Tag: math


Gödel’s incompleteness theorems – Wikipedia, the free encyclopedia

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an “effective procedure” (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, a corollary of the first, shows that such a system cannot demonstrate its own consistency.


Jenn 3d

Jenn is a toy for playing with various quotients of Cayley graphs of finite Coxeter groups on four generators. Jenn builds the graphs using the Todd-Coxeter algorithm, embeds them into the 3-sphere, and stereographically projects them onto euclidean 3-space. (The models really live in the hypersphere so they looked curved in our flat space.) Jenn has some basic motion models governing the six degrees of freedom of rotation of the hypersphere.


An Introduction to Dynamical Systems and Chaos

This tutorial will develop the basic ingredients necessary for modeling and understanding simple (and not so simple) non-linear dynamical systems. The goal of these exercises are to demonstrate you that you can develop significant insight into the behavior of complicated non-linear systems with just a little math, a little art and a little modeling software. By themselves, these tools can lead to frustration. However, when combined in the right ways they can give you surprising powers of understanding. The purpose of this tutorial is to give you practice and guidance into the basic tricks of the trade so that when you are done you will be able to


Lockharts Lament

Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.


No, really, pi is wrong: The Tau Manifesto by Michael Hartl

Welcome to the Tau Manifesto. This manifesto is dedicated to one of the most important numbers in mathematics, perhaps the most important: the circle constant relating the circumference of a circle to its linear dimension. For millennia, the circle has been considered the most perfect of shapes, and the circle constant captures the geometry of the circle in a single number. Of course, the traditional choice of circle constant is pi—but, as mathematician Bob Palais notes in his delightful article “pi Is Wrong!”,1 pi is wrong. It’s time to set things right.


Phi: 1.618. The best source to the golden section, golden mean, divine proportion, Fibonacci series and phi, 1.618. Explore its application to art, design, life, beauty, mathematics, geometry, stock markets, theology, cosmology and more.

GoldenNumber.Net exists to share information on the pervasive appearance of Phi, 1.618 … , the Golden Ratio in life and the universe. Its goal is to present a broad sampling of phi related topics in an engaging and easy-to-understand format and to provide an online community, in which new findings about Phi can be shared.


Karlheinz Essl: Lexikon-Sonate – algorithmic music generator

Lexikon-Sonate is an interactive realtime composition environment for musical composition and live performances. It takes advantage of composition algorithms that has been developed by Karlheinz Essl since 1985. With this algorithmic music generator on can easily create fascinating and complex musical structures on the fly. Furthermore, Lexikon-Sonate is an infinite music installation that can run on a computer for years without repeating itself. Finally, Lexikon-Sonate can be used as an instrument for live performance of electronic music.


A flood of flat-sevenths

According to many pop-musicologists the flat-seventh chord, or subtonic, can be regarded as one of the marks of the Beatles’ experimental period. Some of them even view the way in which the group handled this chord in their songs as one of their real musical innovations. On the Beatles’ 1966 album Revolver, this chord is paired to a lavish use of quartal harmonies. Is this peculiar chord, along with the quartal harmonies, responsible for the album’s meditative atmosphere? Answering this question, Ger Tillekens here takes a closer look at the flat-seventh in songs like “Taxman,” “I’m Only Sleeping,” “Love You To” and “Here, There, and Everywhere.”

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