Category Archives: Mathematics


   

Walking LEGO Machines

Walking LEGO Machine 

On Philippe ”Philo” Hurbain’s LEGO site, I found these amazing 8-legged LEGO walkers.  Here you can find a movie of this walker… its very impressive!

These are designed after Theo Jansen‘s amazing walking kinetic sculptures that are powered by wind or solar power. Here are some great videos of these Theo Jansen’s kinetic sculptures.

Here are two computer graphics studies of Theo Jansen’s mechanism.

Enjoy,
Kevin Knuth
Albany NY


   

Wolfram Web Resources

I received this brochure from Wolfram Research (Mathematica) describing their web resources.  I am going to lose it… so I will remember it here in my Online Cortex.

wolfram.com
http://www.wolfram.com
Wolfram.com is updated regularly with information on new breakthroughs and developments for Mathematica users and technology enthusiasts around the world.  This site features demonstrations and videos on the core Wolfram products, a secure and reliable online store, and a variety of valuable resources for professionals, academics, and students in any field.

The Wolfram Demonstrations Project
http://demonstrations.wolfram.com
The Wolfram Demonstrations Project is an open-code resource that brings to life concepts in science, technology, mathematics, art, finance, and a remarkable range of other areas. From elementary education to front-line research, topics span an ever-growing array of categories, with new interactive visualizations added each day by Mathematica users.

Wolfram Mathematica Documentation Center
http://reference.wolfram.com
All Mathematica documentation is now deployed both in-product and online, featuring over 50,000 carefully chosen examples, animations and tutorials, as well as over 100,000 links.  Used every day as an indispensible reference guide, for Mathematica users, the uniquely powerful documentation system is also ideal for those new to the system who want to learn more about Mathematica’s 2500+ high-level functions for visualization, programming, and computation.

Wolfram Mathworld
http://mathworld.wolfram.com
MathWorld is the web’s most extensive mathematics resource.  Assembled over more than a decade and updated regularly with contributions from the world’s mathematics community.  MathWorld features downloadable Mathematica notebooks, interactive applets, and nearly 13,000 entries on topics ranging from pre-algebra to calculus, number theory, and more.

Library
http://library.wolfram.com
A vast colelction of Mathematica-related material, including thousands of articles, references, and courseware programs, conveniently indexed and organized.

Integrals
http://integrals.wolfram.com
A unique webMathematica-powered site for instantly solving integrals online.

Functions
http://functions.wolfram.com
The worlds largest collection of formulas and visualizations about mathematical functions.

Wolfram Science
http://wolframscience.com
The official website of Stephen Wolfram’s A New Kind of Science, with full-text and enhanced features.

Tones
http://tones.wolfram.com
Original music mined from the computational universe, featuring free-downloads and a generator to create your own personal tones.

StephenWolfram.com
http://stephenwolfram.com
A homepage from Mathematica creator, A New Kind of Science author, and Wolfram Research CEO Stephen Wolfram.

MaxEnt 2007

Friday marked the closing session of the 27th International Workshop on Bayesian and Maximum Entropy Methods in Science and Engineering (MaxEnt 2007).  I had the great pleasure to host this year’s meeting in the lovely city of Saratoga Springs.  We had approximately 100 participants from almost 25 countries spanning all six of the populated continents!

This year marked the 50th anniversary of Ed Jaynes’ ground-breaking 1957 paper “Information Theory and Statistical Mechanics” where he introduces the idea that Statistical Mechanics is an Inferential Theory.  This paper led to the concept of Maximum Entropy, which is used to assign priors in Bayesian Probability Theory, but also, was shown by Adom Giffin at this meeting to be consistent with Bayesian learning.

I was very pleased to have had a distinguished array of invited speakers including: Shun-ichi Amari (RIKEN, JAPAN), Jose M. Bernardo (Universitat de Valencia, SPAIN), Tony Bell (Redwood Neuroscience Institute, USA), Philip Goyal (Perimeter Institute, CANADA), Phil Gregory (University of British Columbia, CANADA), and Stephen Roberts (Oxford, UK).  I will blog over the next few days about some of the ideas that were presented at this meeting.

