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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[1,3] (the domain of existence of f and g) gives the well-known anarchist logo

Anarchy

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.)