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As visually discussed here, a set of equations has recently been popping up as graffiti in Belgium. The equations define five functions of one variable, namely:

$f(x) = 2+ -(x-2)2 + 1 g(x) = 2- -(x-2)2 + 1 h(x) = 3x-3 i(x) = -3x+9 j(x) = 0,2x+1,7$

Plotting the five functions with $x\in \left[1,3\right]$ (the domain of existence of $f$ and $g$) gives the well-known anarchist logo

Anarchy

As mathematicians, we can take this a step further and define an Anarchist curve, by finding the implicit form of the plot of each of the function, and then bringing them together.

In this case, $f,g$ together define the circle, with equation

$(x-2)2+(y-2)2=1$

or rather (fully implicit):

$(x-2)2+(y-2)2-1=0.$

The three functions $h,i,j$ describe the ‘A’ shape. We first rewrite $j$ in a nice form as

$j(x)=x/5+17/10,$

and then write the implicit equation for each of them, multiplying the one for $j$ by $10$ to get rid of the fractions:

$h : y-3x+3 =0 i : y+3x-9 =0 j : 10y-2x -17 =0$

We can now multiply all the left hands together, obtaining:

$(y-3x+3)(y+3x-9)(10y-2x-17)=0$

which is the equation for the ‘A’.

If we then multiply this for the left-hand side of the implicit equation for the circle, we have the Anarchist curve

$((x-2)2+(y-2)2-1)(y-3x+3)(y+3x-9)(10y-2x-17)=0$

(which, for the record, is currently missing from the list of known curves in the Wolfram Alpha database.)

(There's a few more we could draw similarly, but that's for another time.)