$\frac{\max(1,xy)}{y} = x$: Did I just create a fractal? [closed]

fractals maxima-minima desmos

https://www.desmos.com/calculator/6hfyqc6ks9

Did I just create a fractal? Again, the function is $\frac{\max(1,xy)}{y} = x$.

Here's what it looks like on Desmos

Here's what it looks like on WolframAlpha

Even uncannier version, from the alternative form with "sgn" that Wolfram provided: https://www.desmos.com/calculator/rikex4bvld


You have definitely created a strange fractal-like behavior, but not in the equation itself -- rather in Desmos's plotting software :)

The problem with this equation is that it is true not for a one-dimensional curve of points, but rather for the two-dimensional space of points $(x,y)$ where $xy \ge 1$. Algebraically: $$ \frac{\max(1, xy)}{y} = x \iff \max(1, xy) = xy \iff xy \ge 1. $$

The true plot of this is as follows:

enter image description here

Graphing software -- including Desmos -- is usually optimized for the case of plotting a curve, rather than a 2-dimensional space of points. It is likely that is the reason for the weird behavior.

FWIW, Desmos does warn us that something might be up with this plot, as shown by the following warning in your screenshot:

enter image description here


I have identified the regions as being delimited by powers of $2$ and graphs of $y=2^k/x$. The issue is with the implementation of their max function. If you change it to $\max(0,xy)/y=x$, you can extend the pattern all the way down to the origin: Desmos link enter image description here The clean lines make up the true "fractal". But I think this is the end of the line for me, because I have no idea why max would have an issue understanding that if $a\approx 2,b\approx 2$ then clearly, $\max(ab,0)=ab$.

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