlines and introduced the successful Selectric typewriter. In 1963, IBM employees and computers helped. A year it moved its corporate headquarters from New York City to Armonk, New York; the latter half of the 1960s saw IBM continue its support of space exploration, participating in the 1965 Gemini flights, 1966 Saturn flights and 1969 lunar mission. On April 7, 1964, IBM announced the first computer system family, the IBM System/360, it spanned the complete range of commercial and scientific applications from large to small, allowing companies for the first time to upgrade to models with greater computing capability without having to rewrite their applications. It was followed by the IBM System/370 in 1970. Together the
Distributed.net is a distributed computing effort, attempting to solve large scale problems using otherwise idle CPU or GPU time. It is governed by Distributed Computing Technologies, Incorporated, a non-profit organization under U. S. tax code 501. Distributed.net is working on RC5-72, OGR-28. The RC5-72 project is on pace to exhaust the keyspace in just under 150 years, although the project will end whenever the required key is found. Both problems are part of a series: OGR is part of an infinite series. Distributed.net has decided to sponsor the original prize offer for finding the key as a result. In 2001, distributed.net was estimated to have a throughput of over 30 TFLOPS. By 2009, throughput was estimated to be much higher. A coordinated effort was started in February 1997 by Earle Ady and Christopher G. Stach II of Hotjobs.com and New Media Labs, as an effort to break the RC5-56 portion of the RSA Secret-Key Challenge, a 56-bit encryption algorithm that had a $10,000 USD prize available to anyone who could find the key.
This initial effort had to be suspended as the result of SYN flood attacks by participants upon the server. A new independent effort, named distributed.net, was coordinated by Jeffrey A. Lawson, Adam L. Beberg, David C. McNett along with several others who would serve on the board and operate infrastructure. By late March 1997 new proxies were released to resume RC5-56 and work began on enhanced clients. A cow head was selected as the icon of the project's mascot; the RC5-56 challenge was solved on October 1997 after 250 days. The next project was the RC5-64 challenge which took nearly five years to complete before the correct key was found on July 14, 2002 decrypting the message to the plaintext "some things are better left unread". "DNETC" is the file name of the software application which users run to participate in any active distributed.net project. It is a command line program with an interface to configure it, available for a wide variety of platforms. Distributed.net refers to the software application as the "client".
As of May 2009, 32-bit Windows on Intel x86 is the most used configuration, with Linux on Intel x86 in second place, Mac OS X on PowerPC in third place. Portions of the source code for the client are publicly available, although users are not permitted to distribute modified versions themselves. Distributed.net's RC5-72 project is available on the BOINC client through the Moo! Wrapper project. In recent years, most of the work on the RC5-72 project has been submitted by clients that run on the GPU of modern graphics cards. Although the project had been underway for 6 years when the first GPUs began submitting results, as of March 2018, GPUs represent 78% of all completed work units, complete nearly 93% of all work units each day. NVIDIAIn late 2007, work began on the implementation of new RC5-72 cores designed to run on NVIDIA CUDA-enabled hardware, with the first completed work units reported in November 2008. On high-end NVIDIA video cards, upwards of 600 million keys/second has been reported.
Considering a high end single CPU working on RC5-72 may achieve 50 million keys/second, the CUDA advancement represents a performance increase of 1000%. As of March 2018, CUDA clients have completed more than 8% of all work on the RC5-72 project, complete 29% of all work units each day. ATISimilarly, near the end of 2008, work began on the implementation of new RC5-72 cores designed to run on ATI Stream-enabled hardware; some of the products in the Radeon HD 5000 and 6000 series provide key rates in excess of 1.8 billion keys/second. As of March 2018, Stream clients have completed more than 52% of all work on the RC5-72 project, complete more than 10% of all work units each day. OpenCLAn OpenCL client entered beta testing in late 2012 and was released in 2013; as of March 2018, OpenCL clients have completed 17% of all work on the RC5-72 project, complete 53% of all work units each day. No breakdown of OpenCL production by GPU manufacturer exists. CurrentRSA Lab's 72-bit RC5 Encryption Challenge — In progress, 5.058% complete as of 5 March 2018 Optimal Golomb Rulers — In progress, ~43.91% complete as of 5 March 2018CryptographyRSA Lab's 56-bit RC5 Encryption Challenge — Completed 19 October 1997.
