Let’s play the following “chaos game”. Consider a regular triangle and take some initial point in it. Now pick one of the three vertices at random, connect it with the current point, and mark this line’s midpoint as the new point. If you repeat this process many times, what will the resulting shape look like? You might expect to find the points simply shattered everywhere, without much structure, but surprisingly, the points will in fact create a Sierpinski fractal.
The Fourier series approximation for the Step Function (or Square Wave) is 1/n * Sin(n*pi*x)/L for odd values of n. Here is the expansion, plotted up to n=49. Below are the individual waves and above is the progression of their added values!
Dividing by 7 yields this fascinating play of the numbers 1,4,2,8,5 and7. I am not going to spoil the fun by letting you on the pattern that emerges and other captivating properties that you might discover along the way.
Although feel free to write to us if you found anything that marveled you and we would definitely share it with the world.
Who knew division could be this much fun. am i right?
A 5-Cell, or Pentachoron, is a highly symmetrical four dimensional solid shape. Here you can see it projected into three (well, technically perspective-projected into two) dimensions, rotating about its “ZW” axis.
Here’s an alphabetical list of all available free books. Note that many of the links will bring you to an external page, usually with more info about the book and the download links. Also, the links are updated as frequently as possible, however some of them might be broken. Broken links are constantly being fixed. In case you want to report a broken link, or a link that violates copyrights, use the contact form.
A Beginner’s Guide to Mathematica
A Brief Introduction to Particle Physics
A First Course in General Relativity
A New Astronomy
A No-Nonsense Introduction to General Relativity
A Popular History of Astronomy During the Nineteenth Century, Fourth Edition
A Review of General Chemistry
A Simple Guide to Backyard Astronomy
A Text Book for High School Students Studying Physics
A Tour of Triangle Geometry
About Life: Concepts in Modern Biology
Advanced Mathematics for Engineers
Advanced Microwave Circuits and Systems
Advances in Computer Science and IT
Advances in Evolutionary Algorithms
Advances in Geoscience and Remote Sensing
Advances in Haptics
Advances in Human Computer Interaction
Age of Einstein
Aging by Design
AMPL: A Modeling Language for Mathematical Programming
An Introduction to Elementary Particles
An Introduction to Higher Mathematics
An Introduction to Many Worlds in Quantum Computation
An Introduction to Mathematical Reasoning
An Introduction to Mathematics
An Introduction to Proofs and the Mathematical Vernacular
An Introduction to Relativistic Quantum Mechanics
Analysis 1 (Tao T)
Analysis 2 (Tao T)
Astronomy for Amateurs
Astronomy with an Opera-Glass
Automation and Robotics
Basic Algebra, Topology and Differential Calculus
Basic Concepts of Mathematics
Basic Concepts of Thermodynamics
Basic Concepts of Thermodynamics Chapter 1
Basic Ideas in Chemistry
Basic Math: Quick Reference eBook
Basic Mathematics for Astronomy
Basic Positional Astronomy
Basic Principles of Classical and Statistical Thermodynamics