Diagramming: Connecting Cognitive Systems to Improve Reasoning Valeria Giardino Introduction: A Manipulation Experiment I shall start from an example. Consider the following exercise. Take a common A4 sheet, fold it in half along the longer side, and then open it up again. Now there is a fold in the middle of the sheet. Orient the rectangular sheet in such a way that it has its longer side towards you. Hold firm the bottom left corner and grab the bottom right corner, folding it so that it touches the fold in the middle and the figure obtained is a trapeze (Figure 1a). Then, grab the bottom left corner and fold it on the other side of the virtual line that the previous fold has created (Figure 1b). At this point, you should recognize a triangle: two of its sides are already there. To obtain the last one, refold what is left outside the area of the triangle so that it will be not visible anymore (Figure 1c). Figure 1a Figure 1b Figure 1c The experiment has come to its end: there was a rectangle, and now we have a triangle, which is equilateral by construction. Hence, a rectangle (but indeed