# The Buffon’s Needle Problem

Attached is a short write-up on the very interesting geometric probability problem commonly referred to as the Buffon’s Needle problem.  The solution gives a general outline for a Monte Carlo method of approximating pi. neato!\

An R implementation of the monte carlo simulation is:

```#Enter the desired number of trials as n

Buffon_Needle<-function(n){
L=2;D=L;

thetas<-c(rep(0,n))
for(i in 1:n){thetas[i]=runif(1,0,pi/2)}

X<-c(rep(0,n))
for(i in 1:n){X[i]=runif(1,0,L/2)	}

COS<-c(rep(0,n)); COS=cos(thetas)
Compare<-c(rep(0,n))
Compare=(X/(2/L))
Crossover<-c(rep(0,n))
for(i in 1:n){
if(COS[i]>=Compare[i]){Crossover[i]=1}
if(COS[i]<Compare[i]){Crossover[i]=0}
}
Total=sum(Crossover)
Pi_Est=(2*n)/Total

cat("For ",n," trials the estimate of pi is: ",Pi_Est)
}
########################################

Buffon_Needle(100)

```

A SAS implementation of the monte carlo simulation is:

```%macro Buffon_Needle(n=);
%let D=2; %let L=2;

data BN;
do i=1 to &n;
angle=rand("Uniform",0,3.14159/2);
X=rand("Uniform",0,&D/2);
COS=cos(angle);
L=&L;
output;
end;

data BN; set BN;
if(COS >=(X/(2/L))) then Cross = 1;
else Cross = 0;

proc sql;
create table Results as
select (sum(t1.Cross)) as Total_Crosses
from work.BN t1;
quit;

data Results; set Results;
n=&n;
pi_est = (2*&n)/Total_Crosses;

run;

%mend;

%Buffon_Needle(n=1000000);

```

# Scale of the UNIVERSE!!

Linked is an extremely well done little application that takes viewers on a journey from the micro- micro- micro- micro- micro- scopic to the MACRO- MACRO- MACRO- MACRO- MACRO- SCOPIC… and everywhere in between.

# Entire Functions, Cauchy’s Integral Formula, & Polynomials

A beautiful result (here) which reveals an interesting connection between entire functions, Cauchy’s integral formula and polynomials.

# The Fundamental Theorem of Finite Abelian Groups

Attached is a proof of the fundamental theorem of finite Abelian groups, which says that every finite Abelian group can be written as a direct sum of cyclic subgroups of prime power order.

While the theorem is VERY beautiful, the proof is… well, accurate, but like many of the proofs from abstract algebra it is somewhat less than illuminating.

# The Riemann Hypothesis Video (Clay Institute)

The Riemann Hypothesis says that all of the non-trivial zeros of the analytic continuation of the Riemann zeta function lie on the critical line z=1/2+iy.

This (only slightly technical) video gives a nice description of the conjecture as well as some of the important consequences with regards especially to the distribution of the prime numbers.

Video (here).

# Contour Integrals Are Amazing!

Here is a recently calculated contour integral which I find pretty interesting.