In this project, you will approximate Pi using several methods described in Chapter $2$ of the

textbook.

You must submit your work by $11$:$59$ pm on the due date $$ there will be no extensions.

By completing this project, you will gain

$$ practice implementing three algorithms for approximating $($Archimedes$,$ Wallis, and

Monte Carlo$)$

$$ practice following API requirements $($function signatures$)$

$$ experience using turtle graphics to visualize the Monte Carlo solution method

$$ practice importing modules and functions from Python Standard Library $($math$,$ random$)$

and your code $($different parts of this project$)$

$$ some knowledge of deterministic vs$.$ non$-$deterministic algorithms

$$ practice Python functions with different return types, side effects, bool, selection

statements, loops, and strings

When writing Python code, follow the PEP$8.$ Use whitespace between operators and

operands, use descriptive variable names, and add appropriate in$-$code comments $($why$,$ not

what$).$ Do diligent parameter checking, e$.$g$.,$ do not assume that your function will receive a

positive int argument because you expect a positive int.

How to succeed:

$$ Start early

$$ Go to office hours

$$ Ask questions on Discord or email.

All problems in this project are based on PPC Ch$.2.$ Make sure

$$ You may have other helper methods called from the required functions

$$ Include docstring documentation for all the required functions

$2.$ Instructions

$2\mathrm{.}1.[30$ pts$]$ The Archimedes Method

Requirements: Implement the Archimedes method $($Sec$.2\mathrm{.}4$ in PPC$)$ using the following

function signature. In your function, return the computed approximation of $($do not change

the name or type of argument and return value$).$ Feel free to use the book version or come up

with your own.

def pi$\_$arch$($num$\_$sides: int$)->$ float

Function signatures are informative; they contribute to documenting your code. Signatures

describe the input parameters' data types and the return value. Signatures do not perform any

data type validation.

What to submit: Upload arch.py on Coding Rooms.

$2\mathrm{.}2.[30$ pts$]$ The Wallis Method

Requirements: Implement the Wallis $$ approximation method $($Sec$.2\mathrm{.}5\mathrm{.}3$ in PPC$)$ using the

following function signature. In your function, return the computed approximation of $($do not

change the name or type of argument and return value$).$ Feel free to use the book version or

come up with your own.

def pi$\_$wallis$($num$\_$pairs: int$)->$ float

What to submit: Upload wallis.py on Coding Rooms.

$2\mathrm{.}3.[30$ pts$]$ Monte Carlo Simulation

Requirements: Implement the Monte Carlo approximation of $($Sec $2\mathrm{.}6$ in PPC$)$ using the

following function signature. In your function, return the computed approximation of $($do not

change the name or type of argument and return value$).$ Feel free to use the book version or

come up with your own.

Since you use pseudo$-$random numbers, you may get different answers each time you call the

function, so initialize random with a fixed seed $($see example in Part $5).$ Use $42$ as the random

seed in all experiments so all our results match $($read https:$//$tinyurl$.$com$/$UQLUE$42).$

def pi$\_$mc$($num$\_$darts: int$)->$ float

What to submit: Upload mc$.$py on Coding Rooms.

$2\mathrm{.}4.[60$ pts$]$ Putting All Together

While the textbook provides solutions and simple implementations to the previous problems, in

this part, you get to design a method for evaluating the different approaches implemented in

Parts $1-3.$

Your task is to implement a function that computes and prints the required input parameters

for each algorithm, given an error tolerance err$\_$tol $($a floating$-$point value between $0$ and $1).$

E$.$g$.,$ if err$\_$tol is $0\mathrm{.}001,$ how many sides does the Archimedes algorithm need to produce an

error of $0\mathrm{.}001$ or smaller?

Recall that an accurate value of $$ is available through the math.pi constant, but remember that

the smaller the value for err$\_$tol, the longer each method will take. Feel free to implement

any helper functions you need.

Requirements: Function signature $$ input one float number and return a list of three values.

Ensure you import the Python modules you created for Parts $1,2,$ and $3,$ e$.$g$.,$ import arch.

def all$\_$pi$($err$\_$tol: float$)->$ list:

Example. These are the results as printed on a proposed solution to this problem.

$>>>$ print$($all$\_$pi$(0\mathrm{.}1))$

Archimedes: num$\_$sides $=8($Differs by $0\mathrm{.}08012519466907486)$

Wallis: num$\_$pairs $=8($Differs by $0\mathrm{.}0910026575342835)$

Monte Carlo: num$\_$darts $=5($Differs by $0\mathrm{.}05840734641020706)$

$[8,8,5]$

When you run your code repeatedly with the same err$\_$tol input, you get the same values for

Archimedes and Wallis $($i$.$e$.,$ those functions are deterministic$).$ Still, you may obtain different

answers for Monte Carlo $($a non$-$deterministic function$).$ The Monte Carlo produces different

results because it uses pseudo$-$random numbers for the darts' positions and gets different sets of

points with every run.

To ensure you get the same result every time you run, you can initialize the pseudo$-$random

number generator with a fixed seed first $($use $42$ as your seed$)$ before calling your mc$.$pi$\_$mc

function, e$.$g$.,$