ZLover4Life

No, this isn't the crazy in-depth stuff that nobody's going to understand... there'd be no point to posting something like that. Instead, these are some images from my exploration of fractals (specifically, fractals generated by Newton's Method or iterations). Fractals are infinitely repeating patterns, and can be found in nature in snowflakes, crystals, broccoli, and other things. I found these to be interesting, so I thought some of you might feel the same.

If you have any understanding of math, these are basically generated by discretizing (breaking up into pieces) part of the complex plane (600 points across and down for each section means 360000 points) and evaluating every single point to either see which root the iteration process goes to (via Newton's Method), or see what happens if you iterate the function over and over again with the new point.

First image is using my own program, the rest are using an in-development program called "Polynomiography" created by some of my professors. Polynomiography is much faster than my program, but I prefer my color control over that of Polynomiography (Polynomiography selects new colors for each root or you can choose a color code, whereas I can either tell it to shift between two colors or I can define each color individually). If you'd like to see any of these in a different color or zoomed in on a certain area, just let me know and I can make it happen. I'll probably be playing around with it all night as I have no other pressing work to do, so if I encounter other interesting ones, I'll post them up.

cos(x) iterated in black and white


cos(x) iterated:


sin(x) iterated:


zoomed in on part of sin(x) with red added:


zoomed in on upper left arm of previous fractal:


sin(x) root graph using iterations of Newton's Method:


colors shifted:


iterations of x^2-1


zoomed in on top piece


(x^5-2)(x^4+1):


I typically have no interest in the complex plane (other than its applications to Quantum Physics), but it is pretty damn cool to see what happens when we start playing around in it.

ZLover4Life

stupid 10 pic limit

SpeciallySpiked

ah, very cool. In one of my classes we're playing with Fourier Series to get sin and cos to fit different types of frequencies. Not the same but just saying I like these pictures

ZLover4Life

What kind of class is it (title)? Sounds like a Physics course... if someone had told me how applicable the trig functions were (obviously sound frequencies, but also AC electricity and just about anything that has waves), I might have been more interested in it. Applied statistics is my concentration, but I dabble in lots of areas of mathematics.

SpeciallySpiked

The class is Building Energy Lab. A lot of the math work is in the data analysis with different systems in buildings. So you're dead on with AC Electricity and of course other types of frequencies. Also some of the different types of instrumentation we do require following voltage patterns and predicting some behaviors. Woo~!

KasbeKZ

much different than the sin graphs i'm used to seeing lol. good stuff. thanks for posting it.