DE sign: (Deconstructing in-order to find new meanings)
A blogging space about my personal interests; was made during training in Stockholm #Young Leaders Visitors Program #Ylvp08 it developed into a social bookmarking blog.
I studied #Architecture; interested in#Design #Art #Education #Urban Design #Digital-media #social-media #Inhabited-Environments #Contemporary-Cultures #experimentation #networking #sustainability & more =)
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p.s. sharing is usually out of interest not Blind praise. This is neither sacred nor political.
1.61803398875 Design Hoax? Is the Golden Ratio Fact or Fiction?
I've already posted a video lecture by Stanford Mathematics Professor, Keith Devlin titled Mathematical Thinking, it discussed where the Golden Ration has proved valid and where it was a complete fiction...
There have been several articles also posted on the subject entailing that the famous Golden Ratio is a complete historical design hoax! that was passed on by many architects, builders and made an ideal of by many throughout history!? With no scientific background what so ever... Following will embed article share by Co.Design & Arch Daily Enjoy...
In the world of art, architecture, and design, the golden ratio has earned a tremendous reputation. Greats like Le Corbusier and Salvador Dalí have used the number in their work. The Parthenon, the Pyramids at Giza, the paintings of Michelangelo, the Mona Lisa, even the Apple logo are all said to incorporate it.
It’s bullshit. The golden ratio’s aesthetic bona fides are an urban legend, a myth, a design unicorn. Many designers don’t use it, and if they do, they vastly discount its importance. There’s also no science to really back it up. Those who believe the golden ratio is the hidden math behind beauty are falling for a 150-year-old scam.
First described in Euclid’s Elements 2,300 years ago, the established definition is this: two objects are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The value this works out to is usually written as 1.6180. The most famous application of the golden ratio is the so-called golden rectangle, which can be split into a perfect square, and a smaller rectangle that has the same aspect ratio as the rectangle it was cut away from. You can apply this theory to a larger number of objects by similarly splitting them down.
In plain English: if you have two objects (or a single object that can be split into two objects, like the golden rectangle), and if, after you do the math above, you get the number 1.6180, it’s usually accepted that those two objects fall within the golden ratio. Except there’s a problem. When you do the math, the golden ratio doesn’t come out to 1.6180. It comes out to 1.6180339887… And the decimal points go on forever.
“Strictly speaking, it’s impossible for anything in the real-world to fall into the golden ratio, because it’s an irrational number,” says Keith Devlin, a professor of mathematics at Stanford University. You can get close with more standard aspect ratios. The iPad’s 3:2 display, or the 16:9 display on your HDTV all “float around it,” Devlin says. But the golden ratio is like pi. Just as it’s impossible to find a perfect circle in the real world, the golden ratio cannot strictly be applied to any real world object. It’s always going to be a little off.
It’s pedantic, sure. Isn’t 1.6180 close enough? Yes, it probably would be, if there were anything to scientifically support the notion that the golden ratio had any bearing on why we find certain objects like the Parthenon or the Mona Lisa aesthetically pleasing.
But there isn’t. Devlin says the idea that the golden ratio has any relationship to aesthetics at all comes primarily from two people, one of whom was misquoted, and the other of whom was just making shit up.
The first guy was Luca Pacioli, a Franciscan friar who wrote a book called De Divina Proportione back in 1509, which was named after the golden ratio. Weirdly, in his book, Pacioli didn’t argue for a golden ratio-based theory of aesthetics as it should be applied to art, architecture, and design: he instead espoused the Vitruvian system of rational proportions, after the first-century Roman architect, Vitruvius. The golden ratio view was misattributed to Pacioli in 1799, according to Mario Livio, the guy who literally wrote the book on the golden ratio. But Pacioli was close friends with Leonardo da Vinci, whose works enjoyed a huge resurgence in popularity in the 19th century. Since Da Vinci illustrated De Divina Proportione, it was soon being said that Da Vinci himself used the golden ratio as the secret math behind his exquisitely beautiful paintings.
One guy who believed this was Adolf Zeising. “He’s the guy you really want to burn at the stake for the reputation of the golden ratio,” Devlin laughs. Zeising was a German psychologist who argued that the golden ratio was a universal law that described “beauty and completeness in the realms of both nature and art… which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical.”
He was a long-winded guy. The only problem with Zeising was he saw patterns where none exist. For example, Zeising argued that the golden ratio could be applied to the human body by taking the height from a person’s navel to his toes, then dividing it by the person’s total height. These are just arbitrary body parts, crammed into a formula, Devlin says: “When measuring anything as complex as the human body, it’s easy to come up with examples of ratios that are very near to 1.6.”