I also was extremely pleased with the Tutorials, which were presented by John Skilling, Jose Bernardo, Ariel Caticha, Carlos Rodriguez, and myself.  They dealt heavily with the foundations of probability theory and connected heavily with information theory, geometry, and order theory.  Rather than being traditional tutorials, he majority of these talk presented new ideas and new results!

Every year the MaxEnt meeting takes on its own personality.  In 2005 in San Jose, the focus was on sampling methods, and I was pleased to have John Skilling’s contribution on Nested Sampling in my volume.  This year, the focus was on the Foundations of Probability Theory, Information Geometry, Entropy and Bayes, Lattices and Measures, Levels and Loops, Information and Physics, and Quantum Mechanics.  The 2007 Proceedings Volume will contain a large number of cutting-edge papers on these exciting topics.  The collective atmosphere of the meeting seemed to indicate that this community is close to making some exciting breakthroughs.

Many of us at the meeting, myself included, were extremely disappointed by the fact that some individuals were unable to attend due to visa problems.  One such individual, who has been prominent in our community, was denied entry due to the fact that he had a dual French-Iranian citizenship.  These policies enacted by our politicians are as damaging to the scientific community and the advances that we work to provide for humanity, as they are to the individuals and their well-being.  It is high time that our nations stop acting like children.

Next year, MaxEnt 2008 will be hosted by Julio M. Stern on the beaches near Sao Paolo BRAZIL.  MaxEnt 2009 will be hosted by Paul Goggans in University, Mississippi USA, and MaxEnt 2010 will be held in Grenoble FRANCE.

Kevin Knuth
Albany NY

Lorentz Attractors, Topology and Modular Flows

While working with POV Ray, I stumbled on this great article from the American Mathematical Society’s Monthly Essays.  This article examines the Lorenz attractor and its periodic orbits.  It turns out that the periodic orbits form knots, but not just any knots.  They comprise only a very small subset of possible knots.

 Lorenz attractor

The article continues to examine modular flows on lattices, and the periodic orbits that result.  It turns out that the knots defined by these periodic orbits on the modular flow coincide with the Lorenz knots!

Kevin Knuth
Albany NY

Double-Angle Formulas

To display equations, I recently installed LaTeX into WordPress by using a Latexrender plugin and it works great!  I am going to try it out by blogging about a relatively easy way to derive the double angle formulas for the sine and cosine.  The double angle formulas relate \sin( 2\theta) and  \cos( 2\theta)  to \sin( \theta) and  \cos( \theta)

Let’s be honest, who wants to waste their cortical space remembering formulas like this.  Instead, you can derive them pretty easily by remembering a few more elementary facts.  First, you can write any complex number as:

e^{i\theta}=\cos(\theta)+i\sin(\theta).

Next, you must remember how to work with exponents:

x^{a} x^{b} = x^{a+b}

and

(x^a)^2 = x^{2a}

Now we put it all together.
We start by writing

e^{i\theta}=\cos(\theta)+i\sin(\theta)

We then square both sides

(e^{i\theta})^2=(\cos(\theta)+i\sin(\theta))^2

Now use the rule for exponents on the left-hand side, and multiply out the right-hand side:

(e^{i 2\theta})=\cos^2(\theta)+2i\cos(\theta)\sin(\theta)-\sin^2(\theta)

Now write the left-hand side in terms of sines and cosines again:

\cos(2\theta)+i\sin(2\theta)
      =\cos^2(\theta)+2i\cos(\theta)\sin(\theta)-\sin^2(\theta)

Last, equate the real parts and the imaginary parts and you have your double-angle formulas:

\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta)
\sin(2\theta) = 2\cos(\theta)\sin(\theta)

Easy as \pi!
Kevin Knuth
Albany NY