diff --git a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
index 9cf254cee86..c1e8c3b965a 100644
--- a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
+++ b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
@@ -297,6 +297,20 @@ public:
   }
 
   const Covering_sheets & number_of_sheets() const { return _cover; }
+  const std::pair<Vertex_handle, Offset> original_vertex(
+      const Vertex_handle v) const {
+    return (virtual_vertices.find(v) == virtual_vertices.end()) ?
+      std::make_pair(v,Offset()) : virtual_vertices.find(v)->second;
+  }
+  const std::vector<Vertex_handle>& periodic_copies(
+      const Vertex_handle v) const {
+    CGAL_triangulation_precondition(number_of_sheets() != make_array(1,1,1) );
+    CGAL_triangulation_precondition(
+	virtual_vertices.find(v) == virtual_vertices.end());
+    CGAL_triangulation_assertion(
+	virtual_vertices_reverse.find(v) != virtual_vertices_reverse.end());
+    return virtual_vertices_reverse.find(v)->second;
+  }
 
   bool is_extensible_triangulation_in_1_sheet_h1() const;
   bool is_extensible_triangulation_in_1_sheet_h2() const;
diff --git a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_hierarchy_3.h b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_hierarchy_3.h
index 6f7732d036c..462658e2e47 100644
--- a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_hierarchy_3.h
+++ b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_hierarchy_3.h
@@ -135,6 +135,12 @@ public:
 	hints[level] = v->cell();
 	  
 	v->set_down (prev);
+	if (hierarchy[level]->number_of_sheets()[0] != 1) {
+	  std::vector<Vertex_handle> vtc 
+	    = hierarchy[level]->periodic_copies(v);
+	  for (int i=0 ; i<vtc.size() ; i++) vtc[i]->set_down(prev);
+	}
+
 	prev->set_up (v);
 	prev = v;
       }
@@ -327,6 +333,11 @@ insert(const Point &p, Cell_handle start)
 	                                    positions[level].li,
 	                                    positions[level].lj);
     vertex->set_down(previous);// link with level above
+    if (hierarchy[level]->number_of_sheets()[0] != 1) {
+      std::vector<Vertex_handle> vtc 
+	= hierarchy[level]->periodic_copies(vertex);
+      for (int i=0 ; i<vtc.size() ; i++) vtc[i]->set_down(previous);
+    }
     previous->set_up(vertex);
     previous=vertex;
     level++;
@@ -363,6 +374,11 @@ insert(const Point &p, Locate_type lt, Cell_handle loc, int li, int lj)
 	    positions[level].li,
 	    positions[level].lj);
       vertex->set_down(previous);// link with level above
+      if (hierarchy[level]->number_of_sheets()[0] != 1) {
+	std::vector<Vertex_handle> vtc 
+	  = hierarchy[level]->periodic_copies(vertex);
+	for (int i=0 ; i<vtc.size() ; i++) vtc[i]->set_down(previous);
+      }
       previous->set_up(vertex);
       previous=vertex;
       level++;
@@ -405,6 +421,10 @@ move_point(Vertex_handle v, const Point & p)
     else {
 	old->set_up(w);
 	w->set_down(old);
+	if (hierarchy[l]->number_of_sheets()[0] != 1) {
+	  std::vector<Vertex_handle> vtc = hierarchy[l]->periodic_copies(w);
+	  for (int i=0 ; i<vtc.size() ; i++) vtc[i]->set_down(old);
+	}
     }
     if (u == Vertex_handle())
 	break;