Miscellaneous: Bow and Arrow

 The following equation has a graph that looks like a bow and an arrow:

$y=\sin\left(y\right)\cos^{-1}\left(x\right)$

Desmos demonstration:

Background: I was trying to check if an integral from a physics problem seems to converge. And as I try to rotate the graph I get, this thing appears (by mistake, I guess)

My Imperfect Pitch

 I want to make a computer program where it will give random tones and I will guess the pitch, then it will collect the data and generates a graph that shows how inaccurate my hearing is... haha

On a side note, I will be using python (as an exercise).

This is fast! Result:

The number on the color bar is just color code

Reference: 
Making a sound on python
wave file
storing the data
graphing

Connect-$n$ Game on a \(k \times k\) Board

Definition: two players, the first to place n stones in consecutive line (horizontal, vertical, diagonal) wins. A real-life examples are tic tac toe (n=3, k=3) and Gomoku (n=5, k=18). My question is, for what n and k does the game have a must-win strategy? A compromised question is, does there exist a general algorithm to approach the optimal strategy?

Mandelbrot Set Visualization

I'm working on a Java project to create a graph visualization for the famous Mandelbrot set. The logic is fairly straightforward, but I have difficulty finding a way to determine if a series converges or not- without burning out the computer CPU. What's even trickier is to determine the speed of divergence/convergence and color it accordingly.

[3/8/2021] Update: I finish the programming for the graphic-making part, and now it's time to improve my algorithm! The below image uses a very primitive method but it's the first one and is still exciting. Basically, for a point on the complex plane, I calculate the difference between two consecutive terms for the first 5 terms and sum the difference. If the sum is negative (decreasing on average), the point is black; if positive, then the value of the sum is used as the RGB code numbers. If exceeding 255, the point will be white.

[Hours later]

Here it is! I didn't even understand what I was doing on the last one. For a series to converge, we only need that at infinity two consecutive terms will be infinitely close. Here I compare the absolute difference between the 15th and 16th terms, and the absolute difference is the value of the RBG code. Again exceeding 255 will be white.

[3/9/2021]

Absolute difference between 30th and 31st terms < 10 is black, else is white.

[final update]
In May I used the escape time algorithm found online; meanwhile I enable the "zooming in & out" function.

details 1

details 2

    

Proving that Algebraic Numbers Form a Field

as mentioned in this article,  I will prove that $Q[\alpha]$ is a field. The other axioms of fields are rather trivial here, so I will just do the following two.

Proof of a Relation Between Cross Product and Dot Product

This comes from an extra-credit question in my math class. 

Prove: \( \vec a \times (\vec b \times \vec c) = (\vec a * \vec c)\vec b - (\vec a * \vec b)\vec c \)

I can prove it alternatively by manipulating the components algebraically with determinant calculations, but such an approach doesn't make any intuitive sense, and after all, it's boring.

How I edit math equations

The Area Under a Parabola and the Volume of a Pyramid

This is a sketch of an idea I had in 2017. At the time, I didn't know the technical details of calculus but only had an understanding of its philosophy.