Un exemple de non dérivabilité en géométrie du triangle

Un exemple de non-dérivabilité en géométrie du triangle Jacques Dixmier, Jean-Pierre Kahane et Jean-Louis Nicolas? 15 juin 2007 Abstract. Let T be a triangle in a Euclidean plane. If f(T ) denotes the triangle whose vertices are the midpoints of the edges of T , and if we iterate the function f , the situation is simple : all triangles fn(T ) are homothetic and tend to the centroid of T . But, if g(T ) denotes the triangle whose vertices are the feet of the altitudes of T , the problem is not so easy. We shall see that gn(T ) tends to a point L(T ), a new point geometrically linked to T and that L(T ) is a continuous function, in fact hölderian, but is everywhere non-differentiable, hence the title of this paper. In part 1, the existence of L(T ) is proved and its coordinates are calculated in a simple system of axes tied to T . If the circle ?(T ) circumscribed to T is fixed, T depends on three angles ?, ?, ?. By rotation, we may require that ? + ? + ? = 0 so that L(T ) becomes a function L(?, ?) of two variables, and the coordinates of L(T ) become trigonometric series of lacunary type.

• triangles fn


• propriétés de régularité et d'irrégularité de séries


• séries d'exponentielles imaginaires


• o? ?