Welcome to Project Iterra

This is a collection of programs that model different mathematical phenomena. To quickly find the right file, just look up the file name below and put it in the repo's 'Go to file' button on top.

News! 📰

• [16/09/22] Following the advice of industry experts, I will now be including unit tests and integration tests when and where applicable for all future code added to this repo. The first of this will be my simulator for random walks found in the .\RandomWalk folder :)
• [16/09/22] Just found out that as of 19 May 2022, GitHub now supports the rendering of $\LaTeX$ in markdown files! Prepare yourself for a whole new breed of Iterra documentation.

Estimating the shape of a bubble film for a wire loop

Still in development. Here are our messy, inaccurate but pretty preliminary results:

p-Norms in different Lebesgue space

See pNorms directory here for more details.

Brachistochrone Plots

The Brachistochrone curve solves a problem posed by Jean Bernoulli in 1696. The problem translates to:

"If two points A and B are given in a vertical plane, to assign a mobile particle Mthe path AMB along which, descending under its own weight, it passes from the point A to B in the briefest possible time."¹

The code in ClassicalMech\brachistochrone.py plots the brachistochrone curves from $A(0,0)$ to several different B locations.

Brownian Motion using the Weiner Process

The random, jittery movement of dust particles suspended in a fluid, first noticed by Robert Brown and explained by Albert Einstein, can be attributed to random collisions with air molecules in all directions. According to the Weiner Process, since most of the collisions tend to cancel eachother out, the net impulse on the dust particle can be modelled to be normally distributed about $0$.²

The dust particle starts at the red dot and ends at the cyan dot.

The code used to generate this path can be found in Brownian.py.

Legendre Polynomials

The first 5 solutions to Legendre's differential equation.

$$ (1-x^2) \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} - 2x\frac{\mathrm{d} y}{\mathrm{d}x} + n(n+1)y = 0 $$

$$ n =0, 1,2, 3, ... $$

The first plot renders the polynomial expansion of these solutions as a function of x, while the second plot renders them as functions of the trigonometric ( $\sin x$, $\cos x$ & $\tan x$ )) and hyperbolic ( $\sinh x$, $\cosh x$ & $\tanh x$ ) functions. The code is in LegendrePoly.py

Random Walk with partially reflecting boundaries

Our agent, 🤑, is on a street. It lives in a world where time is discrete. Each second it has a probability $p$ of moving right, and $1-p$ of moving left. If it reaches an end of a street, it will stay there until it is fortunate enough to take a step away from the boundary. ³

The Collatz Conjecture

Start with any positive integer, $n$. If the number is even, half it. If the number is odd, triple it and add one. The Collatz conjecture conjecture says the sequence will reach 1 for all $n$. Why this happens is still an open problem in mathematics.

The Collatz conjecture has been modelled in assembly language (Little Man Computer) - for a bit of a challenge - in Collatz\Collatz.asm.

Summation Formulae

These model summation formulae for constants and the first n natural numbers (along with their squares and cubes). The code can be found in SumFormulae.py

Farey's Algorithm

Not the complete algorithm for generating Farey sequences, but it kind of works - there are some extra terms in the sequence it spits out. The code can be found in Farey.py

Dependencies

All necessary dependencies have been included in requirements.txt. The file can be read to install the required dependencies into your machine or virtual environment using the following command on your command line interface (typically PowerShell or Command Prompts for Windows, Terminal for MacOS):

pip install -r requirements.txt

References

¹ L.N. Hand & J.D. Finch, Analytical Mechanics 1st ed. (1999)

² R.T. Durrett, Probability Theory and Examples 5th ed. (2019)

³ G.F. Lawler, Introduction to Stochastic Processes 2nd ed. (2006)