How might we prevent this with better cable design?

This is a project about applying maths to solve real world problems.

Real world problems of a certain broad type.
(Not necessarily headphone cable tangles, but perhaps something analogous;
or something related, like how to achieve good washing results
by a good design for a washing machine,
or how to minimize the risk of umbilical chord tangling in the womb* ...)

So, the first question is, what kind of challenge are we going to try
to apply maths to?
-- For definiteness let's think about the cable design problem.

In which case the _aim_ is to design a better (less tangly) cable.

(The project need not actually achieve this aim, but it should try to
take a helpful step in the right direction.)

This challenge is going to start with the problem of understanding _why_ the
cable tangles.

And perhaps even before that, the problem of establishing exactly what
it means to say that the cable tangles!

(It might be a good idea
for a project student or team working in this area
to begin by getting some quantitative
information on real cable tangling -- that is, by doing some
controlled experiments with real cables.
The exact details of the experiment are not
so important here.
The idea is just to get a feel for what really
happens when a cable tangles.)

The next question is: how do we apply maths to help us with this challenge?!

This is the problem of ``mathematical modelling''.

A good place to look for clues is in the way maths already models
knots and tangles -- in the Theory of Knots and Tangles (which
already exists!
See e.g. the book
Introduction to Knot Theory by R H Crowell and R H Fox).

Then we can see if this existing maths can be adjusted to suit our
cable modelling problem.

Next we will need to think about what is happening physically (i.e. mechanically) with our cable? ...But let's leave that until we've thought about how to model the cable `statically'.

Project themes:

- What are mathematical knots and tangles? That is, what is the formal definition of a mathematical knot?
- In what ways are they similar or different to real (static) knots?
- How can they be worked with mathematically?
- How can their "pictorial" representation be used in rigorous calculations?
- How do they connect with their applications?
- What are the dynamics, the mechanics, of real knots? How can _these_ be modelled?
- What is entropy and how can it help us?
- ...

Keywords: knots, tangles, energy and entropy

References:

- Introduction to Knot Theory by R H Crowell and R H Fox here!
- Geometric Topology in Dimensions 2 and 3 by E E Moise

Links:

- What is a mathematical knot?
- Draft FAQ
- Ranicki's links page
- Ranicki's page
- Burde -- Zieschang book : "Knots"
- entropy (click on "Heat and Thermodynamics"; then "Entropy"; then the 4th definition... or go here )
- prelude to entropy here (go to 1. Introduction)
- to follow
- Overleaf...

page owner: paul