parametric equation

3

The image is not mine. It is a fantastic creation by bigblueboo that has caught some attention outside of the usual math tumblverse. You should definitely check out eir blog and if you like this post you should (also?) reblog the original. With that out of the way:

Modeling. Mathematical modeling is the art of translating real systems, often physical or economic, into the language of mathematics in an attempt to predict future behavior of the system. Doing this often requires making many simplifying assumptions which are “unwarranted” from a purely logical perspective, but make the problems tractable. Therefore mathematical modeling is a distinct (but related) skill from the modern conception of the practice of mathematics.

A “parametric equation” is difficult to define exactly, but it is often (as it is here) a method for producing general surfaces or curves that cannot be described by functions because they do not pass the vertical line test. More specifically for this example, it is a function from a line segment into a higher-dimensional space.

This object has inspired me to get off my lazy butt and start producing content for the blog again. It also happens to be an excellent source of mathematical content: I’m probably going to be doing a daily series of posts about it for a while. I know I have content for at least three days and probably a few more besides.

It is vaguely related to epicycles, but the name “generalized epicycle” is my own invention. If you know an actually recognized name, I would love to know about it!

(EDIT: There is an old version of this post in which the constant terms were omitted and there were some flopped sin/cos symbols. I’m sorry! These haven’t been edited by someone else unlike many of the proofs I post so they’re bound to be a little rough around the edges every so often.)

youtube

Remeber that video the other day? Here it is in 1080p, zoomed out enough that you can see the full structure of the mandeldrop. Watch it in fullscreen with the quality set to 1080p; or something, because youtube’s compression made it terrible, otherwise.

Explanation: It’s still me turning the mandelbrot set inside out. I do this by parametrizing the equation so blah blah blah.

Use this math I worked out to do it yourself if you want.

z^2 + (n/f) * c + (1-(n/f)) * t

where n is the number of the frame being rendered, and f is the total number of frames to be rendered, and t is some arbitrary transformation of c, in this case, c^-1

I’ll write a post for non-mathematicians at some point. One that doesn’t use the sort of impenetrable jargon that idiots who need to feel smart love. I’m about halfway done with it, actually.

Oh also, the reason why the quaternion mandelbrot set looks like poorly-lathed cylindrical shite is because quaternion algebra doesn’t prune preperiodic points from rotation. Instead it make them continuous across 360 degrees. I just figured this out the other day, and consequently respect my fellow countryman’s invention more. They’re almost correct.

arxiv.org
[1601.07230] Extending the Coyote emulator to dark energy models with standard $w_0$-$w_a$ parametrization of the equation of state

[ Authors ]
L. Casarini, S. A. Bonometto, E. Tessarotto, P.-S. Corasaniti
[ Abstract ]
We discuss an extension of the Coyote emulator to predict non-linear matter power spectra of dark energy (DE) models with a scale factor dependent equation of state of the form w = w_0 + ( 1 - a )w_a . The extension is based on the mapping rule between non-linear spectra of DE models with constant equation of state and those with time varying one originally introduced in ref. [25]. Using a series of N-body simulations we show that the spectral equivalence is accurate to sub-percent level across the same range of modes and redshift covered by the Coyote suite. Thus, the extended emulator provides a very efficient and accurate tool to predict non-linear power spectra for DE models with w_0 - w_a parametrization. According to the same criteria we have developed a numerical code that we have implemented in a dedicated module for the CAMB code, that can be used in combination with the Coyote Emulator in likelihood analyses of non-linear matter power spectrum measurements. All codes can be found at https://github.com/luciano-casarini/PKequal.