A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, given any computer algorithm, a Turing machine capable of simulating that algorithm's logic can be constructed; the machine operates on an infinite memory tape divided into discrete "cells". The machine positions its "head" over a cell and "reads" or "scans" the symbol there; as per the symbol and its present place in a "finite table" of user-specified instructions, the machine writes a symbol in the cell either moves the tape one cell left or right either proceeds to a subsequent instruction or halts the computation. The Turing machine was invented in 1936 by Alan Turing, who called it an "a-machine". With this model, Turing was able to answer two questions in the negative: Does a machine exist that can determine whether any arbitrary machine on its tape is "circular", thus by providing a mathematical description of a simple device capable of arbitrary computations, he was able to prove properties of computation in general—and in particular, the uncomputability of the Entscheidungsproblem.
Thus, Turing machines prove fundamental limitations on the power of mechanical computation. While they can express arbitrary computations, their minimalistic design makes them unsuitable for computation in practice: real-world computers are based on different designs that, unlike Turing machines, use random-access memory. Turing completeness is the ability for a system of instructions to simulate a Turing machine. A programming language, Turing complete is theoretically capable of expressing all tasks accomplishable by computers. A Turing machine is a general example of a CPU that controls all data manipulation done by a computer, with the canonical machine using sequential memory to store data. More it is a machine capable of enumerating some arbitrary subset of valid strings of an alphabet. A Turing machine has a tape of infinite length on which it can perform write operations. Assuming a black box, the Turing machine cannot know whether it will enumerate any one specific string of the subset with a given program.
This is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing. The Turing machine is capable of processing an unrestricted grammar, which further implies that it is capable of robustly evaluating first-order logic in an infinite number of ways; this is famously demonstrated through lambda calculus. A Turing machine, able to simulate any other Turing machine is called a universal Turing machine. A more mathematically oriented definition with a similar "universal" nature was introduced by Alonzo Church, whose work on lambda calculus intertwined with Turing's in a formal theory of computation known as the Church–Turing thesis; the thesis states that Turing machines indeed capture the informal notion of effective methods in logic and mathematics, provide a precise definition of an algorithm or "mechanical procedure". Studying their abstract properties yields many insights into computer science and complexity theory. In his 1948 essay, "Intelligent Machinery", Turing wrote that his machine consisted of:...an unlimited memory capacity obtained in the form of an infinite tape marked out into squares, on each of which a symbol could be printed.
At any moment there is one symbol in the machine. The machine can alter the scanned symbol, its behavior is in part determined by that symbol, but the symbols on the tape elsewhere do not affect the behavior of the machine. However, the tape can be moved back and forth through the machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore have an innings; the Turing machine mathematically models a machine. On this tape are symbols, which the machine can read and write, one at a time, using a tape head. Operation is determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write a 1. In the original article, Turing imagines not a mechanism, but a person whom he calls the "computer", who executes these deterministic mechanical rules slavishly. More a Turing machine consists of: A tape divided into cells, one next to the other; each cell contains a symbol from some finite alphabet. The alphabet contains one or more other symbols.
The tape is assumed to be arbitrarily extendable to the left and to the right, i.e. the Turing machine is always supplied with as much tape as it needs for its computation. Cells that have not been written before are assumed to be filled with the blank symbol. In some models the tape has a left e
A cellular automaton is a discrete model studied in computer science, physics, complexity science, theoretical biology and microstructure modeling. Cellular automata are called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessellation structures, iterative arrays. A cellular automaton consists of a regular grid of cells, each in one of a finite number of states, such as on and off; the grid can be in any finite number of dimensions. For each cell, a set of cells called. An initial state is selected by assigning a state for each cell. A new generation is created, according to some fixed rule that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood; the rule for updating the state of cells is the same for each cell and does not change over time, is applied to the whole grid though exceptions are known, such as the stochastic cellular automaton and asynchronous cellular automaton. The concept was discovered in the 1940s by Stanislaw Ulam and John von Neumann while they were contemporaries at Los Alamos National Laboratory.
