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The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations: Yikes. Rather than deciphering it symbol-by-symbol, let's experience the idea. Here's a plain-English metaphor: What does the Fourier Transform do?
This online math course develops the mathematics needed to formulate and analyze probability models for idealized situations drawn from everyday life. Topics include elementary set theory, techniques for systematic counting, axioms for probability, conditional probability, discrete random variables, infinite geometric series, and random walks. Applications to card games like bridge and poker, to gambling, to sports, to election results, and to inference in fields like history and genealogy, national security, and theology. The emphasis is on careful application of basic principles rather than on memorizing and using formulas. Free online math lectures The Quicktime and MP3 formats are available for download, or you can play the Flash version directly.
The concept of infinity is hard to grasp because it is an abstraction. There are no tangible objects in our lives that are truly infinite in number so we really have nothing to compare it to. The only way to get an infinite number of anything is by invoking infinity elsewhere, which doesn’t really clarify matters much. For example, the number of elementary particles in our visible universe, although immensely large, is still a finite number. If we assume that the density of particles in the universe is roughly the same everywhere and further postulate that the universe is of infinite size , then we can arrive at an infinite number of particles.
How to Teach With Technology Learn how to create custom multimedia products for your classroom or ecourse and other great ideas for the classroom. Explore this online course at
“For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art.” - M.C. Escher Contents Page 1 Introduction The Art of Alhambra Our Area of Focus Our Aim Page 2 The Principals behind Tessellations - Translation - Rotation - Reflection - Glide Reflection
6174 is known as Kaprekar's constant [ 1 ] [ 2 ] [ 3 ] after the Indian mathematician D. R. Kaprekar . This number is notable for the following property:
If you are the first to solve this problem I will give you complete set of the Richard Feynman Lectures on Physics (read up upon the Feynman Lectures on Physics ). Question: Take a square of arbitrary side length and construct four new squares with a side length half of the original positioned in the corners a quarter of the way down and across the two sides creating the corner. This would look like: Then in each of the new squares in the furthest corner from the first square create a new square with half the side length a quarter of the way down and across the two sides creating the corner. Repeat this step for each newly constructed square to it infinity.
(This page is entirely factually accurate. It is neither a joke nor a satire nor a collection of fallacious proofs. All these proofs are genuine and the results are true. Thanks.)
Here's the "Troll Pi" or "Pi equals 4" image. Here's the breakdown, as simple as I can make it. All of the following facts are true: The picture describes a sequence of curves.
3 Dimensional Fractals and the Search for the 'true' 3D Mandelbrot Three Dimensional Fractal Mappings Years ago, many fractal enthusiasts viewed these intricate and majestic patterns and thought "Wouldn't this be incredible in 3-D?". Dozens of methods have been developed to view fractals in an arbitrary number of dimensions. Quaternions are 3D shadows of 4D Julia Sets, which, if sliced in a plane, reveal the corresponding 2 dimensional Julia Set. Other attractive developments include 'Quasi-Fuchian" fractals and the recent popularity of the 'Mandelbulb " 3D Mandelbrot - which was exclusively discovered and implemented collaboratively by the inquisitive genius folks at Fractalforums.com . Go there to see what is literally the cutting edge of fractal discovery on a daily basis. Really!
This is a simulation of a network of people playing Prisoner's Dilemma. Red are cooperators; blue are defectors. Credit: UC3M Researchers at Carlos III University of Madrid and the University of Zaragoza theoretically predict, in a scientific study, that contact networks have no influence on cooperation among individuals.
If you’ve not heard about Benford’s Law before, you’re in for a real treat with this post. Before we get into the theory, however, indulge with me in a little thought experiment. (Gedanken) Gedanken Experiment Actual results
"As far as I'm concerned, the funny thing about five is that it's not three, four or six." ~ Professor John Hunton. Does "Penrose tiling" ring a bell? It should if you've been reading this blog for awhile because this phenomenon played a role in the 2011 Nobel Prize in Chemistry. When the Nobel was awarded, I mentioned that, in my opinion, that body of research was a brilliant combination of chemistry, physics and maths.
Imagine I present you with a line of cards labelled 1 through to n , where n is some incredibly large number. I ask you to remove a certain number of cards – which ones you choose is up to you, inevitably leaving ugly random gaps in my carefully ordered sequence. It might seem as if all order must now be lost, but in fact no matter which cards you pick, I can always identify a surprisingly ordered pattern in the numbers that remain. As a magic trick it might not equal sawing a woman in half, but mathematically proving that it is always possible to find a pattern in such a scenario is one of the feats that today garnered Endre Szemerédi mathematics' prestigious Abel prize . The Norwegian Academy of Science and Letters in Oslo awarded Szemerédi the one million dollar prize today for "fundamental contributions to discrete mathematics and theoretical computer science".
A Sierpinksi carpet is one of the more famous fractal objects in mathematics. Creating one is an iterative procedure. Start with a square, divide it into nine equal squares and remove the central one.