Clearly we can solve problems by thinking about them. Sometimes we have the impression that in doing so we use words, at other times diagrams or images. Often we use both. What is going on when we use mental diagrams or images? This question is addressed in relation to the more general multi-pronged question: what are representations, what are they for, how many different types are they, in how many different ways can they be used, and what difference does it make whether they are in the mind or on paper? The question is related to deep problems about how vision and spatial manipulation work. It is suggested that we are far from understanding what is going on. In particular we need to explain how people understand spatial structure and motion, and how we can think about objects in terms of a basic topological structure with more or less additional metrical information. I shall try to explain why this is a problem with hidden depths, since our grasp of spatial structure is inherently a grasp of a complex range of possibilities and their implications. Two classes of examples discussed at length illustrate requirements for human visualisation capabilities. One is the problem of removing undergarments without removing outer garments. The other is thinking about infinite discrete mathematical structures, such as infinite ordinals. More questions are asked than answered. 1 We can think with diagrams Consider the trick performed by Mr Bean (actually the actor Rowan Atkinson): removing his (stretchable) underpants without removing his trousers.1 Is that really possible? Think about it 1The first draft of this paper located Mr Bean in a launderette. Toby Smith corrected me, pointing out that the shy Mr Bean was on the beach, and wished to remove his underpants then put on his swimming trunks, both without removing his trousers. On 29th July 1995 I posted Mr Bean’s problem as a followup to a discussion of achievements of AI in several internet news groups (comp.ai, comp.ai.philosophy, sci.logic, sci.cognitive) and received a number of interesting and entertaining comments. Chris Malcolm pointed out the similarity with the bra and sweater problem, i.e. removing a bra without removing the sweater worn above it. Readers are invited to reinvent the jokes that were then posted, about which problem was easier for whom under which conditions. In particular, someone pointed out the distinction between difficulty due to unfamiliarity vs difficulty due to being distracted. if you haven’t previously done so.2 Is it possible to remove the underpants without removing the trousers, leaving the waistband of the trousers constantly around the person’s waist, allowing only continuous changes of shape of the body and the underpants and trousers, e.g. stretching, bending, twisting, but with no separation of anything into disconnected parts, no creation of new holes, etc.? Does it matter whether the waistband of the trousers is tight or not? Many people can answer this question by thinking about it and visualising the processes required, even if they have not seen Rowan Atkinson’s performance. A harder question is: in how many significantly different ways can the underpants be removed? 2 Some comments on the underpants problem It is easier to consider the underpants being distorted, ignoring who does it and how, than trying to work out all the contortions of posture Mr Bean would have to go through to produce the appropriate sequence of changes. If we abstract away from the problem of how the wearer makes the transformations happen we can suppose Mr Bean remains rigid and still and someone else pulls and stretches his underpants perhaps using long thin tongs where necessary. (Is it obvious that this change makes no difference to the main problem? Why?) Even with this abstraction there are several different ways of thinking about the underpants problem. Some use only topological relationships preserved under all continuous transformations, including those which change size, shape and distances. Some also use metrical relationships involving shape and size. We can also use topological relationships with structural features of under-specified metrical relationships. Thinking purely topologically is quite hard to do, since it involves finding the most general way to characterise the relationship between Mr. Bean and his garments in the initial and final states. From that point of view the start and end states are equivalent and there is no problem for Mr Bean to solve. So it cannot be the right way to think about the problem of how to do it. Most people do not think like that. They conceptualise the problem in a largely qualitative but partly metrical fashion, including various ways the underpants might stretch and fold. We shall see that it is useful to combine different abstractions. 2.1 How many distinct solutions are there? Most people at first see only two symmetrically related solutions to the problem. One involves stretching the left side of the underpants down through the left trouser leg, over the foot and back up the left leg, leaving only the right leg through its hole. The underpants can then be slid down the right leg and out. A similar solution starts on right side, with the underpants emerging through the left trouser leg. If the waist band of the trousers is loose there are several more pairs of symmetrically related solutions, e.g. sliding one side of the underpants up over the head and down the other side and out through the leg, or sliding the central (leg-divider) part of the underpants down inside a leg then over the foot and up the same leg on the outside, then out past the waist band, over the head and down the other leg. It is easier to visualise than to describe! Another pair of solutions 2I have previously given audiences the task of finding out how many possible numbers of intersection (or tangent) points there can be between a triangle and a circle in the same plane. It is easier than Mr Bean’s problem, but many people miss out some cases unless prompted.