Self-intersection of vector valued function A vector valued function $r(t)$ has the following coordinates:
$$x = 4\cos\left(\frac12t\right)+2\cos(2t)+\cos(4t)\\
y = 4\sin\left(\frac12t\right)+2\sin(2t)+\sin(4t)$$
I have to find the $t$-values where the curve crosses/intersects itself.
Using Maple I've not been able to find any solutions. I've tried a lot of combinations of setting the $x$- and $y$-coordinates equal each other, with no luck.
 A: Self-intersection happens when two different values of $t$ produce the same coordinates $[x,y]$. Finding self-intersections amounts to solving the system of two equations with two unknowns
$$x(t)=x(s), \quad y(t)=y(s) \tag1$$ 
where $t$ and $s$ are two values of the parameter, which we want to be distinct ($t\ne s$, more precisely $t-s$ should not be a multiple of $4\pi$.) Since you mention Maple, I understand that a numeric solution is desirable; which is very good because I don't have hopes for an algebraic solution. Thus, fsolve is in order. Since fsolve returns only one solution, and we don't want it to give us $t=s$, a little change of notation is called for. 
Write $t+h$ instead of $s$ above, and specify an interval for $h$ that does not include $0$ or any other multiples of $4\pi$.  By changing this interval, I was able to find all self-intersections, marked below (except I did not mark mirror images):  

Here is the Maple code.
eqns:=[4*cos(t/2)+2*cos(2*t)+cos(4*t) = 4*cos((t+h)/2)+2*cos(2*(t+h))+cos(4*(t+h)), 4*sin(t/2)+2*sin(2*t)+sin(4*t) = 4*sin((t+h)/2)+2*sin(2*(t+h))+sin(4*(t+h))];
fsolve(eqns, {t=0..4*Pi, h=0.1..1});

Output:  $h=0.3625428535$,  $t=1.124653409$. This corresponds to the tiniest loop in the picture, because $h$ is small. 
fsolve(eqns, {t=0..4*Pi, h=1..2});

Output: $h = 1.725224282$, $t = 5.420573166$. This is the intersection on the left. 
fsolve(eqns, {t=0..4*Pi, h=2..3});

Output: $h = 2.383110138$, $t = 0.7155905393$. This comes from the loop that contains another loop.
A: You don't want to set the $x$ and $y$ coordinates equal to each other. For example, the system
$
\begin{cases}
x(t) = t\\
y(t) = t
\end{cases}
$
has $x=y$ everywhere but it never crosses itself.
Each of $x$ and $y$ in your system has period of $4\pi$, so it will cross itself.
