Structures in the plot of the “squareness” of numbers (This is based on an earlier MSE posting,
"Structures in the plot of the “squareness” of numbers.")

My main question is to explain the structural features of this plot:



This is a plot of what I call the squareness ratio $r(n)$ of a natural number $n$ (or simply the "squareness").
The squareness $r(n)$ is the largest ratio $\le 1$ that can be obtained
by partitioning the factors of $n$ into two parts and forming the ratio of their products.
A perfect square has squareness $1$.
A prime $p$ has squareness $1/p$.
In a sense, the squareness measures how close is $n$ to a perfect square.

The squareness ratios for the first ten number $n=1,\ldots,10$ are
$$1,\frac{1}{2},\frac{1}{3}
   ,1,\frac{1}{5},\frac{2}{3},\frac{1}{7},\frac{1}{2},1,\frac
   {2}{5} \;.$$
A more substantive example is $n=12600=2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7$,
for which the largest ratio is 
$$\frac{3 \cdot 5 \cdot 7}{2^3 \cdot 3 \cdot 5}=\frac{7}{8}=0.875 \;.$$
One more example: $n = 2^2 \cdot 3^2  \cdot 5^2  \cdot 7  \cdot 11 = 69300$,
$$r(n) = \frac{2^2 \cdot  3^2 \cdot  7}{5^2 \cdot  11}=\frac{252}{275} \approx 0.92 \;.$$
The only feature of this plot—the rays through the origin—that is
evident to me
is explained by the fact that, for a prime $p$,
for $n = k p$ and $k=1,\ldots,p$, the squareness ratio is $k/p$, and so those
ratios lie on a line through the origin of slope $\frac{1}{p^2}$.
MSE user PattuX emphasized that similar rays occur for particular composite $n$.
Several other features could use explanation:
(1) The discernable density change at $r=\frac{1}{2}$.
(2) The (apparent) hyperbolas.
(3) The interesting structure near $r=1$, both negative (hole-)curves and positive (dot-)curves:




Detail: $35 K \le n \le 60K$ (approximately), near $r=1$.


I welcome explanations for (1), (2), (3), and other apparent features of the plot.
This is to satisfy curiosity; it is far from my expertise.


Added(1): As per Gerhard Paseman's request, the plot with only odd $n$ ratios:



       

Squareness ratio $r(n)$ for odd $n$ only; even $n$ not plotted.


Added(2): The landscape is rather different for larger $n$
(in accordance with Lucia's insights):



       

Squareness ratio $r(n)$ for $900{,}000 \le n \le 1{,}000{,}000$.


 A: You're asking about the distribution of $d^2/n$ where $d$ is the 
largest divisor of $n$ below $\sqrt{n}$.  This is closely related to work on the multiplication table problem, from which it follows that the square-ness ratio is usually close to 0. So the observed patterns are eventually insignificant (at least to first order, they may be significant in lower order terms) and eventually the plot will just be really concentrated at the bottom.  
To expand a bit, Kevin Ford has shown (following earlier work of Erdos and Tenenbaum) that the number of integers up to size $x$ with a divisor in $y$ to $2y$ is 
$$ 
H(x;y, 2y) \asymp \frac{x}{(\log y)^{\delta} (\log \log y)^{3/2}} 
$$ 
with 
$$
\delta = 1 -\frac{1+\log \log 2}{\log 2} = 0.08607\ldots .
$$ 
Using this with $y= \sqrt{x}/2$, $\sqrt{x}/4$, $\ldots$, $\sqrt{x}/2^k$, we see that only $\ll k x /((\log x)^{\delta}(\log \log x)^{3/2})$ integers below $x$ have a divisor in $(\sqrt{x}/2^k, \sqrt{x})$.  Taking $k= 2\log \log x$ (for example), it follows readily that $\ll x/(\log x)^{\delta}$ numbers below $x$ have a square-ness ratio larger than $1/\log x$.  
A: At least some part of the features may be explained by plotting $\frac1{r(n)}$, it looks like this:

It is more or less clear that the slopes are $\frac1{k^2}$, $k=1,2,3,...$
(So the original plot is a superimposition of the corresponding hyperbolas)
A: Just an additional comment: because it was discussed, whether the structures shall be visible when n increases, I've thought, it would possibly be interesting to rescale the axes. One additional plot, the original values recomputed, but $n$- and $r(n$)-axes logarithmically scaled for display gives this image:     
 
$\qquad \qquad $ ($ \small n  \to \log_{10}(n) $ ,  $  \small r(n) \to \log_{10}(r(n)) $  where $ \small n=1 \ldots 100 000$, $ \small 0 \lt r(n) \le 1$)