While studied by some throughout the 1950s and 1960s, it was not until the 1970s and Conway's Game of Life, a two-dimensional cellular automaton, that interest in the subject expanded beyond academia. In the 1980s, Stephen Wolfram engaged in a systematic study of one-dimensional cellular automata, or what he calls elementary cellular automata. Wolfram published A New Kind of Science in 2002, claiming that cellular automata have applications in many fields of science; these include cryptography. The primary classifications of cellular automata, as outlined by Wolfram, are numbered one to four, they are, in order, automata in which patterns stabilize into homogeneity, automata in which patterns evolve into stable or oscillating structures, automata in which patterns evolve in a chaotic fashion, automata in which patterns become complex and may last for a long time, with stable local structures. This last class are thought to be computationally universal, or capable of simulating a Turing machine.
Special types of cellular automata are reversible, where only a single configuration leads directly to a subsequent one, totalistic, in which the future value of individual cells only depends on the total value of a group of neighboring cells. Cellular automata can simulate a variety of real-world systems, including biological and chemical ones. One way to simulate a two-dimensional cellular automaton is with an infinite sheet of graph paper along with a set of rules for the cells to follow; each square is called a "cell" and each cell has two possible states and white. The neighborhood of a cell is the nearby adjacent, cells; the two most common types of neighborhoods are the von Neumann neighborhood and the Moore neighborhood. The former, named after the founding cellular automaton theorist, consists of the four orthogonally adjacent cells; the latter includes the von Neumann neighborhood as well as the four diagonally adjacent cells. For such a cell and its Moore neighborhood, there are 512 possible patterns.
For each of the 512 possible patterns, the rule table would state whether the center cell will be black or white on the next time interval. Conway's Game of Life is a popular version of this model. Another common neighborhood type is the extended von Neumann neighborhood, which includes the two closest cells in each orthogonal direction, for a total of eight; the general equation for such a system of rules is kks, where k is the number of possible states for a cell, s is the number of neighboring cells used to determine the cell's next state. Thus, in the two dimensional system with a Moore neighborhood, the total number of automata possible would be 229, or 1.34×10154. It is assumed that every cell in the universe starts in the same state, except for a finite number of cells in other states. More it is sometimes assumed that the universe starts out covered with a periodic pattern, only a finite number of cells violate that pattern; the latter assumption is common in one-dimensional cellular automata.
Cellular automata are simulated on a finite grid rather than an infinite one. In two dimensions, the universe would be a rectangle instead of an infinite plane; the obvious problem with finite grids is. How they are handled will affect the values of all the cells in the grid. One possible method is to allow the values in those cells to remain constant. Another method is to define neighborhoods differently for these cells. One could say that they have fewer neighbors, but one would have to define new rules for the cells located on the edges; these cells are handled with a toroidal arrangement: when one goes off the top, one comes in at the corresponding position on the bottom, when one goes off the left, one comes in on the right. This can be visualized as taping the left and right edges of the rectangle to form a tube taping the top and bottom edges of the tube to form a torus. Universes of other dimensions are handled similarly; this solves bounda
James Gary Propp is a professor of mathematics at the University of Massachusetts Lowell. In high school, Propp was one of the national winners of the United States of America Mathematical Olympiad, an alumnus of the Hampshire College Summer Studies in Mathematics. Propp obtained his AB in mathematics in 1982 at Harvard. After advanced study at Cambridge, he obtained his PhD from the University of California at Berkeley, he has held professorships at seven universities, including Harvard, MIT, the University of Wisconsin, the University of Massachusetts Lowell. Propp is the co-editor of the book Microsurveys in Discrete Probability and has written more than thirty journal articles on game theory and probability, recreational mathematics, he lectures extensively and has served on the Mathematical Olympiad Committee of the Mathematical Association of America, which sponsors the USAMO. In the early 90s Propp lived in Boston and in Arlington, Massachusetts. In 1996, Propp and David Wilson invented coupling from the past, a method for sampling from the stationary distribution of a Markov chain among Markov chain Monte Carlo algorithms.
