[r-t] RE: Various
Robin Woolley
robin at robinw.org.uk
Sun Nov 21 09:02:00 UTC 2004
Sorry for the delay!
Firstly, Don Morrison's point: "What does "uncountably finite" mean?"
As it's obviously a pure-maths term, I spent some time unsuccessfully trying
to locate it in my first year notes. The nearest I came to it was the Heine
Borel theorem. Further thought yielded the following:
As pure-mathematicians never DO anything, they just think about doing
things, I think that the use of 'countable' means that there is a simple
algorithm to count the members of a set. They would never dirty their hands
by actually counting things, 1, 2, 3,... etc. As Fraleigh says, "never
underestimate a theorem that counts things". That the set is finite there is
no doubt, since at any stage, there is a finite set of parental methods.
(The countability or not of methods at a given stage was addressed very
early in this theory list.) So the sub-set of parents which do work at only
two stages is therefore finite, but there's no formula for the size of the
set.
Nice to see someone has the patience to read my non-sound-bite e-mails. What
about the main points, Don? b.t.w., I don't know Jim Phillips. How does he
interpret (G)B7 (and its partner (G)B8)?
Looking at extensions themselves. Classical extension had (2n-3) possible
paths below (worst case scenario) and 2(n-2) possible paths above, giving a
maximum set size of 2(n-2)(2n-3), which is of the order of n^2. Quantum
extension (extension by modes) has (n-1)(n-2) both above and below which is
of the order of n^4. It's interesting to see the increase in the limit of
possible extension paths quantified. Classical extension of minor can be
carried out by hand on the back of the RW address label.
As regards a program to do this, is it more efficient to generate the 400
extensions of any minor parent, pick off the ones with (say) pb lead ends
and then apply the restrictions, or apply the restrictions then generate the
extensions?
Here's a case of 'uncountably finite' in that there's no algorithm to give
the number of true extensions, but it is finite.
Back to the old topic:
Part of the MC's justification for (G)B1 is "This new restriction gives us
the confidence to introduce several new constructions without the risk that
they will increase the number of unsatisfactory extensions". If an
extension is 'unsatisfactory', which I take to mean almost quite, but not
totally, unlike the parent, would any ringers wish to disregard it as an
extension anyway? (Or am I being naive on this?) Taking Roker as my usual
example, the two extensions to royal aren't particularly similar to the
parent.
PABS et al. mention a truncated icosahedron - consisting of pentagons
and hexagons. A simple counting algorithm tells us that a polyhedron
consisting of just pentagons and hexagons must have exactly 12 pentagons.
Definitely pure maths! The truncated icosheadron is, of course, one of the
13 Archimedean solids.
On families, Sir Martin Rees, the Astronomer Royal, reminds us that, whilst
there is an infinite set of plane regular figures, three dimensions give us,
as pointed out already, just five regular solids. In four dimensions there
are six, but just three at all possible higher stages.
Regards
Robin.
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