RSA Lab's 56-bit DES-II-1 Encryption Challenge — Completed 23 February 1998 RSA Lab's 56-bit DES-II-2 Encryption Challenge — Ended 15 July 1998 RSA Lab's 56-bit DES-III Encryption Challenge — Completed 19 January 1999 CS-Cipher Challenge — Completed 16 January 2000. RSA Lab's 64-bit RC5 Encryption Challenge — Completed 14 July 2002. Golomb rulersOptimal Golomb Rulers — Completed 13 October 2004 Optimal Golomb Rulers — Completed 24 October 2008 Optimal Golomb Rulers — Completed 24 February 2009 Optimal Golomb Rulers — Completed 19 February 2014 RSA Secret-Key Challenge Golomb Ruler DES Challenges Brute f
Berkeley Open Infrastructure for Network Computing
The Berkeley Open Infrastructure for Network Computing, an open-source middleware system, supports volunteer and grid computing. Developed to support the SETI@home project, it became generalized as a platform for other distributed applications in areas as diverse as mathematics, medicine, molecular biology, environmental science, astrophysics, among others. BOINC aims to enable researchers to tap into the enormous processing resources of multiple personal computers around the world. BOINC development originated with a team based at the Space Sciences Laboratory at the University of California and led by David Anderson, who leads SETI@home; as a high-performance distributed computing platform, BOINC brings together about 311,742 active participants and 834,343 active computers worldwide processing on average 26.431 PetaFLOPS as of 9 June 2018. The National Science Foundation funds BOINC through awards SCI/0221529, SCI/0438443 and SCI/0721124. Guinness World Records ranks BOINC as the largest computing grid in the world.
BOINC code runs on various operating systems, including Microsoft Windows, macOS, Android and FreeBSD. BOINC is free software released under the terms of the GNU Lesser General Public License. BOINC was developed to manage the SETI@home project; the original SETI client was a non-BOINC software for SETI@home. As one of the first volunteer grid computing projects, it was not designed with a high level of security; as a result, some participants in the project attempted to cheat the project to gain "credits," while some others submitted falsified work. BOINC was designed, in part; the BOINC project started in February 2002, the first version was released on April 10, 2002. The first BOINC-based project was Predictor@home launched on June 9, 2004. In 2009, AQUA@home deployed multi-threaded CPU applications for the first time, followed by the first OpenCL application in 2010; as of 2 January 2018, 37 BOINC projects are active. In essence, BOINC is software that can use the unused CPU and GPU cycles on a computer to do scientific computing—what one individual does not use of his/her computer, BOINC uses.
In late 2008, BOINC's official website announced that Nvidia had developed a system called CUDA that uses GPUs for scientific computing. With NVIDIA's assistance, some BOINC-based projects now have applications that run on NVIDIA GPUs using CUDA. Beginning in October 2009, BOINC added support for the ATI/AMD family of GPUs also; these applications run from 2 to 10 times faster than the former CPU-only versions. In 7.x preview versions, GPU support was added for computers using Mac OS X with AMD Radeon graphic cards. BOINC consists of a server system and client software that communicate with each other to distribute and process work units and return the results. BOINC can be controlled remotely by remote procedure calls, from the command line, from the BOINC Account Manager. BOINC Manager has two "views": the Advanced View and the Simplified GUI; the Grid View was removed in the 6.6.x clients. The appearance of the Simplified GUI is user-customizable, in that users can create their own designs. A BOINC app exists for Android, allowing every person owning an Android device – smartphone and Kindle – to share their unused computing power.
The user is allowed to select the research projects they want to support, if it is in the app's available project list. By default, the application will allow computing only when the device is connected to a WiFi network, is being charged, the battery has a charge of at least 90%. Only some of the BOINC projects are available, including Asteroids@home, Collatz Conjecture, Einstein@home, Enigma@home, LHC@home, Moo! Wrapper, Quake Catcher Network, Rosetta@home, SETI@home, theSkyNet POGS, Universe@Home, World Community Grid and Yoyo@home. A BOINC Account Manager is an application that manages multiple BOINC project accounts across multiple computers and operating systems. Account managers were designed for people who are new to BOINC or have several computers participating in several projects; the account manager concept was conceived and developed jointly by GridRepublic and BOINC. Current account managers include: BAM! GridRepublic Charity Engine Dazzler The BOINC Credit System is designed to avoid cheating by validating results before granting credit.
A credit management system helps to ensure that users are returning results which are both scientifically and statistically accurate. Online distributed computing is entirely a volunteer endeavor. For this reason, projects are dependent on a complicated and variable mix of new users, long-term users, retiring users. There are about 35 projects listed, of which about half yield published reports; the licensing of the projects varies. Since 2013, the cryptocurrency Gridcoin has been associated with BOINC as a remunerative coin. Gridcoin uses a modified proof-of-stake timestamping system called proof-of-research to reward participants for computational work completed on BOINC; the proof-of-research system was implemented on October 11, 20
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Martin Gardner was an American popular mathematics and popular science writer, with interests encompassing scientific skepticism, philosophy and literature—especially the writings of Lewis Carroll, L. Frank Baum, G. K. Chesterton, he is recognized as a leading authority on Lewis Carroll. The Annotated Alice, which incorporated the text of Carroll's two Alice books, was his most successful work and sold over a million copies, he had a lifelong interest in magic and illusion and was regarded as one of the most important magicians of the twentieth century. He was considered the doyen of American puzzlers, he was a versatile author, publishing more than 100 books. Gardner was best known for creating and sustaining interest in recreational mathematics—and by extension, mathematics in general—throughout the latter half of the 20th century, principally through his "Mathematical Games" columns; these appeared for twenty-five years in Scientific American, his subsequent books collecting them. Gardner was one of the foremost anti-pseudoscience polemicists of the 20th century.
