MY JOURNAL
Where I Write About Anything And Everything
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A while back I wrote a blog entry comparing various copyleft and permissive licenses and why I generally preferred the latter. In that article I explained how I felt copyleft could be useful, but felt that it was too far reaching. Since that time, I have done a lot of thinking, and I have come to a change of heart. Thus I am relicensing all of my content, as well as any future content I may create. I know this may inconvenience some people (all the better that I change earlier rather than later) but I truly feel that this is the best option for my work moving forward.
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So it's been a while since I last posted anything to this blog, and I really want to change that. I recently created a series of mathematical animations, all based on the same function. But I wanted to do more than just share these animations, I wanted to explain how I made them and the interesting features I discovered in the process. I thought about making a video for this purpose, as there are plenty of educational channels on YouTube nowadays. But I’m not that great at public speaking, and I did not want my presentation to detract from the animations themselves. Thus I decided to write this blog post.
In this blog entry I attempt to explain how you can build a polynomial up from its roots. That is the points where the function equals zero. I then explain how this same technique can be used to build rational functions, and what this means in the complex plane. This is something I use to generate a lot of the abstract mathematical art that I share on Tumblr, Flickr, and Deviant Art. So if you are interested in duplicating my art style, or just reading the rest of my blog, this entry is essential.
Root finding methods are an important tool to have in any mathematics library. For those who don't know, root finding methods are, by definition, methods that find the zeros (or roots) of some arbitrary function. That is, given a function and some initial parameters, theses methods will find a solution that when plugged back into the function yields zero.
This might sound like a rather specific case, but this problem of finding zeros can be generalized to more exotic problems. For example, computing the inverse of a given function, or finding the intersection point between two curves, can both be obtained using root finding. Furthermore, despite their practical applications, root finding methods can generate beautiful fractals when applied to the complex plane. I have always been a proponent of the free exchange of ideas and information, supporting organizations like Creative Commons and the Open Source Initiative whenever possible. However, what most people don't know is that there are two distinct types of free content on the internet. The distinction between these two types is a crucial and ongoing argument over what it means exactly for content to be 'free'.
One thing that my math teachers always said: "You can't take the logarithm of a negative number". Well it turns out that you can, but the answer is complex, in every sense of the word. I’m going to assume you already know a bit about complex numbers and logarithms.
In order to take the log of a complex number, we must re-write the number in it's polar form. However this form is not unique. In particular, the parameter 'n' can be any integer value, and we still get a valid result. Furthermore, the result of the log function is dependent on which value of 'n' we choose. This means that the log function is multivalued. In fact an infinite number of complex values could satisfy the log function. Have you ever wondered how long the tracks are in Super Mario Kart, or how fast Bubsy Bobcat is running compared to Sonic, or how far it is to walk between Onett and Forside in Earthbound? Well, maybe not, but if you have I have the perfect program to answer your questions!
Making my own website, is something I have dreamed about for quite some time. Even as a kid, I thought it would be nice to have a place to showcase all my various ROM hacks and modules for games that had level editors. Now that I am making my own games from scratch, and trying to get more involved with the furry and artistic communities, I felt like the time was right to start making that website.
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Jake DawsonHello there. Welcome to my personal blog. I hope you enjoy your stay! Categories
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