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

[Aaron Sloman]

Thanks for the replies, including one from Norman Megill not posted to the list.
He sent me another example involving rubber bands which I've added to my web
page here (with his permission):
https://www.cs.bham.ac.uk/research/projects/cogaff/misc/rubber-bands.html#Megill

1. Timothy Chow wrote::

> The standard approach is to find a link invariant that distinguishes the
> two configurations.
>
> https://en.wikipedia.org/wiki/Knot_invariant
>
> I don't know offhand a specific invariant that will work for your problem
> but it seems likely that many of the standard invariants will fit the
> bill.

I'll have to try to find time later to investigate the standard invariants. For
now I just want to respond that I have an ulterior motive in investigating such
examples, namely trying to understand what sorts of brain mechanisms could have
enabled ancient mathematicians to make discoveries about impossible spatial
structures and processes, or about necessary features of structures and
processes, long before the development of "modern" mathematical treatments of
knots.

I think some of the brain mechanisms are also shared by some non-human animals
with high spatial intelligence, even if they cannot reflect on or communicate
with others about their thinking, because they don't share the adult human
meta-cognitive mechanisms. (That also seems to apply to very young children with
high spatial intelligence used when playing with toys, manipulating food and
clothing, etc.)

There are reasons to believe that "standard" theories in neuroscience and
psychology, based on trainable neural networks, are false, because such
mechanisms cannot discover, or even represent impossiblity or necessity.
I suspect (as I think Turing conjectured) that chemical forms of information
processing are used. Indirect evidence comes from newly hatched animals with
spatial intelligence used detecting, catching and consuming food, without having
to go through a training/learning process. Their competences must somehow have
been produced by chemical processes in the eggs, prior to hatching.

I suspect that explaining how the mechanisms work will be very difficult.

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2. Norman Megill <nm at alum.mit.edu> wrote to me privately with an image attached,
which I have now added to my rubber-band document (with his permission)

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/rubber-bands.html#Megill

It illustrates a different problem about rubber bands, which some readers may
find entertaining.

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3. Alex Galicki wrote:

> 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

Thanks for the reference, but I think you misunderstood my example.

My problem (closing the chain of rubber bands) was not *defined* in
the manner you supposed.

The comment that cutting and joining could produce the desired end result was
merely an observation not a specification of the problem.

The problem I specified was about whether it is possible to transform a chain of
linked rubber bands to a closed (circular) chain without any cutting and
joining. There are many transformations that *can* be produced by cutting and
joining but can also be produced without cutting and joining: e.g. take two
chains of linked rubber bands and combine them into a single chain.
(I assume it's obvious how to join them both with and without any cutting.)

I suspect that the impossibility of closing the chain without cutting anything
can be understood using ancient brain mechanisms that existed before the
development of formal Knot Theory. If ancient (human) brain mechanisms are up to
the task that may be because they have some way of (a) exhaustively
investigating a continuous space of processes, (b) knowing that the
investigation was exhaustive.

A simple case of that is knowing that two closed loops cannot become linked
merely by moving them around in space without opening and closing either loop,
and without allowing any part of one loop to pass through the material of the
other loop. Separating linked closed loops is likewise impossisble.

Fairly young children seem to have that understanding, though as far as I know
there's nothing in current neuroscience that explains how their brains detect
the impossibility.

Current neuroscience is also incapable of explaining most ancient mathematical
competences. I think Turing understood the problem, which could explain why he
began to investigate chemical mechanisms shortly before he died. In that case
his 1952 paper on chemistry-based morphogenesis was merely a report on an
incidental discovery, without revealing his real motivation for studying
chemistry -- hinted at obscurely in his Mind 1950 paper.

I'll be discussing these issues with Tufts University neuroscientist Mike Levin
at a conference next wednesday. Zoom link not yet available. If anyone is
interested email me and I'll pass on the information when I get it.

--------------
Aaron
http://www.cs.bham.ac.uk/~axs