But it didn’t matter if it was made up or not. Zeising’s theories became extremely popular, “the 19th-century equivalent of the Mozart Effect,” according to Devlin, referring to the belief that listening to classical music improves your intelligence. And it never really went away. In the 20th century, the famous Swiss-French architect Le Corbusier based his Modulor system of anthropometric proportions on the golden ratio. Dalí painted his masterpiece The Sacrament of the Last Supper on a canvas shaped like a golden rectangle. Meanwhile, art historians started combing back through the great designs of history, trying to retroactively apply the golden ratio to Stonehenge, Rembrandt, the Chatres Cathedral, and Seurat. The link between the golden ratio and beauty has been a canard of the world of art, architecture, and design ever since.
In the real world, people don’t necessarily prefer the golden ratio.
Devlin tells me that, as part of an ongoing, unpublished exercise at Stanford, he has worked with the university’s psychology department to ask hundreds of students over the years what their favorite rectangle is. He shows the students collections of rectangles, then asks them pick out their favorite one. If there were any truth behind the idea that the golden ratio is key to beautiful aesthetics, the students would pick out the rectangle closest to a golden rectangle. But they don’t. They pick seemingly at random. And if you ask them to repeat the exercise, they pick different rectangles. “It’s a very useful way to show new psychology students the complexity of human perception,” Devlin says. And it doesn’t show that the golden ratio is more aesthetically pleasing to people at all.
Devlin’s experiments aren’t the only ones to show people don’t prefer the golden ratio. A study from the Haas School of Business at Berkeley found that, on average, consumers prefer rectangles that are in the range of 1.414 and 1.732. The range contains the golden rectangle, but its exact dimensions are not the clear favorite.
The designers we spoke to about the golden ratio don’t actually find it to be very useful, anyway.
Richard Meier, the legendary architect behind the Getty Center and the Barcelona Museum of Contemporary Art, admits that when he first started his career, he had an architect’s triangle made that matched the golden ratio, but he had never once designed his buildings keeping the golden ratio in mind. “There are so many other numbers and formulas that are more important when designing a building,” he tells me by phone, referring to formulas that can calculate the maximum size certain spaces can be, or ones that can determine structural load.
Alisa Andrasek, the designer behind Biothing, an online repository of computational designs, agrees. “In my own work, I can’t ever recall using the golden ratio,” Andrasek writes in an email. “I can imagine embedding the golden ratio into different systems as additional ‘spice,’ but I can hardly imagine it driving the whole design as it did historically… it is way too simplistic.”
Giorgia Lupi of Accurat, the Italian design and innovation firm, says that, at best, the golden ratio is as important to designers as any other compositional rule, such as the rule of thirds: maybe a fine rule-of-thumb, but one that good designers will feel free to reject. “I don’t really know, in practice, how many designers deliberately employ the golden ratio,” she writes. “I personally have never worked with it our used it in my projects.”
Of the designers we spoke to, industrial designer Yves Béhar of Fuseproject is perhaps kindest to the golden ratio. “I sometimes look at the golden ratio as I observe proportions of the products and graphics we create, but it’s more informational than dogmatic,” he tells me. Even then, he never sets out to design something with the golden ratio in mind. “It’s important as a tool, but not a rule.”
Even designers who are also mathematicians are skeptical of the golden ratio’s use in design. Edmund Harriss is a clinical assistant professor in the University of Arkansas’ mathematics department who uses many formulas to help generate new works of art. But Harriss says that the golden ratio is, at best, just one of many tools at a mathematically inclined designer’s fingertips. “It is a simple number in many ways, and as a result it does turn up in a wide variety of places…” Harriss tells me by email. “[But] it is certainly not the universal formula behind aesthetic beauty.”
If the golden ratio’s aesthetic merit is so flimsy, then why does the myth persist?
Devlin says it’s simple. “We’re creatures who are genetically programmed to see patterns and to seek meaning,” he says. It’s not in our DNA to be comfortable with arbitrary things like aesthetics, so we try to back them up with our often limited grasp of math. But most people don’t really understand math, or how even a simple formula like the golden ratio applies to complex system, so we can’t error-check ourselves. “People think they see the golden ratio around them, in the natural world and the objects they love, but they can’t actually substantiate it,” Devlin tells me. “They are victims to their natural desire to find meaning in the pattern of the universe, without the math skills to tell them that the patterns they think they see are illusory.” If you see the golden ratio in your favorite designs, you’re probably seeing things.
