Arithmetic-analytic representation of Peano curve (Q2292319)

Summary: In this work, we obtained a nonmatrix analytic expression for the generator of the Peano curve. Applying the iteration method of fractal, we established a simple arithmetic-analytic representation of the Peano curve as a function of ternary numbers. We proved that the curve passes each point in a unit square and that the coordinate functions satisfy a Hölder inequality with index \(\alpha =1/2\), which implies that the curve is everywhere continuous and nowhere differentiable.