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From: RogerB on 12 Apr 2010 19:25
On Apr 11, 1:44 pm, Cheng Cosine <asec...(a)gmail.com> wrote:
> Hi:
>
>  We have in 2D polar coordinates and in 3D spherical coordinates. But
> do we have other parametric expressions for a hyper-ball in a
> dimension higher than 3?
>
>  Thanks,

Abelian group Cayley tables define algebras that "conserve" the
determinants of the symbolic table as "sizes". If the sizes are
quadratic, they define polar duals wher the size are squared radii.

E.g. C3 is {{a,b,c},{b,c,a},{c,a,b}} and conserves o = (a+b+c) and rr
= ((a-b)^2+(b-c)^2+(c-a)^2)/2. The cartesian-to-polar mapping is
{o,rr,theta=Arctan[[2*a-b-c,-Sqrt[3]*(b-c)]}. The polar-to-cartesian
mapping is {a=(o+2*Sqrt[rr]*Cos[theta])/3, b=(o+2*Sqrt[rr]*Cos[theta
+2*Pi/3)/3, c=(o+2*Sqrt[rr]*Cos[theta-2*Pi/3])/3}. Angles are additive
on vector multiplication, so the duals allow the definition of powers
and roots.

I have published (in Wolfram MathSource) over 20 similar results for
some larger groups, and for algebras "folded" from them. Folding gives
tables 1/2 or 1/3 as large as the group, by an equivalence
relationship that replaces a central subgroup by generalised signs.

The C3C4 group is particularly interesting. It has 2 linear sizes, o1
& o2, and 5 quadratic sizes, giving 5 (radius,angle) pairs {p,a1} etc.
These convert to 3-phase, 4-phase, 6-phase, and two 12-phase
sinusoids. The mappings are long expresseion, including terms such as
c= o1+o2+p Cos[a1]+q Cos[a2-w]+r Cos[a3-w]-2s r3 Sin[a4]-2t r3
Sin[a5], where w=2Pi/3 and r3=Sqrt[3]. o1 & o2 are "offsets"; they
displace the centre of the "hyperball" from the origin.

Non-Abelian tables (such as most Clifford algebras) do not give
additive angles. As it is difficult to find compact expressions for
the sizes of groups with over 12 elements, I have only defined a few
larger polar duals.

RogerB
From: spudnik on 14 Apr 2010 16:44
tripolar co-ordinates are not euler direction cosines,
which is just a homogenous form of vectorial direction
in space.

> Non-Abelian tables (such as most Clifford algebras) do not give
> additive angles. As it is difficult to find compact expressions for
> the sizes of groups with over 12 elements, I have only defined a few
> larger polar duals.

thus:
darn; I thought, from the header,
you were using a multiplier of 7 ... and that
made me realize, the professors who do that,
are subverting the "big Oh" and "little oh" formalism.

that partition of the triplet is so important,
vuz Brun's constant!

> for x,y > 7, twins(x+y) <= twins(x) + twins(y)
> where twins is the prime twins counting function,
> where 3,5,7 is considered as 2 twins.

thus:
just because it was British,
I'd assume that the folks at E.Anglia did this, on purpose.
"global" warming is almost & assiduously all computerized
simulacra,
and extremely limited reporting, about glaciers e.g.
> >http://www.abc.net.au/unleashed/stories/s2868937.htm

thus:
to recap my reply to the TEDdies comments (as I am still
listening to B.Greene's pop-sci talk ... zzzz),
first of all,
Minkowski made a silly slogan about ordinary phase-space,
then he died. thank you!
> http://www.ted.com/talks/lang/eng/brian_greene_on_string_theory.html

thus:
they were just at the library auditorium,
selling the electromags to cure depression.... beats the heck
out of electroconvulsing, but I missed the refreshments!

thus:
I didn't get the gist of the CBS reportage, although it seemed
to be literate & wikipediaized (yeeha .-)
seemed like "more decimal points," although
there was a (wikip.) bibliographic note referring to Dicke --
I think, it was his paper that Einstein saw on one
of his rare visits to his Caltech office, and pooh-poohed,
regarding the predominant redshifitng of the heavens.

thus:
and, if at the centerof Sun is an iron core,
the theory might have to be revized (don't laugh;
not only was this a mainstream theory at one time,
it may not have been laid to rest (in current research)).

thus:
Rob, you uneducated, global-warmed-over bog-creature --
did you create any oil, today?...
seriously, that was amuzing about the cancellation-of-submission.
reminds me
of the time that Popular Science made an on-the-wayside attack
upon S. Fred Singer; at the time they were owned by Times-Mirror,
the then-owner of the LAtribcoTimes. the article was nominally and
visually an aggrandizement of three professors (and taht could
have included one of my own, at UCLA) of a theory about climate,
which had been celebrated already (I think) with a Nobel.
they included a mug-shot of the good doctor,
along with no mention of his vitae; alas!

thus:
the Skeptics were a Greek cult in the Roman Pantheon,
along with the Peripatetics, the Gnostics, the Solipsists etc.
ad vomitorium; as long as the Emperor was the Top doG,
you were left to your beliefs (til, of course,
Jesus -- after it became the state church).

thus:
virtually all of "global" warming -- strictly a misnomer, along
with Arrhenius 1896 "glasshouse gasses," except to first-order --
is computerized simulacra & very selective reporting, although
a lot of the latter is just a generic lack of data (that is,
historical data for almost all glaciers -- not near civilization).
I say, from the few that I casually *am* familiar with,
that *no* database shows "overall" warming --
not that the climate is not changing, rapidly,
in the Anthropocene.

thus:
instead, we should blame Pascal for discovering,
experimentally, his "plenum," which he thought was perfect. I mean,
it's always good to have a French v. English dichotomy,
with a German thrown-in for "triality."
> of Newton's "action at a distance" of gravity,
> via the re-adumbration of his dead-as-
> a-doornail-or-Schroedinger's-cat corpuscle,
> "the photon." well, and/or "the aether,"
> necessitated by "the vacuum."

--Light: A History!
http://21stcenturysciencetech.com

--NASCAR rules on rotary engines!
http://white-smoke.wetpaint.com
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