For more than 150 years, the Golden Ratio has been one of the main tenets of design, informing generations of architects, designers, and artists. From Le Corbusier to Apple, Vitruvius to Da Vinci, the ratio purportedly dictates which forms will be found aesthetically pleasing. Yet mathematicians and designers have grown skeptical of the practical applications of the Golden Ratio, with Edmund Harriss of the University of Arkansas' mathematics department putting it at its most simple: "It is certainly not the universal formula behind aesthetic beauty." Writing for Fast Co. Design, John Brownlee collates sources as diverse as the mathematics department at Stanford University to Richard Meier, laying out the case against what may just be design's greatest hoax. Read the full article here.
BBC Universe Documentary The Great Math Mystery BBC Documentary
Fibonacci Melody: Greg Sheehan at TEDxSydney
Greg Sheehan is one of Australia's premier and most innovative percussionists widely regarded internationally as a leader in his field. As a performer, he is significantly represented in the last three decades of Australian contemporary music as both a live band member and studio musician on hundreds of recordings.
Greg has created a melody from the first 8 numbers in the Fibonacci sequence and performs it in the Concert Hall of the Sydney Opera House for TEDxSydney 2013.
In the spirit of ideas worth spreading, TEDx is a program of local, self-organized events that bring people together to share a TED-like experience. At a TEDx event, TEDTalks video and live speakers combine to spark deep discussion and connection in a small group. These local, self-organized events are branded TEDx, where x = independently organized TED event. The TED Conference provides general guidance for the TEDx program, but individual TEDx events are self-organized.* (*Subject to certain rules and regulations)
The Secret Mathematicians - Professor Marcus du Sautoy
Professor du Sautoy examines the way that Mathematics has overtly and covertly inspired some of the greatest artists. He examines how they might be considered as secret mathematicians: http://www.gresham.ac.uk/lectures-and...From composers to painters, writers to choreographers, the mathematician's palette of shapes, patterns and numbers has proved a powerful inspiration. Artists can be subconsciously drawn to the same structures that fascinate mathematicians as they hunt for interesting new structures to frame their creative process. Professor du Sautoy will explore the hidden mathematical ideas that underpin the creative output of well-known artists and reveal that the work of the mathematician is also driven by strong aesthetic values.
The transcript and downloadable versions of the lecture are available from the Gresham College Website: http://www.gresham.ac.uk/lectures-and...
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,500 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Introduction to Mathematical Thinking & History #Stanford #MIT #TEDx #Lectures
Published on Mar 18, 2014
Richard Brown is the Director of Undergraduate Studies in the Department of Mathematics at Johns Hopkins University. However, it wasn't until after he completed his undergraduate studies in architecture that he chose to enter the field of mathematics. Brown is the editor of the book 30-Second Mathematics and has been awarded and nominated for multiple teaching awards.
In the spirit of ideas worth spreading, TEDx is a program of local, self-organized events that bring people together to share a TED-like experience. At a TEDx event, TEDTalks video and live speakers combine to spark deep discussion and connection in a small group. These local, self-organized events are branded TEDx, where x = independently organized TED event. The TED Conference provides general guidance for the TEDx program, but individual TEDx events are self-organized.* (*Subject to certain rules and regulations)
Published on Dec 11, 2012
(October 1, 2012) Keith Devlin gives an overview of the history of mathematics. He discusses how it has evolved over time and explores many of its practical applications in the world.
Originally presented in the Stanford Continuing Studies Program.
(October 8, 2012) Professor Keith Devlin dives into the topics of the golden ratio and fibonacci numbers. Originally presented in the Stanford Continuing Studies Program.
Published on Dec 11, 2012
(October 15, 2012) Professor Keith Devlin looks at how algebra, one of the most foundational concepts in math, was discovered.
Originally presented in the Stanford Continuing Studies Program.
Published on Dec 11, 2012
(October 22, 2012) Professor Keith Devlin discusses how calculus is truly one of the most useful discoveries of all time.
Originally presented in the Stanford Continuing Studies Program.
(October 29, 2012) Keith Devlin concludes the course by discussing the development of mathematical cognition in humans as well as the millennium problems.
Originally presented in the Stanford Continuing Studies Program.
License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu Subtitles are provided through the generous assistance of Jimmy Ren.
Uploaded on Nov 8, 2010
October 23, 2010 - Professor Margot Gerritsen illustrates how mathematics and computer modeling influence the design of modern airplanes, yachts, trucks and cars. This lecture is offered as part of the Classes Without Quizzes series at Stanford's 2010 Reunion Homecoming.
