A partial index of discussion notes is in
The claim is that such discoveries are related to the perception of positive and negative affordances, of sorts discussed by James Gibson, but go much further than Gibson's examples and categories. (He focused too much on online intelligence and not enough on offline intelligence, used in planning and designing, for example.)
In this discussion of 'toddler theorems'
I also suggest that many animals, and young children, including pre-verbal toddlers, are capable of making and using such discoveries (about what is and is not possible) but without possessing the type of meta-cognitive information processing architectures/mechanisms required for noticing that the discoveries have been made or that they are used.
They may also understand various possibilities and impossibilities without having the linguistic competences required to describe, or discuss them. Some evolutionarily older form of representation and manipulation of information, shared with other animals, is probably involved internally.
Moreover, even richer architectures are required for individuals to understand WHY certain things are necessarily true or impossible, i.e. necessarily false. And still more mechanism is required for engaging in discussion with other individuals about these things or teaching them to others who are able to ask questions, argue and propose counter arguments or counter examples.
An extended discussion of such phenomena related to pictures of impossible
objects or situations can be found in
Offers of help, including better pictures, are welcome.
FIG 1: Two sheets of paper, the same size and shape originally, one folded along a long line, the other along a short line. Each has one fold, though there are additional curves produced by gravity.
FIG 2: One sheet of paper, with several lines drawn across it in different orientations. The paper is folded along only one line, across the width of the sheet. Could it simultaneously be folded across two lines?
FIG 3: Yes: in this case there are two folds: one is, as before, across the width. The second fold runs along the shortest line shown in the previous picture. Note that that line does not cross or meet the line of the first fold on the surface of the paper: when the sheet is flat the first fold line and the new short one converge to a point off the sheet, on the right.
FIG 4: Here is yet another fold, not along the shortest line drawn but along the next shortest line which also does not meet the original fold line on the sheet of paper: the intersection between the original long fold-line and the new one is beyond the right edge, further from the edge than the previous intersection.
NOTE: There is a longer line drawn on the paper which was straight when the paper was flat and which goes across the current fold line. The intersection point can be seen very clearly.
Could the paper be folded along the original long fold line and also along the line just mentioned which crosses the long line?
Answer: YES, provided that the second fold is done while the paper is folded flat, along the first line. In that case another fold could be made along the line that cuts across the longer line diagonally. However the new fold would not go along the full length of the second line.
WHY NOT? When a sheet of paper has a straight line L1, along which it is to be folded, and another line L2 crosses L1, then if the paper is folded along L1, that will create a bend in line L2, and in general the two parts of the bent line will no longer be co-linear in 3-D space.
You should be able to convince yourself of that either by imagining the fold happening or doing it with a piece of paper -- i.e. first fold the paper flat along one line, then try to fold it along another line that crosses the first fold diagonally. You will be able to make the second fold lie on only one portion of the second line, unless the second line is perpendicular to the first fold line. WHY IS IT IMPOSSIBLE?
When the fold is not completely flat, the possibilities are even more restricted. If L2 is originally perpendicular to L1 (when the paper is flat), then the fold along L1 will produce a bend in L2, and if the paper is folded flat along L1, then the two parts of L2 will be adjacent in space: they will be co-planar, lying in the plane perpendicular to the fold.
However if L2 is not perpendicular to L1 when the paper is flat, like the line seen drawn across the fold in Fig 4, then folding the paper flat along L1 will cause the two parts of L2 to form a V shape, with one arm of the V on the far side of the paper. So the two parts of the bent line L2 cannot be adjacent when the paper is folded flat along L1.
If the paper is folded along a line, but not flat, e.g. so that the two parts meet at a right angle, then no other fold in the paper is possible crossing the fold since the new fold crossing the old one will have to form a straight line, and with the paper folded at right angles there is no straight line in the paper that crosses the line of the fold: at the intersection point the line will have to bend.
I previously wrote:
The impossibility described here is probably very familiar to people who do a lot of wall-paper hanging in spaces where walls meet, e.g. in alcoves, or near window openings. This can be dealt with by cutting "V" shapes out of the paper before attaching it or by cutting a slit and allowing one portion of the paper to be pasted over another portion."
However, when I discussed this with Auke Booij, he pointed out that the two folds can meet in the surface of the paper, as long as there is a third fold through the intersection point. The following two pictures demonstrate that he is right:
FIG 5: A sheet of paper with three intersecting lines, for folds.
FIG 6: The sheet of paper shown in FIG 5 above, folded along the three intersecting lines. Note that one fold is in the opposite direction to the other two. The original impossibility theorem concerning two folds was mis-stated: it did not allow for the impossibility to be "removed" by a third fold.