Contrary to many MCMC algorithms, coupling from the past gives in principle a perfect sample from the stationary distribution. His papers have discussed the use of surcomplex numbers in game theory. Propp was a member of the National Puzzlers' League under the nom Aesop, he was recruited for the organisation by colleague Henri Picciotto, cruciverbalist and co-author of the league's first cryptic crossword collection. Propp is the creator of the "Self-Referential Aptitude Test", a humorous multiple-choice test in which all questions except the last make self-references to their own answers, it was created in the early 1990s for a puzzlers' party. Propp is the author of Tuscanini, a 1992 children's book about a musical elephant, illustrated by Ellen Weiss. In 2015 he was elected as a fellow of the American Mathematical Society "for contributions to combinatorics and probability, for mentoring and exposition." He is married to research psychologist Alexandra Gubin. They have a daughter Eliana. Propp's website Jim Propp at the Mathematics Genealogy Project
Simple DirectMedia Layer
Simple DirectMedia Layer is a cross-platform software development library designed to provide a hardware abstraction layer for computer multimedia hardware components. Software developers can use it to write high-performance computer games and other multimedia applications that can run on many operating systems such as Android, iOS, Mac OS X, Windows. SDL manages video, input devices, CD-ROM, shared object loading and timers. For 3D graphics, it can handle an OpenGL or Direct3D context. A common misconception is that SDL is a game engine. However, the library is suited to building games directly, or is usable indirectly by engines built on top of it; the library is internally written in C and depending on the target platform, C++ or Objective-C, provides the application programming interface in C, with bindings to other languages available. It is free and open-source software subject to the requirements of the zlib License since version 2.0, with prior versions subject to the GNU Lesser General Public License.
Under the zlib License, SDL 2.0 is available for static linking in closed-source projects, unlike SDL 1.2. SDL is extensively used in the industry in both small projects. Over 700 games, 180 applications, 120 demos have been posted on the library website. Sam Lantinga created the library, first releasing it while working for Loki Software, he got the idea while porting a Windows application to Macintosh. He used SDL to port Doom to BeOS. Several other free libraries were developed to work alongside SDL, such as SMPEG and OpenAL, he founded Galaxy Gameworks in 2008 to help commercially support SDL, although the company plans are on hold due to time constraints. Soon after putting Galaxy Gameworks on hold, Lantinga announced that SDL 1.3 would be licensed under the zlib License. Lantinga announced SDL 2.0 on 14 July 2012, at the same time announcing that he was joining Valve Corporation, the first version of, announced the same day he joined the company. Lantinga announced the stable release of SDL 2.0.0 on 13 August 2013.
SDL 2.0 is a major update to the SDL 1.2 codebase with a different, not backwards-compatible API. It replaces several parts of the 1.2 API with more general support for multiple input and output options. Some feature additions include multiple window support, hardware-accelerated 2D graphics, better Unicode support. Support for Mir and Wayland was added in SDL 2.0.2 and enabled by default in SDL 2.0.4. Version 2.0.4 provided better support for Android. SDL is a wrapper around the operating-system-specific functions; the only purpose of SDL is to provide a common framework for accessing these functions for multiple operating systems. SDL provides support for 2D pixel operations, file access, event handling and threading, it is used to complement OpenGL by setting up the graphical output and providing mouse and keyboard input, since OpenGL comprises only rendering. A game using the Simple DirectMedia Layer will not automatically run on every operating system, further adaptations must be applied; these are reduced to the minimum, since SDL contains a few abstraction APIs for frequent functions offered by an operating system.