Margot Gerritsen, PhD, is an Associate Professor of Energy Resources Engineering, with expertise in mathematical and computational modeling of energy and fluid flow processes. She teaches courses in energy and the environment, computational mathematics and computing at Stanford University.
Margot Gerritsen is a professor of energy resources engineering and the director of the Institute for Computational and Mathematical Engineering. After receiving her master's degree in applied mathematics in the Netherlands, Gerritsen moved to the United States in search of "hillier and sunnier places." She received her doctorate in scientific computing and computational mathematics at Stanford, and later became a faculty member at the University of Auckland in New Zealand. Gerritsen specializes in renewable and fossil energy production and in computational mathematics. She is also active in coastal ocean dynamics and yacht design.
In the spirit of ideas worth spreading, TEDx is a program of local, self-organized events that bring people together to share a TED-like experience. At a TEDx event, TEDTalks video and live speakers combine to spark deep discussion and connection in a small group. These local, self-organized events are branded TEDx, where x = independently organized TED event. The TED Conference provides general guidance for the TEDx program, but individual TEDx events are self-organized.* (*Subject to certain rules and regulations)
Published on Sep 10, 2013
Margot Gerritsen (Stanford Computational Math) on "Linear Algebra: the incredible beauty of a branch of math with a bad reputation" at a USF LASER - with special thanks to Tim Davis and his beautiful matrix collection
Ein Porträt der Mathematikerin Olga Holtz, Professorin für angewandte Mathematik in Berlin. Dieses Video wurde produziert für WeltWissen - 300 Jahre Wissenschaften in Berlin, Martin-Gropius-Bau, Berlin, von uncertainty-film, d2010.
History of Mathematics in 50min
Published on Sep 21, 2012
GRCC Mathematics Professor John Dersch reviews many historical innovations in math.
MATH HISTORY SERIES OF LECTURES
Uploaded on Mar 13, 2011
Pythagoras' theorem is both the oldest and the most important non-trivial theorem in mathematics.
This is the first part of the first lecture of a course on the History of Mathematics, by N J Wildberger, the discoverer of Rational Trigonometry. We will follow John Stillwell's text Mathematics and its History (Springer, 3rd ed). Generally the emphasis will be on mathematical ideas and results, but largely without proofs, with a main eye on the historical flow of ideas. A few historical tidbits will be thrown in too...
In this first lecture (with two parts) we first give a very rough outline of world history from a mathematical point of view, position the work of the ancient Greeks as following from Egyptian and Babylonian influences, and introduce the most important theorem in all of mathematics: Pythagoras' theorem.
Two interesting related issues are the irrationality of the 'square root of two' (the Greeks saw this as a segment, or perhaps more precisely as the proportion or ratio between two segments, not as a number), and Pythagorean triples, which go back to the Babylonians. These are closely related to the important rational parametrization of a circle, essentially discovered by Euclid and Diophantus. This is a valuable and under-appreciated insight which high school students ought to explicitly see.
In fact young people learning mathematics should really see more of the history of the subject! The Greeks thought of mathematics differently than we do today, and all students can benefit from a closer appreciation of the difficulties which they saw, but which we today largely ignore.
This series has now been extended a few times--with more than 35 videos on the History of Mathematics.
Uploaded on Mar 10, 2011
Pythagoras' theorem is both the oldest and the most important non-trivial theorem in mathematics.
This is the second part of the first lecture of a short course on the History of Mathematics, by N J Wildberger at UNSW (MATH3560 and GENS2005). We will follow John Stillwell's text Mathematics and its History (Springer, 3rd ed). Generally the emphasis will be on mathematical ideas and results, but largely without proofs, with a main eye on the historical flow of ideas. A few historical tidbits will be thrown in too...
In this first lecture (with two parts) we first give a very rough outline of world history from a mathematical point of view, position the work of the ancient Greeks as following from Egyptian and Babylonian influences, and introduce the most important theorem in all of mathematics: Pythagoras' theorem.
Two interesting related issues are the irrationality of the 'square root of two' (the Greeks saw this as a length, but not as a number), and Pythagorean triples, which go back to the Babylonians. These are closely related to the important rational parametrization of a circle, essentially discovered by Euclid and Diophantus.
The Greeks thought of mathematics differently than we do today, and all students can benefit from a closer appreciation of the difficulties which they saw, but which we today largely ignore.
