This document is
A PDF version may be added later.
A partial index of discussion notes is in
Suppose you grasp the two end bands firmly, one in each hand: there will then be many movements you can perform without letting go: shaking the bands, stretching them (i.e. pulling the ends further apart) twisting them, looping the chain over a door handle and pulling, winding them round a candlestick, etc. Is there any action you can perform while holding the two ends that will cause the rubber bands to come apart without breaking or being cut? If not, why not?
The bands have been joined in such a way that each band except the left-hand end band has an asymmetric shape: at the left hand end the last band forms a simple loop, while at the right hand end there are two loops, with the next band going through them.
To maintain the pattern of connections in the picture, you would have to add new bands to the two ends in different ways. Can you visualise what would be required to add a band at either end? How would the processes have to be constrained if you wished to maintain the regular pattern of connections?
How might the chain look different if you added bands without maintaining the regular pattern? Would it be possible to produce alternating types of connection between bands?
What shapes can be formed by connecting a new band to a non-end band in the chain? E.g. could you form a "K" shape instead of a linear shape? Could the bands be linked to form a branching tree shape, with three branches added at every node?
Is it possible to join the two end bands of the chain in the same way as the pairs of adjacent bands are joined, i.e. by pushing a loop in one end band through the other end band then pulling the rest of the band through its loop? This would close up the chain of bands to form a large loop.
Figure 2 shows what a chain of rubber bands stretched around another object, and joined out of sight to form a closed loop would look like!
If that cannot be done, then why not?
Many more examples, of different types of impossibility, are presented here:
Could a current AI learning engine be trained to distinguish things that are possible from those are impossible (although they can be described, or depicted)?
It is impossible for statistics-based forms of learning (e.g. those used in Deep Learning), whose form of learning produces useful probability estimates, ever to discover that something is impossible, or that certain features of an object or process makes possession of other features by that particular object or process impossible. It requires a different sort of cognitive mechanism about which, as far as I know, current neuroscience tells us nothing.
There are AI theorem provers that can tell whether a particular formula is provable within a system of axioms and rules, for certain classes of formulae. This requires the theorem prover to incorporate "meta-knowledge" about its own operation. Proving that there is no proof whose length is less than N steps can typically be done by exhaustive search. Demonstrating that there is no proof of any length is usually much more difficult. Questions of this sort can lead to undecidability results in mathematics, logic and AI.
Presumably there are also undecidability theorems waiting to be proved regarding mechanisms in human brains, though they may depend on which resources are available outside the brain.
Question for mathematicians:
Is this impossibility a known theorem in 3-D topology, or a special case of a known theorem? What sort of proof of impossibility of closing the loop would satisfy mathematical standards of rigour?
How can non-mathematicians find it obvious that closing the loop is impossible
(perhaps after a little thought). I have asked a few, who did not take long to
decide it was impossible, though they did not find it easy to say why not?
Why is the impossibility hard to explain?
What is the role of the assumption that no part of a rubber band can pass
through another part of a rubber band?
Why is the impossibility hard to explain?
What is the role of the assumption that no part of a rubber band can pass through another part of a rubber band?
Related question: what kind of visual mechanism, or reasoning mechanism, makes it possible to discover that IF the rubber bands are indefinitely stretchable (so that size and thickness differences do not produce obstacles) THEN:
(a) it is always possible to link an isolated pair of rubber bands using the procedure mentioned (producing a two-link chain)?I am assuming that anyone who reads and understands (c) will agree that the process is impossible, but most non-mathematicians (and some mathematicians) will find it difficult to explain why. If I am wrong and it is possible, please let me know how. If possible please provide a video of the process, or a sequence of snap-shots.
(b) it is always possible to extend such a chain by linking a new isolated rubber band to an end band?
(c) it is never possible to transform such a (linear) chain into a looped chain by pushing a loop in one end band through the other end band then pulling the rest of the first band through its loop?
An interesting argument for impossibility
In July 2017, I received an interesting suggestion outlining a potential proof of impossibility of closing the chain (without cutting and rejoining portions of a rubber band) from Leila Sloman and David Sherman (Stanford University)
They suggested starting from the observation that if the loop were closed as hinted in Figure 2, it could not be "opened up" (without cutting and rejoining a band) to form a linear chain of the sort depicted in Figure 1.
I agree that this does somehow "feel" impossible in a way that is different from perceiving the impossibility of starting with an open chain as in Figure 1 and transforming it to an closed chain without cutting and joining.
A summary of their argument as I understand it:
A configuration consisting of a closed loop of linked rubber bands cannot be opened up by unlooping one of the rubber bands, since attempting to do that that would end up pulling that band back in a circle to the same place it started.
Try to visualise undoing one of the links depicted in Figure 2.
The fact that the loop cannot be undone would need to be proven rigorously. How could this be done? Is there a relevant known theorem about knots?
Impossible process vs impossible structure
NB I am not saying that it is impossible for a complete circular chain of rubber bands to exist: merely that it is impossible to produce one from a collection of existing rubber bands without introducing a temporary discontinuity in any of the bands.
If the final band is somehow "grown" in place, or a band is cut, looped as required, and then the cut fused so that the join is invisible, then, the result could be a complete circular chain of linked rubber bands. So it's not a type of object that's claimed here to be impossible, but a type of process.
Can any currently existing automated theorem prover cope?
Is there any automated theorem prover that can make the sort of discovery described above and prove the impossibility? I assume all the deep-learning technology is irrelevant, since it does not yield knowledge about what's impossible, or necessarily the case, or mathematical implications.
Many aspects of Euclidean geometry are concerned not with static structures (despite the frequent use of static diagrams) but with invariant features of processes that produce or modify structures. For example, invariant aspects of areas, or changes of areas, of triangles as the triangular shape (and/or position, orientation, or size) vary are summed up in theorems about areas of planar triangles as illustrated here:
Hidden Depths of Triangle Qualia (Especially their areas.)A separate file The Triangle Sum Theorem discusses ways of demonstrating that changing the size or shape of a planar triangle will not alter the sum of the interior angles.
Old and new proofs concerning the sum of interior angles of a triangle.
(More on the hidden depths of triangle qualia.)
I suspect biological evolution produced mechanisms for (proto-mathematical?) reasoning about deformable structures that are not exactly planar, do not involve infinitely thin or infinitely long, or perfectly straight lines, long before the development of Euclidean geometry.
However, not all the organisms that can use such mathematical reasoning are
aware that they are doing so and can discuss their reasoning with others. That
includes pre-verbal human toddlers, as illustrated in
Meta-Morphogenesis and Toddler Theorems: Case Studies
I think this is all connected with what James Gibson famously referred to as perception of "affordances" (including opportunities and obstacles for possible actions), though I don't know whether he ever noticed the connection with ancient mathematical discoveries.
J. J. Gibson,
The Ecological Approach to Visual Perception,
Houghton Mifflin, Boston, MA, 1979,
It took me a couple of weeks to notice this.
This document one of many presenting examples of perception of possibilities and impossibilities (involving geometry, topology, and numbers) on this web site. Here are some more of them (including those already mentioned above):
Another challenge for automated reasoning systems -- discover/invent a 3-D
ontology in order to explain/understand sensed 2-D phenomena:
This is part of the Turing-inspired Meta-Morphogenesis project:
A long term project: specify a Super-Turing reasoning machine capable of making
these discoveries and explaining them: