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

[Aaron Sloman]

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