Mathematics Lectures. Mathematics. Triangle Dissection Paradox. From Wolfram MathWorld. MacTutor History of Mathematics. George W. Hart.
Abstract Algebra - Free Harvard Courses - StumbleUpon. Weierstrass functions - StumbleUpon. Weierstrass functions are famous for being continuous everywhere, but differentiable "nowhere".
Here is an example of one: It is not hard to show that this series converges for all x. In fact, it is absolutely convergent. It is also an example of a fourier series, a very important and fun type of series. It can be shown that the function is continuous everywhere, yet is differentiable at no values of x. Here's a graph of the function. You can see it's pretty bumpy. Below is an animation, zooming into the graph at x=1. Wikipedia and MathWorld both have informative entries on Weierstrass functions. back to Dr. An Architecture for Combinator Graph Reduction Download free online e-book chm pdf. Imagining the Tenth Dimension - A Book by Rob Bryanton - StumbleUpon.
Mathematics. Virtual Reality Polyhedra. Powerful constraints on any structure that inhabits it.
Welcome to this collection of thousands of virtual reality polyhedra for you to explore. I hope you enjoy playing with them as much as I enjoyed making them. There are hundreds here which have never been illustrated in any previous publication. Polyhedra have an enormous aesthetic appeal and the subject is fun and easy to learn on one's own. One can appreciate the beauty of this image without knowing exactly what its name means --- the compound of the snub disicosidodecahedron and its dual hexagonal hexecontahedron --- but the more you know about polyhedra, the more beauty you will see. This site is a free self-contained easy-to-explore tutorial, reference work, and object library for people interested in polyhedra. I believe the best way to learn about polyhedra is to make your own paper models or other models.
Fibonacci in Nature - StumbleUpon. The Fibonacci numbers play a significant role in nature and in art and architecture.
We will first use the rectangle to lead us to some interesting applications in these areas. We will construct a set of rectangles using the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, and 34 which will lead us to a design found in nature. You will need a ruler, protractor, and compass. Start by drawing two, unit squares (0.5 cm is suggested) side by side. Next construct a 2-unit by 2-unit square on top of the two, unit squares. Your construction will look like this: Now, with your compass, starting in the unit squares, construct in each square an arc of a circle with a radius the size of the edge of each respective square (Your arcs will be quarter circles.).
This spiral construction closely approximates the spiral of a snail, nautilus, and other sea shells. We will next consider the use by architects and artists throughout history of the "Golden Ratio" and other geometric shapes based upon these ratios. Prime numbers. Version for printing Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.
The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers. A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28. By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved.
Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. In about 200 BC the Greek Eratosthenes devised an algorithm for calculating primes called the Sieve of Eratosthenes. There is then a long gap in the history of prime numbers during what is usually called the Dark Ages. Calculus Mega Cheat Sheet - StumbleUpon. K-MODDL & Tutorials & Reuleaux Triangle - StumbleUpon. If an enormously heavy object has to be moved from one spot to another, it may not be practical to move it on wheels.
Instead the object is placed on a flat platform that in turn rests on cylindrical rollers (Figure 1). As the platform is pushed forward, the rollers left behind are picked up and put down in front. An object moved this way over a flat horizontal surface does not bob up and down as it rolls along. The reason is that cylindrical rollers have a circular cross section, and a circle is closed curve "with constant width. " What does that mean? Is a circle the only curve with constant width? How to construct a Reuleaux triangle To construct a Reuleaux triangle begin with an equilateral triangle of side s, and then replace each side by a circular arc with the other two original sides as radii (Figure 4).
Foundations of Mathematics. Mathematics.