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Re: Conway on Trilinear vs Barycentric
Posted:
Aug 11, 1998 10:04 PM
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>Prof. Chapman will have to speak for himself, but this is an
>inaccurate paraphrase of my actual statement that quadrays are
>essentially just barycentric coordinates. Since they obviously
>differ in detail - including the one that gives barycentric
>coordinates their name! - they are certainly not a subclass
>thereof.
>
>Brian M. Scott
Hmmmm. I find this somewhat confusing Brian. "Essentially
barycentric" but "differ in detail - including the one that
gives barycentric coordinates their name!... not a subclass
thereof." Some might define this "detail" to be "essential"
given it sounds like nomenclature is at stake -- i.e. do we
call them "essentially barycentric" if differing in some
detail that essentially defines "barycentric"?
To get more specific, here's an excerpt from a draft of an
article I'm writing for a professionals' magazine (my signed
Writer's Agreement keeps me from disclosing which one).
I post it here for peer review -- not wanting to mislead
my readers:
If this all seems pretty exotic, nothing like what you
remember from high school geometry class, it is. To the best
of my knowledge quadrays have only been on the scene for
about 20 years, were invented by a number of individuals
working both collaboratively and solo (e.g. David Chako,
D. Lloyd Jarmusch, Josef Hasslberger and myself). [SNIP]
Why use quadrays? Note the simple, whole number coordinates
I get for my tetrahedron (above). Many other shapes come out
with similarly simple data -- reason enough for a mathematician
to toy with this gizmo. And given the ideas are familiar
enough to high school students with some background in
Cartesian coordinates and vector addition (putting arrows
tip-to-tail), why not clue them in as well? The ideas relate
back to more conventional topics, while keeping minds flexible,
reminding kids (and their teachers) that xyz isn't the only
game in town.
Comments welcome by private email: pdx4d@teleport.com (or
post here if so moved). Should I be telling my readers anything
about barycentric coordinates i.e. mentioning their inventors,
talking about how quadrays are "essentially the same thing"?
For obvious reasons I don't want to just lift Dr. Chapman's
"barycentrics made difficult" characterization, as the whole
point of my article is how easy and accessible-to-kids is
this NeoCartesian game.
Kirby
PS: as to the sci.math roots of this query (re how to taxonomize
quadrays among many things -- and I still prefer "NeoCartesian" to
"essentially barycentric") here's an excerpt from Dr. Chapman's
post to that newsgroup of July 29, 1998, full text archived courtesy
of Deja News.
Major reference:
http://www.teleport.com/~pdx4d/quadrays.html
=====
Kirby Urner (quoted by RC):
Seems like sometimes mathematicians give lip service to how
consistency and precision are the hallmarks of their
game-playing, but then turn up their noses if someone comes
along with a consistent and precise symbol game that just
doesn't happen to be the same as theirs.
Robin Chapman (commenting):
Urner's quadray system (or barycentrics made difficult) is
easily translated to Cartesians and vice versa. The only
difference is that Cartesians are easier to use. The easiest
way of doing quadray computations is to translate to
Cartesians, do the computations in Cartesians, and translate
back!
Kirby Urner (quoted by RC):
...not essentially barycentric, I disagree...
Brian Scott (quoted by RC, responding to KU):
Eh? Of course they are: the basic idea's exactly the same.
Robin Chapman (responding to BS):
Well said, Brian.
=====
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