  A systematic procedure to numerically compute a horseshoe map is presented.
This new method uses piecewise functions and expresses the required operations
by means of elementary transformations, such as translations, scalings,
projections and rotations. By repeatedly combining such transformations,
arbitrarily complex folding structures can be created. We show the potential of
these horseshoe piecewise maps to illustrate several central concepts of
nonlinear dynamical systems, as for example the Wada property.