Uploaded on Mar 17, 2011
The ancient Greeks loved geometry and made great advances in this subject. Euclid's Elements was for 2000 years the main text in mathematics, giving a careful systematic treatment of both planar and three dimensional geometry, culminating in the five Platonic solids. Apollonius made a thorough study of conics. Constructions played a key role, using straightedge and compass.
This is one of a series of lectures on the History of Mathematics by Assoc. Prof. N J Wildberger at UNSW.
Uploaded on Mar 18, 2011
The ancient Greeks loved geometry and made great advances in this subject. Euclid's Elements was for 2000 years the main text in mathematics, giving a careful systematic treatment of both planar and three dimensional geometry, culminating in the five Platonic solids. Apollonius made a thorough study of conics: ellipse, parabola and hyperbola. Constructions played a key role, using straightedge and compass.
This is one of a series of lectures on the History of Mathematics by Assoc. Prof. N J Wildberger at UNSW.
Uploaded on Mar 24, 2011
The ancient Greeks studied squares, triangular numbers, primes and perfect numbers. Euclid stated the Fundamental theorem of Arithmetic: that a natural number could be factored into primes in essentially a unique way. We also discuss the Euclidean algorithm for finding a greatest common divisor, and the related theory of continued fractions. Finally we discuss Pell's equation, arising in the famous Cattle-problem of Archimedes.
Uploaded on Mar 28, 2011
We discuss primarily the work of Eudoxus and Archimedes, the founders of calculus. Archimedes in particular discovered formulas that are only found in advanced calculus courses, concerning the relations between the volumes and surface areas of a sphere and a circumscribing cylinder. We also discuss his work on the area of a parabolic arc, Heron's formula (improved using ideas of Rational Trigonometry), hydrostatics, and the Principle of the Lever. He was a true genius.
Uploaded on Apr 3, 2011
After the later Alexandrian mathematicians Ptolemy and Diophantus, Greek mathematics went into decline and the focus shifted eastward. This lecture discusses some aspects of Chinese, Indian and Arab mathematics, in particular the interest in number theory: Pell's equation, the Chinese remainder theorem, and algebra. Most crucial was the introduction of the Hindu-Arabic number system that we use today.
We also discuss the influence of probably the most important problem of the mathematical sciences from a historical point of view: understanding the motion of the night sky, in particular the planets. This motivated work in trigonometry, particularly spherical trigonometry, of both Indian and Arab mathematicians.
Prominent mathematicians whose work we discuss include Sun Zi, Aryabhata, Brahmagupta, Bhaskara I and II, al-Khwarizmi, al-Biruni and Omar Khayyam.
If you are interested in supporting my YouTube Channel: here is the link to my Patreon page: https://www.patreon.com/njwildberger?... You can sign up to be a Patron, and give a donation per view, up to a specified monthly maximum.
Uploaded on Apr 3, 2011
After the later Alexandrian mathematicians Ptolemy and Diophantus, Greek mathematics went into decline and the focus shifted eastward. This lecture discusses some aspects of Chinese, Indian and Arab mathematics, in particular the interest in number theory (Pell's equation, the Chinese remainder theorem, and algebra. Most crucial was the introduction of the Hindu-Arabic number system that we use today.
We also discuss the influence of probably the most important problem of the mathematical sciences from a historical point of view: understanding the motion of the night sky, in particular the planets. This motivated work in trigonometry, particularly spherical trigonometry, of both Indian and Arab mathematicians.
Prominent mathematicians whose work we discuss include Sun Zi, Aryabhata, Brahmagupta, Bhaskara I and II, al-Khwarizmi, al-Biruni and Omar Khayyam.
Uploaded on Apr 10, 2011
We now move to the Golden age of European mathematics: the period 1500-1900, in this course on the History of Mathematics. We discuss hurdles that the Europeans faced before this time and how they emerged, with the help of Arab algebra and translations of Greek works, to harness the Hindu-Arabic number system and a host of novel symbols including Vieta's new use of letters to represent unknowns to tackle new problems.
Quadratic equations had been solved by almost all earlier mathematical civilizations; cubic equations was a natural step, taken by Tartaglia and Cardano and others. Tartaglia also discovered a formula for the volume of a tetrahedron, and Vieta a trigonometric way of solving cubics.
Uploaded on Apr 10, 2011
We now move to the Golden age of European mathematics: the period 1500-1900 in this course on the History of Mathematics. We discuss hurdles that the Europeans faced before this time and how they emerged, with the help of Arab algebra and translations of Greek works, to harness the Hindu-Arabic number system and a host of novel symbols including Vieta's new use of letters to represent unknowns to tackle new problems.
