Make generative art that uses ideas from gradient descent. (e.g. a spline where mean curvature has certain properties.)

Gradient descent is an optimization algorithm used to find local extrema, typically in highly dimensional data, by taking steps in the direction of the gradient. This algorithm can be used to create generative art in several ways. For instance, one could take some functions and display the path of various starting points, similar to solution curves in differential equations. The user could also play around with color, line thickness, and other effects. Another way to use this algorithm is to cast a “shadow” onto a two-dimensional plane, where two of the coordinates would represent pixel position and the others could modify the other previously described parameters.

Keywords: Gradient descent, calculus, differential equations, generative art

https://madeincalifornia.blogspot.com/2012/11/gradient-descent-algorithm.html

Make a pop-up book

Creating a pop-up book can be a creative application of geometry, topology, modeling, and more. For example, one can create a pop-up book using the principles of tessellations, tiling with no gaps or overlaps, to show a repeating pattern unfolding as the page turns. One can attempt to mimic 4D shapes such as hypercubes with complex page folding. Each page can be a new mathematical concept that relies on visualization to get the best understanding like fractals, knots, the torus, and more.

The process for creating this book would first start with planning; we will need to choose all the mathematical concepts we want to represent and map out what page goes where. Next, we will have to study each object/shape we are representing in accordance with paper engineering and pop-ups; this means calculating the folds, cuts, and angles needed for a reliable and accurate pop-up. We will make prototypes which will later turn into final products. 

Keywords: Geometry, Pop-up, Paper Engineering, Mathematical Modeling.

16th Century Math Pop-up Book

Draw a Fractile in PIL

The fractile I will be focusing on is the mandelbrot set fractile. According to Wikipedia, The Mandelbrot set^] is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. The set is defined in the complex plane as the complex numbers c for which the function f_c(z) = z² + c does not diverge to infinity when iterated starting at z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. We assume that the sequence Z_n is not bounded if the modulus of one of its terms is greater than 2.  the mandelbrot set. MIT’s Dr. Kreiger has a YouTube video that elaborates on the Mandelbrot set. https://www.youtube.com/watch?v=NGMRB4O922I 

As a beginner programmer, I think drawing the mandelbrot set makes a rewarding exercise that provides practice with loops and conditionals, and perhaps OOP. Specifically, for this project, I will get more practice with the PIL library in Python. 

The link https://realpython.com/mandelbrot-set-python/ shows the process I intend to follow but with plain Python code(in my case, the PIL library). I will be using the recursive function shown:

I will be using PIL for image processing https://pillow.readthedocs.io/en/stable/  I have no experience so I will be learning the documentation alongside. 

Keywords: the mandelbrot set, modulus, fractile, pillow, infinity, OOP.

A fractile.