Question about closing a chain of elastic bands -- and hatching eggs

[Alex Galicki]

I believe your "link question" is a fairly basic knot theory question.

Your definition of a "*ring* of rubber bands" seems somewhat imprecise
to me. However, if we say that "ring" means the end result of the
procedure you have described ("if the last band added is cut before
the chain is closed and the
cut repaired afterwards"), then such a ring always contains a Hopf
link. Whereas anything you can concoct from n rubber bands without
cutting is always equivalent to the trivial link with n components.
See the following for some introduction to the knot theory:
https://homepages.abdn.ac.uk/r.hepworth/pages/files/Knots_Notes.pdf

On Wed, 8 Sept 2021 at 10:12, Aaron Sloman <A.Sloman at cs.bham.ac.uk> wrote:
>
> It is very easy to link elastic/rubber bands to form an arbitrarily long chain.
>
> Each new band can easily be linked to an  existing end band by pushing part of
> the new band through an end band then pushing/pulling the 'other' end of the new
> band through the the newly protruding portion of itself. (Much easier to show
> than to describe).
>
> Question: is it possible to turn such a *chain* of rubber bands into a *ring* of
> rubber bands by linking the band at one end of the chain to the band at the
> other end, in the same way as new bands are linked to an existing end band?
>
> A little experimentation shows that it is impossible, but it's not clear what
> sort of reasoning mechanism enables the impossiblity to be established.
>
> Is there a "standard" analysis or proof of this sort of impossibility?
>
> (Of course, existence of a "circular" chain of linked rubber bands is not
> impossible -- if the last band added is cut before the chain is closed and the
> cut repaired afterwards.)
>
> This impossibility of closing the chain without cutting is not provable using
> Euclid's axioms/rules, which don't mention processes involving flexible objects.
>
> Some pictures and an inconclusive discussion are available  here:
>
> http://www.cs.bham.ac.uk/research/projects/cogaff/misc/rubber-bands.html
>
> This example is part of a larger investigation into mechanisms capable of
> explaining ancient forms of mathematical reasoning (and related forms of spatial
> intelligence in non-human species, such as squirrels, crows, elephants,
> octopuses,...). Can digital computer based mechanisms produce the required
> forms of intelligence?
>
> E.g. are there computer-based reasoning mechanisms that are able to create
> proofs of spatial impossibility or necessity, such as the rubber-band
> impossibility.
>
> I know there are some topological theorem provers, but I don't know whether
> there are any that can prove or autonomously discover the rubber-band
> impossibility.
>
> Could the processes in human brains that support such discoveries make
> use of chemical mechanisms that are related to mechanisms that control
> enormously complex chemical processes that assemble complete, intricately
> structured, animals inside eggs, starting with a few fairly amorphous volumes of
> chemical matter and a speck of DNA in a cell at the centre?
>
> I suspect much human spatial reasoning will turn out to be related to previously
> unnoticed chemistry-based self-bootstrapping forms of control in eggs, whose
> later stages prior to hatching involve types of complex, multi-layered, highly
> parallel, chemistry-based distributed 'virtual' machinery that combine
> continuous and discrete mechanisms, controlling intricate assembly of a large
> variety of types of physiological components required in a newly hatched chick.
>
> As far as I can tell, these (tiny!) biological assembly mechanisms are unmatched
> by any human-designed forms of computation or control developed so far, e.g. for
> use in factory assembly lines.
>
> Moreover, the abilities of newly hatched animals such as the avocets in this 35
> sec clip from the BBC 2021 Springwatch programme, Episode 5, 1st June,
>
> https://www.cs.bham.ac.uk/research/projects/cogaff/movies/avocets/avocet-hatchlings.mp4
>
> show that chemistry-based assembly processes in eggs can produce not only highly
> complex physical/chemical structures and mechanisms (in the bodies of the
> avocets), but also forms of intelligent control of post-hatching behaviour that
> don't have to be learnt by training neural nets in the environment, as currently
> assumed by many (most?) cognitive scientists and neuroscientists.
>
> The whole Springwatch episode is now on youtube:
> https://www.youtube.com/watch?v=FV6ZHe0CiHw
> The "Avocet Island) section starts around 12min 23sec.
>
> Could the in-egg chemical assembly mechanisms that produce forms of spatial
> intelligence in newly hatched animals be related to development of forms of
> intelligence in young humans who later become mathematicians making discoveries
> in geometry and topology -- like many mathematicians who existed centuries
> before Euclid?
>
> Certainly (non-hybrid) neural nets cannot explain abilities to discover cases of
> necessity or impossibility since those mechanisms merely collect statistics and
> derive probabilities. Necessity and impossibility are not points on a scale of
> probabilities.
>
> Aaron
> http://www.cs.bham.ac.uk/~axs
>
> Aaron Sloman,
> Honorary Professor of Artificial Intelligence and Cognitive Science
> (Retired, but still working full time, on the Meta-Morphogenesis project)
>
> School of Computer Science,
> The University of Birmingham
> Edgbaston
> Birmingham B15 2TT UK