The Beauty of Roots. I feel like talking about some pure math, just for fun on a Sunday afternooon. Back in 2006, Dan Christensen did something rather simple and got a surprisingly complex and interesting result. He took a whole bunch of polynomials with integer coefficients and drew their roots as points on the complex plane. The patterns were astounding! Then Sam Derbyshire joined in the game. He then plotted all the roots using some Java programs, and created this amazing image: You really need to click on it and see a bigger version, to understand how nice it is.
Here’s a closeup of the hole at 1: Note the line along the real axis! Next, here’s the hole at And here’s the hole at Note how the density of roots increases as we get closer to this point, but then suddenly drops off right next to it. But the feathery structures as we move inside the unit circle are even more beautiful! They have a very different character near the point But I think my favorite is the region near the point and in the plane with and/or.
Freebook. Mathematics. Logic. Linearalgebra. Delaunay. Www.iki.fi/sol - Tutorials - Interpolation Tricks. Contents 1. Why 0..1 Range While making demos I've found different interpolation tricks to be extremely valuable. Adding little smoothness to all kinds of movement, be it actual movement of the camera, some object, fading of lights, fading in and out etc, makes things much more enjoyable to watch.
Sharp movements and changes are jarring and should be avoided. (*1) Generally speaking, when making some kind of animation, we know the starting and ending positions, and want to transition between these. Values between 0 and 1 have some rather interesting properties, including the fact that you can multiply any value between 0 and 1 with another value between 0 and 1, and the result is guaranteed to be between 0 and 1. (*2) These properties can be used to tweak the way we move from 0 to 1 in various ways. 2.
Let's say we want to move the variable X between points A and B in N steps. For (i = 0; i < N; i++) { X = ((A * i) + (B * (N - i))) / N; } Or, put another way, this becomes: 3. 4. 5. 6. 7. 8. Permutations. A set V consists of n elements if its elements can be counted 1, 2,..., n. In other words, the set V can be brought into a 1-1 correspondence with the set Often it's more convenient to start counting from 0. Then we get the set Definition A permutation is a 1-1 correspondence of a set V onto itself: Being able to count elements in the set V means the set can be written as However, a set may be counted in many different ways.
For example, a set of two elements can be counted in exactly two ways. In how many ways may one count a set of n elements? The number of permutations of a set of n elements is denoted n! Thus n! How many ways are there to count an empty set, the set with 0 elements? An aside There is just one way to do nothing so that However, the result of this activity is nothing or, in math parlance, 0.
I placed the answer to the question at the bottom of this page. What's 4!? Here is another way to do this. Theorem For all integer n > 0, n! J. Answer to the problem 720! College Algebra Tutorial on The Fundamental Counting Principle. College Algebra Tutorial on Permutations. LectureNotes < Main < TWiki.