# [r-t] Complexity of extents

Philip Earis Earisp at rsc.org
Thu Jun 13 16:05:41 UTC 2013

```I am interested in measuring the "complexity" of extents, with a view to seeing / exploring some "maximally complex" extents.  Please note that I'm not talking about method complexity, metrics to judge how complex or static a method is etc.

Pretty much every extent that has ever been rung consists of simple, often short, repeated building blocks, ie not something that I would call complex. Now I realise this doesn't mean the extent is easy to ring...something like Mermaid Doubles, for example (a principle with 24 changes per division, where the plain course generates the extent) has a fairly featureless line that requires some focus (see http://ringing.org/main/pages/method?match=Mermaid&name-query=Search), but ultimately the extent divides into 5 equal parts and so the extent can hardly be thought of as being maximally complex.

Conversely, an extent can formally be an asymmetric one-part, but still consist of many identical or similar building blocks and hence not really be "complex". Ander Holroyd's 120 of original doubles (http://www.math.ubc.ca/~holroyd/comps/o5.txt) would be an example here, as perhaps would something like a spliced 120 of say St Simon's Doubles-type methods.

Turning to the well-studied case of extents of minimus, Ander has analysed the 10792 possible extents that don't have jump changes at <http://www.math.ubc.ca/~holroyd/minimus.html>. Ignoring rotations, reversals and mirror images this becomes 162 extents, of which 75 are asymmetric 1-parts, which seems a necessary but not sufficient condition when looking for maximal complexity.

I'm not quite clear in my own mind what I'm asking for (!), but, people, here are a few questions:

*         Are there appropriate standard tests / algorithms that can be used in such a scenario, ie here to rank extents in terms of their intrinsic complexity?

*         What is best to measure for this? Minimal repetition of blocks of notation?  Or can this fall down somehow (eg with the spliced doubles examples, which has a lack of identical patterns but still much structure)?

*         Have people got examples of "complex" extents on higher numbers than minimus?

*         How would something that has a definite design-principle like <http://bellringers.net/pipermail/ringing-theory_bellringers.net/2008-March/002011.html> come out on such metrics?