The syntax of SDL is function-based: all operations done in SDL are done by passing parameters to subroutines. Special structures are used to store the specific information SDL needs to handle. SDL functions are categorized under several different subsystems. SDL is divided into several subsystems: Basics Initialization and Shutdown, Configuration Variables, Error Handling, Log Handling Video Display and Window Management, surface functions, rendering acceleration, etc. Input Events Event handling, Support for Keyboard, Mouse and Game controller Force Feedback SDL_haptic.h implements support for "Force Feedback" Audio SDL_audio.h implements Audio Device Management and Recording Threads multi-threading: Thread Management, Thread Synchronization Primitives, Atomic Operations Timers Timer Support File Abstraction Filesystem Paths, File I/O Abstraction Shared Object Support Shared Object Loading and Function Lookup Platform and CPU Information Platform Detection, CPU Feature Detection, Byte Order and Byte Swapping, Bit Manipulation Power Management Power Management Status Additional Platform-specific functionalityBesides this basic, low-level support, there are a few separate official libraries that provide some more functions.
These comprise the "standard library", are provided on the official website and included in the official documentation: SDL_image — support for multiple image formats SDL_mixer — complex audio functions for sound mixing SDL_net — networking support SDL_ttf — TrueType font rendering support SDL_rtf — simple Rich Text Format renderingOther, non-standard libraries exist. For example: SDL_Collide on Sourceforge created by Amir Taaki; the SDL 2.0 library has language bindings for: Ada C C++ C# D Genie Go Haskell Lua Nim OCaml Pascal Perl Python Rust Vala Common Lisp Because of the way SDL is designed, much of its source code is split into separate modules for each operating system, to make calls to the underlying system. When SDL is compiled, the appropriate modules are selected for the target system. Following back-ends are available: GDI back-end for Microsoft Windows. DirectX back-end. Sam Lantinga has stated. Quartz back-end for macOS. Xlib back-end for X11-based windowing system on various operating systems.
OpenGL contexts on various platf
Christopher Gale Langton is an American computer scientist and one of the founders of the field of artificial life. He coined the term in the late 1980s when he organized the first "Workshop on the Synthesis and Simulation of Living Systems" at the Los Alamos National Laboratory in 1987. Following his time at Los Alamos, Langton joined the Santa Fe Institute, to continue his research on artificial life, he left SFI in the late 1990s, abandoned his work on artificial life, publishing no research since that time. Langton is the first-born son of author of books including the Homer Kelly Mysteries, he has two adult sons: Colin. Langton made numerous contributions to the field of artificial life, both in terms of simulation and computational models of given problems and to philosophical issues, he early identified the problems of information and reproduction as intrinsically connected with complexity and its basic laws. Inspired by ideas coming from physics phase transitions, he developed several key concepts and quantitative measures for cellular automata and suggested that critical points separating order from disorder could play a important role in shaping complex systems in biology.
These ideas were explored albeit with different approximations, by James P. Crutchfield and Per Bak among others. While a graduate student at the University of Michigan, Langton created the Langton ant and Langton loop, both simple artificial life simulations, in addition to his Lambda parameter, a dimensionless measure of complexity and computation potential in cellular automata, given by a chosen state divided by all the possible states. For a 2-state, 1-r neighborhood, 1D cellular automata the value is close to 0.5. For a 2-state, Moore neighborhood, 2D cellular automata, like Conway's Life, the value is 0.273. Langton is the first-born son of author of books including the Homer Kelly Mysteries, he has two adult sons: Colin. He is an atheist. Christopher G. Langton. "Artificial Life: An Overview". MIT Press, 1995. Christopher G. Langton. "Artificial Life III: Proceedings of the Third Interdisciplinary Workshop on the Synthesis and Simulation of Living Systems". Addison-Wesley, 1993. Christopher G. Langton.
