The Weierstrass function is known as one of these so-called pathological mathematical objects, continuous everywhere, while nowhere differentiable. In the sequel, we have chosen, first, to concentrate on the unconventional history of this function, a function breaking with the mathematical canons of classical analysis of the XIX th century. We recall that it then took nearly a century for new mathematical properties of this function to be brought to light. It has since been the object of a renewed interest, mainly as regards the box-dimension of the related curve. We place ourselves in this vein, and, thanks to our result of 2018, which shows that this value can be obtained in a simple way, without calling for theoretical background in dynamic systems theory, we put forward the link between the non-differentiability and the value of the box-dimension of the curve.