task try 2

module2/exo1/toy_document_orgmode_python_en.org

 #+TITLE:  On the computation of pi
-#+AUTHOR: Anton Y.
-# #+DATE:   2025-07-07
 #+LANGUAGE: en
-# #+PROPERTY: header-args :eval never-export
 
 #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="http://www.pirilampo.org/styles/readtheorg/css/htmlize.css"/>
-# #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="http://www.pirilampo.org/styles/readtheorg/css/readtheorg.css"/>
-# #+HTML_HEAD: <script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.3/jquery.min.js"></script>
-# #+HTML_HEAD: <script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/js/bootstrap.min.js"></script>
-# #+HTML_HEAD: <script type="text/javascript" src="http://www.pirilampo.org/styles/lib/js/jquery.stickytableheaders.js"></script>
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="http://www.pirilampo.org/styles/readtheorg/css/readtheorg.css"/>
+#+HTML_HEAD: <script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.3/jquery.min.js"></script>
+#+HTML_HEAD: <script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/js/bootstrap.min.js"></script>
+#+HTML_HEAD: <script type="text/javascript" src="http://www.pirilampo.org/styles/lib/js/jquery.stickytableheaders.js"></script>
 #+HTML_HEAD: <script type="text/javascript" src="http://www.pirilampo.org/styles/readtheorg/js/readtheorg.js"></script>
 
-* Asking the math library
+#+PROPERTY: header-args  :session  :exports both
 
-My computer tells me that π is approximatively
+* Asking the math library
+My computer tells me that $\pi$ is /approximatively/
 
-#+begin_src python :results output :session :exports both
+#+begin_src python :results value :session *python* :exports both
 from math import *
 pi
 #+end_src
 
 #+RESULTS:
+: 3.141592653589793
 
 * * Buffon's needle
+Applying the method of [[https://en.wikipedia.org/wiki/Buffon%2527s_needle_problem][Buffon's needle]], we get the *approximation*
 
-Applying the method of Buffon's needle, we get the approximation
-
-#+begin_src python :results output :session :exports both
+#+begin_src python :results value :session *python* :exports both
 import numpy as np
 np.random.seed(seed=42)
 N = 10000
@@ -36,15 +34,15 @@ theta = np.random.uniform(size=N, low=0, high=pi/2)
 #+end_src
 
 #+RESULTS:
+: 3.128911138923655
 
 * Using a surface fraction argument
+A method that is easier to understand and does not make use of the
+$\sin$ function is based on the fact that if $X\sim U(0,1)$ and $Y\sim
+U(0,1)$, then $P[X^2+Y^2\leq 1] = \pi/4$ (see [[https://en.wikipedia.org/wiki/Monte_Carlo_method]["Monte Carlo method" on
+Wikipedia]]). The following code uses this approach: 
 
-A method that is easier to understand and does not make use of the sin
-function is based on the fact that if X∼U(0,1) and Y∼U(0,1), then
-P[X2+Y2≤1]=π/4 (see "Monte Carlo method" on Wikipedia). The following
-code uses this approach:
-
-#+begin_src python :results file :session :var matplot_lib_filename=(org-babel-temp-file "figure" ".png") :exports both
+#+begin_src python :results output file :var matplot_lib_filename="figure_pi_mc2.png" :exports both :session *python* 
 import matplotlib.pyplot as plt
 
 np.random.seed(seed=42)
@@ -65,13 +63,13 @@ print(matplot_lib_filename)
 #+end_src
 
 #+RESULTS:
-[[file:None]]
+[[file:figure_pi_mc2.png]]
 
-It is then straightforward to obtain a (not really good) approximation to π
- by counting how many times, on average, X2+Y2
- is smaller than 1:
+It is then straightforward to obtain a (not really good) approximation
+to $\pi$ by counting how many times, on average, $X^2 + Y^2$ is smaller
+than 1: 
 
-#+begin_src python :results output :session :exports both
+#+begin_src python :results output :session *python* :exports both
 4*np.mean(accept)
 #+end_src