"Life at the Edge of Chaos". in "Artificial Life II", Addison-Wesley, 1991. Christopher G. Langton. "Artificial Life II: Proceedings of the Second Interdisciplinary Workshop on the Synthesis and Simulation of Living Systems". Addison-Wesley, 1991. Christopher G. Langton. "Computation at the edge of chaos". Physica D, 42, 1990. Christopher G. Langton. "Computation at the edge of Chaos: Phase-Transitions and Emergent Computation." Ph. D. Thesis, University of Michigan. Christopher G. Langton. "Is There a Sharp Phase Transition for Deterministic Cellular Automata?", with W. K Wootters, Physica D, 45, 1990. Christopher G. Langton. "Artificial Life: Proceedings of an Interdisciplinary Workshop on the Synthesis and Simulation of Living Systems". Addison-Wesley, 1988. Christopher G. Langton. "Studying Artificial Life with Cellular Automata". Physica D, 22, 1986. Christopher G. Langton. "Self Reproduction in Cellular Automata". Physica D, 10, 1984. About Langton's workA. GaJardo, A. Moreira, E. Goles. "Complexity of Langton's Ant".
Discrete Applied Mathematics, 117, 2002. M. Boden. "The Philosophy of Artificial Life". Oxford University Press, 1996. Stuart Kauffman. Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, 1993. Melanie Mitchell, Peter T. Hraber, James P. Crutchfield. Revisiting the edge of chaos: Evolving cellular automata to perform computations. Complex Systems, 7:89–130, 1993. Melanie Mitchell, James P. Crutchfield and Peter T. Hraber. Dynamics and the "Edge of Chaos": A Re-Examination J. P. Crutchfield and K. Young, "Computation at the Onset of Chaos", in Complexity and the Physics of Information, W. Zurek, editor, SFI Studies in the Sciences of Complexity, VIII, Addison-Wesley, Massachusetts pp. 223–269. Artificial life Langton's loops Langton's ant Cellular Automata Explanation of Langton's Lambda The Swarm development group
In mathematics, the Fibonacci numbers denoted Fn form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is, F 0 = 0, F 1 = 1, F n = F n − 1 + F n − 2, for n > 1. One has F2 = 1. In some books, in old ones, F0, the "0" is omitted, the Fibonacci sequence starts with F1 = F2 = 1; the beginning of the sequence is thus: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Fibonacci numbers are related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are named after Italian mathematician Leonardo of Pisa known as Fibonacci, they appear to have first arisen as early as 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.
Fibonacci numbers appear unexpectedly in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, graphs called Fibonacci cubes used for interconnecting parallel and distributed systems, they appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts. Fibonacci numbers are closely related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. Lucas numbers are intimately connected with the golden ratio; the Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1985. In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long syllables of 2 units duration, juxtaposed with short syllables of 1 unit duration.
Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm + 1. Knowledge of the Fibonacci sequence was expressed as early as Pingala. Singh cites Pingala's cryptic formula misrau cha and scholars who interpret it in context as saying that the number of patterns for m beats is obtained by adding one to the Fm cases and one to the Fm−1 cases. Bharata Muni expresses knowledge of the sequence in the Natya Shastra. However, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala: Variations of two earlier meters... For example, for four, variations of meters of two three being mixed, five happens.... In this way, the process should be followed in all mātrā-vṛttas. Hemachandra is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number... of the next mātrā-vṛtta."
Outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. Using it to calculate the growth of rabbit populations. Fibonacci considers the growth of a hypothetical, idealized rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field. Fibonacci posed the puzzle: how many pairs will there be in one year? At the end of the first month, they mate. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair, making 5 pairs. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month; this is the nth Fibonacci number. The name "Fibonacci sequence" was first used by the 19th
Ian Stewart (mathematician)
Ian Nicholas Stewart is a British mathematician and a popular-science and science-fiction writer. He is Emeritus Professor of Mathematics at the University of England. Stewart was born in 1945 in England. While in the sixth form at Harvey Grammar School in Folkestone he came to the attention of the mathematics teacher; the teacher had Stewart sit mock A-level examinations without any preparation along with the upper-sixth students. He was awarded a scholarship to study at the University of Cambridge as an undergraduate student of Churchill College, where he studied the Mathematical Tripos and obtained a first-class Bachelor of Arts degree in mathematics in 1966. Stewart went to the University of Warwick where his PhD on Lie algebras was supervised by Brian Hartley and completed in 1969. After his PhD, Stewart was offered an academic position at Warwick, he is well known for his popular expositions of mathematics and his contributions to catastrophe theory. While at Warwick, Stewart edited the mathematical magazine Manifold.
