Blender Merge Vertices
So now when you select either object both objects will be selected.
Blender merge vertices. Every island of selected vertices connected by selected edges will merge on its own median center leaving one vertex per island. Adjust the merge distance until the correct number of verts are removed. The options to merge selections together at first and at last appear only when vertex select is the active mode of selection when in edit mode. I think arexma explained it quite well but you seemed to have missed the point that the two objects must be one object.
Merging vertices of course also deletes some edges and faces. But blender will do everything it can to preserve edges and faces only partly. The button in in the left tool shelf hotkey t and when you press it you get a panel for setting the merge distance and up in the info panel header you can see the removed nnn vertices information. To merge elements together first make a vertex edge or face based selection then from the vertex menu click merge vertices alt m vertex merge vertices.
Before this update the only thing you needed to do was to press alt m but that brings up another menu now. Shift select the cube and plane hit ctrl j to join the two objects. Every island of selected vertices connected by selected edges will merge on its own median center leaving one vertex per island. In the merge pop up menu that appears select at first at last at center at cursor or collapse.
So i ve been using blender for quite some time now but since the release of 2 83 i do not know how to merge the vertices at center.