Quadratic equations had been solved by almost all earlier mathematical civilizations; cubic equations was a natural step, taken by Tartaglia and Cardano and others. Tartaglia also discovered a formula for the volume of a tetrahedron, and Vieta a trigonometric way of solving cubics.
Uploaded on May 8, 2011
The development of Cartesian geometry by Descartes and Fermat was one of the main accomplishments of the 17th century, giving a computational approach to Euclidean geometry. Involved are conics, cubics, Bezout's theorem, and the beginnings of a projective view to curves. This merging of numbers and geometry is discussed in terms of the ancient Greeks, and some problems with our understanding of the continuum are observed; namely with irrational numbers and decimal expansions. We also discuss pi and its continued fraction approximations.
Uploaded on May 9, 2011
The development of Cartesian geometry by Descartes and Fermat was one of the main accomplishments of the 17th century, giving a computational approach to Euclidean geometry. Involved are conics, cubics, Bezout's theorem, and the beginnings of a projective view to curves. This merging of numbers and geometry is discussed in terms of the ancient Greeks, and some problems with our understanding of the continuum are observed; namely with irrational numbers and decimal expansions. We also discuss pi and its continued fraction approximations.
Uploaded on May 9, 2011
Projective geometry began with the work of Pappus, but was developed primarily by Desargues, with an important contribution by Pascal. Projective geometry is the geometry of the straightedge, and it is the simplest and most fundamental geometry. We describe the important insights of the 19th century geometers that connected the subject to 3 dimensional space.
Uploaded on May 30, 2011
Calculus has its origins in the work of the ancient Greeks, particularly of Eudoxus and Archimedes, who were interested in volume problems, and to a lesser extent in tangents. In the 17th century the subject was widely expanded and developed in an algebraic way using also the coordinate geometry of Descartes. This is one of the most important developments in the history of mathematics.
Calculus has two branches: the differential and integral calculus. The former arose from the study by Fermat of maxima and minima of functions via horizontal tangents.
The integral calculus computes areas and volumes beyond the techniques of Archimedes. It was developed independently by Newton and Leibnitz, but others contributed too. Newton's focus was on power series, for which differentiation and integration can be done term by term using a formula of Cavalieri, and which gave remarkable new formulas for pi and the circular functions. He had a dynamic view of the subject, motivated in large part by physics.
Leibnitz was more interested in closed forms, and introduced the notation which we use today. Both used infinitesimals, in the form of differentials.
Uploaded on Jun 6, 2011
We discuss various uses of infinite series in the 17th and 18th centuries. In particular we look at the geometric series, power series of log, the Gregory-Newton interpolation formula, Taylor's formula, the Bernoulli's, Eulers summation of the reciprocals of the squares as pi squared over 6, the harmonic series, product expansion of sinx, the zeta function and Euler's product expansion for it, the exponential function, complex values and finally the circular functions too!
Uploaded on Jun 5, 2011
The main historical problem in the history of science is: to explain what is going on with the night sky, in particular what the planets are doing. The resolution of this was the greatest achievement of the 17th century.
The key figures were Copernicus, Galileo, Brahe, Kepler and most famously Isaac Newton. This interesting story, culminating with Kepler's Laws and their explanation by Newton's Laws of Motion and Law of Gravitation, ought to be studied in depth by all undergraduate students of mathematics!
It is notable that the story involves classical geometry in a major way, and gives a great impetus to our study of conic sections and their many remarkable properties.
Uploaded on Jun 5, 2011
The development of non-Euclidean geometry is often presented as a high point of 19th century mathematics. The real story is more complicated, tinged with sadness, confusion and orthodoxy, that is reflected even the geometry studied today. The important insights of Gauss, Lobachevsky and Bolyai, along with later work of Beltrami, were the end result of a long and circuitous study of Euclid's parallel postulate. But an honest assessment must reveal that in fact non-Euclidean geometry had been well studied from two thousand years ago, since the geometry of the sphere had been a main concern for all astronomers.
This lecture gives a somewhat radical and new interpretation of the history, suggesting that there is in fact a much better way of thinking about this subject, as perceived already by Beltrami and Klein, but largely abandoned in the 20th century. This involves a three dimensional linear algebra with an unusual inner product, looked at in a projective fashion. This predates and anticipates the great work of Einstein on relativity and its space-time interpretation by Minkowski.
For those interested, a fuller account of this improved approach is found in my Universal Hyperbolic Geometry (UnivHypGeom) series of YouTube videos.
