Re: webpage update: multimonzo (math terminology warning

Hi Gene,

Well the two implied vector spaces Q and R are embedded
one in the other, so that seems okay to me, as there
is no risk of confusion by having two implied vector
spaces, about what the actual point in the lattice is
that one is referring to - there is a natural embedding
of the one vector space into the other.

Actually what we have here is a module isn't it
if you forget the embedding. I thought there would
be a name for a vector space with a ring as the scalars
instead of a field, and that's what it is:

http://mathworld.wolfram.com/Module.html

So, we have a module embedded in an implied
vector space with the reals as the
natural choice for the scalar field
for the embedding.

Then, if you want to make a one one map
from intervals to the vector space then
- each interval maps to a unique
subset of the vector space so use
that instead. E.g. each interval
corresponds to a line in the vector
space implied by the 2D Tenney lattice,
or a plane in the 3D lattice and so on
- just reduces the dimension by 1.

That's just a matter of taking a geometric
approcah ratehr than an algebraic approach
to the Tenney lattice - and it may be
useful to do that in some situations.
Maybe examining geometric constructions
may lead to interesting ways of proving
results - it is probably the sort of
approach I would follow if I did some
work on this as I've done work before
on multi-dimensional geometry and
it is a natural thing to follow up.

For instance, the non periodic Penrose tiling
based pitch Lattice that Erv Wilson defined
would be a natural thing to explore
in a geometric context, though it would also indeed
have a natural algebraic interpretation too using
the interpretation of a Penrose tiling as
the 2D projection of a slice at a particular
angle through a tiling by hypercubes in five
dimensional space - and then interpreting
the 5D space algebraically.

Robert

hi Robert,

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:

> That's just a matter of taking a geometric
> approcah ratehr than an algebraic approach
> to the Tenney lattice - and it may be
> useful to do that in some situations.

it's *definitely* useful to use geometry
sometimes instead of (or as a complement to)
algebra !!!!!!!

there are some of us (me, for instance) who
really have a hard time with algebra above
the elementary level, but who have no problem
grasping the structure and properties when viewing
it geometrically.

-monz

Hi Monz,

> > That's just a matter of taking a geometric
> > approcah ratehr than an algebraic approach
> > to the Tenney lattice - and it may be
> > useful to do that in some situations.

> it's *definitely* useful to use geometry
> sometimes instead of (or as a complement to)
> algebra !!!!!!!

> there are some of us (me, for instance) who
> really have a hard time with algebra above
> the elementary level, but who have no problem
> grasping the structure and properties when viewing
> it geometrically.

Oh I think that was an example of typical
British understatement :-).

I should say it is often very useful
to take a geometric approach. Geometry
is a modern discipline that got
revitalised with Coxeter's work in
the middle of the twentieth century
and his books on the subject,
and then with the work of
many geometers after that and the publication
of the big survey book on modern geometrical researches
by Grunbaum and Shephard, and is now a very
lively and modern area of maths research.

So there is
no need to feel ashamed and old
fashioned any more if one is doing geometry.

Of course some stalwarts had gone on
doing geometry research before Coxeter
revived it but I gather that before his work it was
considered to be sort of a bit played
out, sort of attitude that the Greeks
and Euclid did most of it and then
a few more details got filled in in the
nineteenth century with hardly any more to do
for a modern geometer except for curved spaces
and other exotic geometric constructions.
Geometers were regarded as sort of strange creatures
living in the past, finding occasional
theorems and results that Euclid left out...

But it's not like that any more! And
as you say the nice thing about geometry
is that it is often very accessible to
some newbies - those who find it easy
to think about things geometrically.

Which isn't true of everyone - some people
find geometrical thinking hard and
algebraic type thnking easy, just
naturally, so it is a matter of temperament
as well as training amongst
mathematicians as much as amongst
non mathematicians. Best to have
both approaches available,
to appeal to everyone, and each will
have its particular strong points
so that it may be easier to prove some
results one way or the other way.

Robert

hi Robert,

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:

> Hi Monz,
>
> ...
> some people find geometrical thinking hard
> and algebraic type thnking easy, just
> naturally, so it is a matter of temperament
> as well as training amongst
> mathematicians as much as amongst
> non mathematicians. Best to have
> both approaches available,
> to appeal to everyone, and each will
> have its particular strong points
> so that it may be easier to prove some
> results one way or the other way.

exactly what i've always believed.

i've always felt that redundant coding
is a good idea: present property relationships
geometrically with different shapes, colors,
volumes, etc., and both algebraically, and
in simple arithmetic if that applies, which
it usually does.

-monz