His 1957 book Fads and Fallacies in the Name of Science became a classic and seminal work of the skeptical movement. In 1976 he joined with fellow skeptics to found CSICOP, an organization promoting scientific inquiry and the use of reason in examining extraordinary claims. Gardner, son of a petroleum geologist father and an educator and artist mother, grew up in and around Tulsa, Oklahoma, his lifelong interest in puzzles started in his boyhood when his father gave him a copy of Sam Loyd's Cyclopedia of 5000 Puzzles and Conundrums. He attended the University of Chicago, where he earned his bachelor's degree in philosophy in 1936. Early jobs included reporter on the Tulsa Tribune, writer at the University of Chicago Office of Press Relations, case worker in Chicago's Black Belt for the city's Relief Administration. During World War II, he served for four years in the U. S. Navy as a yeoman on board the destroyer escort USS Pope in the Atlantic, his ship was still in the Atlantic when the war came to an end with the surrender of Japan in August 1945.
After the war, Gardner returned to the University of Chicago. He attended graduate school for a year there. In 1950 he wrote an article in the Antioch Review entitled "The Hermit Scientist", it was one of Gardner's earliest articles about junk science, in 1952 a much-expanded version became his first published book: In the Name of Science: An Entertaining Survey of the High Priests and Cultists of Science and Present. In the late 1940s, Gardner moved to New York City and became a writer and editor at Humpty Dumpty magazine where for eight years he wrote features and stories for it and several other children's magazines, his paper-folding puzzles at that magazine led to his first work at Scientific American. For many decades, his wife Charlotte, their two sons and Tom, lived in Hastings-on-Hudson, New York, where he earned his living as a freelance author, publishing books with several different publishers, publishing hundreds of magazine and newspaper articles. Appropriately enough—given his interest in logic and mathematics—they lived on Euclid Avenue.
The year 1960 saw the original edition of the best-selling book of The Annotated Alice. In 1979, Gardner retired from Scientific American and he and his wife Charlotte moved to Hendersonville, North Carolina. Gardner never retired as an author, but continued to write math articles, sending them to The Mathematical Intelligencer, Math Horizons, The College Mathematics Journal, Scientific American, he revised some of his older books such as Origami and the Soma Cube. Charlotte died in 2000 and two years Gardner returned to Norman, where his son, James Gardner, was a professor of education at the University of Oklahoma, he died there on May 22, 2010. An autobiography — Undiluted Hocus-Pocus: The Autobiography of Martin Gardner — was published posthumously. Martin Gardner had a major impact on mathematics in the second half of the 20th century, his column was called "Mathematical Games" but it was much more than that. His writing introduced many readers to real mathematics for the first time in their lives.
The column lasted for 25 years and was read avidly by the generation of mathematicians and physicists who grew up in the years 1956 to 1981. It was the original inspiration for many of them to become scientists themselves. David Auerbach wrote: A case can be made, in purely practical terms, for Martin Gardner as one of the most influential writers of the 20th century, his popularizations of science and mathematical games in Scientific American, over the 25 years he wrote for them, might have helped create more young mathematicians and computer scientists than any other single factor prior to the advent of the personal computer. Among the wide array of mathematicians, computer scientists, magicians, artists and other influential thinkers who inspired and were in turn inspired by Gardner are John Horton Conway, Bill Gosper, Ron Rivest, Richard K. Guy, Piet Hein, Ronald Graham, Donald Knuth, Robert Nozick, Lee Sallows, Scott Kim, M. C. Escher, Douglas Hofstadter, Roger Penrose, Ian Stewart, David A. Klarner, Benoit Mandelbrot, Elwyn R. Berlekamp, Solomon W. Golomb, Raymond Smullyan, James Randi, Persi Diaconis, Penn & Teller, Ray Hyman.
His admirers included such diverse people as W. H. Auden, Arthur C. Clarke, Carl Sagan, Isaac Asimov, Richard Dawkins, Stephen Jay Gould, the entire French literary group known as the Oulipo. Salvador Dali once sought him out to discuss four-dimensional hypercubes. Gardner wrote to M. C. Escher in 1961 to ask permission