Published on Apr 29, 2012
After the work of Diophantus, there was something of a lapse in interest in pure number theory for quite some while. Around 1300 Gersonides developed the connection between the Binomial theorem and combinatorics, and then in the 17th century the topic was again taken up, notably by Fermat, and then by Euler, Lagrange, Legendre and Gauss. We discuss several notable results of Fermat, including of course his famous last theorem, also his work on sums of squares, Pell's equation, primes, and rational points on curves. The rational parametrization of the Folium of Descartes is shown, using the technique of Fermat.
We also state Fermat's little theorem using the modular arithmetic language introduced by Gauss.
Published on Apr 29, 2012
The laws of motion as set out by Newton built upon work of Oresme, Galileo and others on dynamics, and the relations between distance, velocity and acceleration in trajectories. With Newton's laws and the calculus, a whole new arena of practical and theoretical investigations opened up to 17th and 18th century mathematicians, such as the Bernoulli family (Johann, Jacob, Daniel, Nicholas, etc), Euler, Huygens and others. Non-algebraic curves played a prominent role, such as the catenary, the shape of a hanging chain, the cycloid, which become famous as both the curve of quickest descent and the curve of equal time descent, and the lemniscate which would play a major role in the theory or elliptic integrals, and gives us our sign for infinity.
We also discuss some other curves that played a role in mechanics, in particular the vibrating string studied by d'Alembert, and the elastica of Euler.
Moving ahead a few centuries, we show that important progress in the theory of curves still happens in modern times, with the discovery of de Casteljau and Bezier, around 1960, of a new way of thinking about curves in terms of control points.
Published on Apr 29, 2012
Complex numbers of the form a+bi are mostly introduced these days in the context of quadratic equations, but according to Stillwell cubic equations are closer to their historical roots. We show how the cubic equation formula of del Ferro, Tartaglia and Cardano requires some understanding of complex numbers even when only real zeroes appear to be involved.
The use of imaginary numbers in calculus manipulations is illustrated with some computations of Johann Bernoulli relating the inverse tan function to complex logarithms, and the connections bewteen tan (na) to tan(a).
The geometrical planar representation of complex numbers goes back to Cotes, Euler and DeMoivre in some form, and then more explicity at the end of the 18th century to Wessel and Argand, and then Gauss.
The Fundamental theorem of algebra is a key undergraduate result that often proves elusive---it was so also for the pioneers of the subject. Euler, Gauss and d'Alembert all struggled with the result, but made progress. Here we outline the ideas behind the proofs of d'Alembert and Gauss.
Published on May 6, 2012
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.
Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a major role in modern physics, due to the work of Einstein.
If you like this topic, and want to learn more, make sure you don't miss Wildberger's exciting new course on Differential Geometry! See the Playlist DiffGeom, at this channel.
Published on May 13, 2012
This video gives a brief introduction to Topology. The subject goes back to Euler (as do so many things in modern mathematics) with his discovery of the Euler characteristic of a polyhedron, although arguably Descartes had found something close to this in his analysis of curvature of a polyhedron. We introduce this via rational turn angles, a renormalization of angle where a full turn has the value one (very reasonable, and ought to be used more!!) The topological nature of the Euler characteristic was perhaps first understood by Poincare, and we sketch his argument for its invariance under continuous transformations.
We discuss the sphere, torus, genus g surfaces and the classification of orientable, and non-orientable closed 2 dimensional surfaces, such as the Mobius band (which has a boundary) and the projective plane (which does not). The interest in these objects resulted from Riemann's work on surfaces associated to multi-valued functions in the setting of complex analysis.
Finally we briefly mention the important notion of a simply connected space, and the Poincare conjecture, solved recently, according to current accounts, by G. Perelman.
If you enjoy this subject, you can have a look at my video series Algebraic Topology. This series has now also been continued, so if you go to the Playlist MathHistory, you will find more videos on the History of Mathematics.
Published on Mar 5, 2014
In the 19th century, the geometrical aspect of the complex numbers became generally appreciated, and mathematicians started to look for higher dimensional examples of how arithmetic interacts with geometry.
A particularly interesting development is the discovery of quaternions by W. R. Hamilton, and the subsequent discovery of octonians by his friend Graves and later by A. Cayley. Surprisingly perhaps the arithmetic of these 4 and 8 dimensional extensions of complex numbers are intimately connected with number theoretical formulas going back to Diophantus, Fibonacci and Euler.
Published on Apr 16, 2014
In the 19th century, the study of algebraic curves entered a new era with the introduction of homogeneous coordinates and ideas from projective geometry, the use of complex numbers both on the curve and at infinity, and the discovery by the great German mathematician B. Riemann that topological aspects of complex curves were intimately connected with the arithmetic of the curves.