Reminder: this paper is part of a larger discussion of abilities of humans (and perhaps future robots) to make discoveries, reflect on them, and discover that they are mathematical truths (e.g. truths of geometry or topology).
Most non-mathematicians will simply treat the impossibility as an empirical discovery, until they reflect on whether there could be a counter example, e.g. where walls are made of different materials making previously impossible configurations possible. (How do you know, in some cases, that the material used cannot make any difference.?)
Challenge to amateur, unwitting, mathematicians
Your challenge now is to realise that impossibilities discovered are not just contingent facts about particular pieces of paper, e.g. the impossibility of just two folds meeting in the same surface, when the surface is not folded flat along the folds.
Likewise, the possibility of providing three folds, provided that one fold is in the opposite direction to the other two, as shown above, is not just a contingent fact about a particular sheet of paper.
Any flat flexible material that can be curved (e.g. into a tube or cone) or folded along a crease, but which cannot be stretched (as cling-film or a rubber balloon can, for example) can be folded along the length of any line drawn in the surface. After that fold there will be various additional folds that can be made, and others that cannot be made because they would cross the first fold in the surface. Additional folds can be made that cross the first fold line outside the surface, i.e. in empty space beyond the edge, as in Figures 3 and 4.
How can you convince yourself of the claimed impossibility? I think this is a non-trivial piece of geometric topology.
There is probably a mathematical publication somewhere that states and proves the theorems introduced informally here. (A reference would be welcome, especially if intelligible to non-mathematicians.)
Why am I presenting this example? Because it nicely illustrates the relationships between mathematical discovery and the perception and use of affordances. My attention was first drawn to this theorem when I was helping my wife, Alison, with a tricky job: she was papering a fairly high ceiling that had been discoloured when rain leaked through the temporary roof above a dormer window during roof repairs.
The discoloured ceiling had a flat horizontal part and at one side a curve upwards towards the higher ceiling above the hallway. This required two strips of wall paper to cover most of the flat horizontal part and a third strip to cover the remainder of the horizontal part and the bit that bent upwards.
For the first two bits, Alison, up the ladder, pressed the paper up onto the ceiling starting at one wall, while I held the pre-pasted paper up on a horizontal rod at the end of a pole which I slowly moved toward the opposite wall, constantly keeping it pressed up against the ceiling, with a length of pre-pasted paper hanging over the rod. The vertical, hanging, length of paper grew shorter as I moved the rod back to the second wall while Alison ensured that the expanding horizontal portion of the paper was flattened against the ceiling.
For the third bit of paper we hit the above theorem: it was not possible for the pasted sheet of paper both to bend/curve up along the line where the ceiling began to slope upward, while the unattached portion of paper hung vertically over the rod I was pressing up against the ceiling: this required two folds in different orientations in the surface of the paper but meeting in the paper: a mathematical impossibility.
The solution was to use a narrower strip of paper that did not go far up beyond the start of the slope, and to move the horizontal rod at some distance from the location where the paper was being pushed up on one side. The pictures below should help readers visualise the situation.
This is an example that contributes to the discussion of connections between
perception of affordances and discoveries in geometry and topology here:
I have discussed the theorem as involving a constraint on what can be done with folds, which involve a sharp edge. But that's a special case of a more general deformation from a planar sheet to one with a curve instead of a sharp fold. The more general mathematical result seems to be that neither two sharp folds nor two smooth curves can be formed from a planar material with the properties of a sheet of paper: namely it can't be stretched or compressed in any direction in the surface: distances, along the paper, between any two points on the paper, are invariant. This is not true of a sheet made from the material used to make balloons: while a balloon is being blown up points on the surface move further apart.
up at the ceiling after two strips of wall-paper had covered part of the
(c) How some of the work was done:
(d) After the narrow third strip of wallpaper had been added, covering the rest
of the unpapered area:
Alas nobody was at hand to take a picture of us attempting the impossible operation with a much wider third strip of wallpaper which would have gone much further up to the right, before the impossibility of bending the paper up the wall at the same time as the not yet stuck portion of paper hung over a horizontal rod (pointing left and right) struck us!
Anthropologists may already, unwittingly, have collected much evidence in their studies of diverse societies with different cultures, and different levels and kinds of technology.
There's also scope for developmental psychologists to investigate examples of
such discoveries in young children, even pre-verbal children, some of them
discussed in this paper on Toddler Theorems:
Of course, understanding what brain mechanisms make such mathematical discoveries possible, and replicating those competences in robots remains a complex collection of challenges.
More examples are discussed in
School of Computer Science
The University of Birmingham