• If $p$ is a prime number, then $\varphi(2p) = p-1$. This may account for one of the high-density lines. The implication though is that the density gets sparser as $n$ goes to infinity. – steven gregory Mar 1 '16 at 4:20
As Steven Gregory says, the second line is numbers where $\varphi(n)$ is very close to $n/2$, and this is made up of numbers of the form $2^ap$ where $p$ is prime and $a\geq 1$. The third line is numbers where $\varphi\approx 2n/3$, made up of numbers of the form $3^ap$ where $a\geq1$. There is a rough fourth line of slope $1/3$, which includes numbers of the form $2^a3^bp$, and a fainter one of slope $4/5$ corresponding to $5^ap$. Obviously as the numbers you have to multiply the primes by to reach a particular line get larger, points on that line get sparser.