Set the Level slider to zero. Click on the green flag and the initiator is drawn. Set the Level slider to one, click on the green flag, and the generator is drawn. It's the shape of the generator that is applied, at a reduced scale, to every line segment at higher levels of the curve. Since the Peano curve passes through every point in a square, the curve has dimension of 2!
The Italian mathematician Giuseppe Peano (who is also renowned for Peano's Postulates of number theory) proved in 1890 that a geometric figure can be generated by a moving point can moving in such a way that in a finite time it will pass through all the points of a square—and yet no one would consider the entire area of a square as simply a curve. In this project, I scripted Peano's method (there are many other methods) for moving the point. Just in case, remember, a mathematical line has no width so don't think 'painting" a square with a brush will work. It won't. Your brush has zero width!