He wrote a column called "Mathematical Recreations" for Scientific American magazine from 1991 to 2001. This followed the work of past columnists like Martin Gardner, Douglas Hofstadter, A. K. Dewdney. Altogether, he wrote 96 columns for Scientific American, which were reprinted in the books "Math Hysteria", "How to Cut a Cake: And Other Mathematical Conundrums" and "Cows in the Maze". Stewart has held visiting academic positions in Germany, New Zealand, the US. Stewart has published more than 140 scientific papers, including a series of influential papers co-authored with Jim Collins on coupled oscillators and the symmetry of animal gaits. Stewart has collaborated with Jack Cohen and Terry Pratchett on four popular science books based on Pratchett's Discworld. In 1999 Terry Pratchett made both Jack Cohen and Professor Ian Stewart "Honorary Wizards of the Unseen University" at the same ceremony at which the University of Warwick gave Terry Pratchett an honorary degree. In March 2014 Ian Stewart's iPad app, Incredible Numbers by Professor Ian Stewart, launched in the App Store.
The app was produced in partnership with Touch Press. The Science of Discworld, with Jack Cohen and Terry Pratchett The Science of Discworld II: The Globe, with Jack Cohen and Terry Pratchett The Science of Discworld III: Darwin's Watch, with Jack Cohen and Terry Pratchett The Science of Discworld IV: Judgement Day, with Jack Cohen and Terry Pratchett Catastrophe Theory and its Applications, with Tim Poston, Pitman, 1978. ISBN 0-273-01029-8. Complex Analysis: The Hitchhiker's Guide to the Plane, I. Stewart, D Tall. 1983 ISBN 0-521-24513-3 Algebraic number theory and Fermat's last theorem, 3rd Edition, I. Stewart, D Tall. A. K. Peters ISBN 1-56881-119-5 Galois Theory, 3rd Edition and Hall ISBN 1-58488-393-6 Galois Theory Errata The Foundations of Mathematics, 2nd Edition, I. Stewart, D Tall. ISBN 978-019870-643-4 Wheelers, with Jack Cohen Heaven, with Jack Cohen, ISBN 0-446-52983-4, May 2004 In 1995 Stewart received the Michael Faraday Medal and in 1997 he gave the Royal Institution Christmas Lecture on The Magical Maze.
He was elected as a Fellow of the Royal Society in 2001. Stewart was the first recipient in 2008 of the Christopher Zeeman Medal, awarded jointly by the London Mathematical Society and the Institute of Mathematics and its Applications for his work on promoting mathematics. Stewart married his wife, Avril, in 1970, they met at a party at a house. They have two sons, he lists his recreations as science fiction, guitar, keeping fish, geology and snorkelling. Ian Stewart at the Mathematics Genealogy Project personal webpage Michael Faraday prize winners 2004–1986 Directory of Fellows of the Royal Society: Ian Stewart Prof Ian Stewart at Debrett's People of Today What does a Martian look like? Jack Cohen and Ian Stewart set out to find the answers Ian Stewart on space exploration by NASA Ian Stewart on Minesweeper one of the Millennium mathematics problems Press release about Terry Pratchett "Wizard Making" of Jack Cohen and Ian Stewart at the University of Warwick Interview with Ian Stewart on the Science of Discworld series Audio Interview with Ian Stewart on April 25, 2007 from WINA's Charlottesville Right Now Podcast series with Ian Stewart on the history of symmetry A Partly True Story published in: Scientific American, Feb 1993 "The Joy of Mathematics – A conversation with Ian Stewart", Ideas Roadshow, 2013 "In conversation with Ian Stewart", Chalkdust Magazine, 2016