Raul wrote:
> Is there a better way of doing this?
> {: +. r.inv j./1 1
Marshall responded:
> You can also use (+/&.:*:) in place of |@j./ ,
> leaving you with -@^.@(+/&.:*:)"1
Raul wrote:
> Experimenting: the - is necessary and the ^. is not necessary.
> (I do not get a hexagon without the minus, I do get a hexagon
> without the ^.).
> Immediately after writing this I realized the - is also
> unnecessary - changing >./ to <./
What I love is that through some simple trig and a few experiments, we got
from {:@+.@(r.^:_1)@(j./) to +/&.:*: .
I suppose I find this particularly gratifying because I spent some time
trying to restate Raul's phrase in terms of simple arithmetic operations,
staying entirely in the real domain, and I eventually reproduced
Marshall's verb. Having spent so much time "simplifying", when I got the
final, irreducible result, I wondered at the need for -@^. , and what its
physical interpretation was.
Raul's original verb could be rendered in English as "the length component
of a polar coordinate (initially specified in Cartesian terms)". Why
should that length be expressed as the negative log of a distance? Why
not, as Don put it, "the raw distance"?
I know there are subtle and beautiful connections between the trigonometric
and exponential functions, and the e hidden in r. is one expression of
that. But I'm still not seeing the fundamental physical interpretation.
In other words, I wasn't surprised with the -@^. disappeared in Raul's use
case; I might've been more surprised if it'd persisted.
Anyone want to help me see it? Maybe the best illustration would be a
concrete use case where the -@^. isn't superfluous - one where where it is
not only necessary, but inevitable?
That is, a use case where -@^. has obvious physical interpretation, when
applied to the distance. Ideally one like Raul's, which ultimately didn't
involve complex numbers (i.e. a real-valued binary [dyadic] operation on
real numbers).
-Dan
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