In this lecture we look at the use of homogeneous coordinates, stereographic projection and the Riemann sphere, circular points at infinity, Laguerre's projective description of angle, curves over the complex numbers and the genus of Riemann surfaces.
This meeting of projective geometry, algebra and topology led the way to modern algebraic geometry.
Published on May 8, 2014
Here we give an introduction to the historical development of group theory, hopefully accessible even to those who have not studied group theory before, showing how in the 19th century the subject evolved from its origins in number theory and algebra to embracing a good part of geometry.
Actually the historical approach is a very fine way of learning about the subject for the first time.
We discuss how group theory enters perhaps first with Euler's work on Fermat's little theorem and his generalization of it, involving arithmetic mod n. We mention Gauss' composition of quadratic forms, and then look at permutations, which played an important role in Lagrange's approach to the problem of solving polynomial equations, and was then taken up by Abel and Galois.
The example of the symmetric group is at the heart of the subject, and so we examine S_3. In the 19th century groups of transformations became to be intimately tied to symmetries of geometries, with the work of Klein and Lie. A nice example that ties together the algebraic and geometric sides of the subject is the symmetry groups of the Platonic solids.
Published on May 11, 2014
Galois theory gives a beautiful insight into the classical problem of when a given polynomial equation in one variable, such as x^5-3x^2+4=0 has solutions which can be expressed using radicals. Historically the problem of solving algebraic equations is one of the great drivers of algebra, with the quadratic equation going back to antiquity, and the discovery of the cubic solution by Italian mathematicians in the 1500's. Here we look at the quartic equation and give a method for factoring it, which relies on solving a cubic equation. We review the connections between roots and coefficients, which leads to the theory of symmetric functions and the identities of Newton.
Lagrange was the key figure that introduced the modern approach to the subject. He realized that symmetries between the roots/zeros of an equation were an important tool for obtaining them, and he developed an approach using resolvants, that suggested that the 5th degree equation was perhaps not likely to yield to a solution. This was confirmed by work of Ruffini and Abel, which set the stage for the insights of E. Galois.
Published on May 15, 2014
We continue our historical introduction to the ideas of Galois and others on the fundamental problem of how to solve polynomial equations. In this video we focus on Galois' insights into how extending our field of coefficients, typically by introducing some radicals, the symmetries of the roots diminishes. We get a correspondence between a descending chain of groups of symmetries, and an increasing chain of fields of coefficients. This was the key that allowed Galois to see why some equations were solvable by radicals and others not, and in particular to explain Ruffini and Abel's result on the insolvability of the general quintic equation.
Published on May 18, 2014
In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
Published on May 18, 2014
In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some serious foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
Published on May 15, 2014
We continue our historical introduction to the ideas of Galois and others on the fundamental problem of how to solve polynomial equations. In this video we focus on Galois' insights into how extending our field of coefficients, typically by introducing some radicals, the symmetries of the roots diminishes. We get a correspondence between a descending chain of groups of symmetries, and an increasing chain of fields of coefficients. This was the key that allowed Galois to see why some equations were solvable by radicals and others not, and in particular to explain Ruffini and Abel's result on the insolvability of the general quintic equation.
Published on Jun 3, 2014
This is the second video in this lecture on simple groups, Lie groups and manifestations of symmetry.
During the 19th century, the role of groups shifted from its origin in number theory and the theory of equations to its role in describing symmetry in geometry. In this video we talk about the history of the search for simple groups, the role of symmetry in tesselations, both Euclidean, spherical and hyperbolic, and the introduction of continuous groups, or Lie groups, by Sophus Lie. Along the way we meet briefly many remarkable mathematical objects, such as the Golay code whose symmetries explain partially the Mathieu groups, the exceptional Lie groups discovered by Killing, and some of the other sporadic simple groups, culminating with the Monster group of Fisher and Greiss.
The classification of finite simple groups is a high point of 20th century mathematics and the cumulative efforts of many mathematicians.
Published on Mar 18, 2015
We review some of the development of number systems from the ancient Greeks, followed by the Indian and then Arabic development of our Hindu-Arabic numeral system. Then we focus on the new directions forged by the European mathematicians of the 15th and 16th centuries, culminating in the work of Simon Stevin, who shaped our current view towards decimal number arithmetic. We examine the idea that Stevin was the father of `real numbers' (this is not really credible) and also look at some of his other achievements, for example in music and physics.
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