Angular Division vs Chord Division - Method II or Mexican or other???

[Bazil]

Hi all,

After reading a few pages of this group I see most of you are way
ahead of me on this but wanted to share anyway.

I had been playing with some simple domes and researching a lot. I was
a bit surprised by the number of different strut lengths in higher
frequency and in particular the variance between shortest and longest
struts. So i put my thinking cap on and thought of dividing an
icosahedron edge by angle, not by even lengths along the chord. I
realised I wasn't the first to describe this method - i think it's
what is referred to as Method II in the dome book 2? - and Fuller and
others described and used the method, especially in high frequency
domes (16V) - the Epcot dome?

Dividing the arc by angle, the struts between each new vertex above
the original icosa edge are the same length. Struts within the
original triangle are necessarily different but symmetries allow for
reductions depending on V.

What I found so far : with 3V there are still 3 different lengths but
the qty of two of them change; 4V has 5 different struts down from 6
with the traditional method; 5V has 7 (I have reduced this to 5 now),
two (4!!!) less than the 9 of the traditional method; 6V has 8, one
less than the 9 of the traditional method.
Also, the difference between shortest and longest struts is less. This
becomes significant as V increases. Whereas the percentage difference
seems to grow as V increases in the traditional case, it seems to stay
about the same with angular division - I have only checked this up to
V6. The resultant "pattern" of the struts is perhaps not as nice -
especially in 3V but evens out with higher frequency. What changes is
the lines of struts adjacent to a hemispherical line become a little
more wavy. Just like the line of division in an odd frequency only a
little more pronounced.

I just read the "6v octahedral hemispere, Mexican method" thread and
am just reading some threads i found searching for "Mexican".

My main point is that I found a couple of ways to get exact (to 14
decimal places) arc intersections - one by direct formula and another
where i couldn't find a formula by - can't remember the mathematical
term - approaching by smaller amounts from either side...

Bryan

[Hector Alfredo Hernández Hdez.]

HI,

Angular division and chord divition are the same thing.
ie exista a function biyective between them

this means that t1=t2  < ==>   chord1=chord2   for angles less than pi/2.


the math problem to solve for "mexican" method its reduced to solve a no-linear equation system.

and for frecuency >= 6  is a system with more equations than freedom degre,
for this reason we need add fredom degres to have a equalt quatity.
A way to add freedom degres is qiben freedom a one or more r.
this is:, some r dont need r=1 necesarily.

"mexican method" is one improvement to Clinton method , with out incertidumbre in radios ...


BY THE WAY, I WILL PROPOSE TO CALL     GHT method insteat of "mexican"   WHAT DO YOU THINK ABOUT IT GUYS?


See you.

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[Hector Alfredo Hernández Hdez.]

DRAW

2012/2/3 Hector Alfredo Hernández Hdez. <hecto...@gmail.com>

[TaffGoch]

Hector,

I must be thick-headed, today.  What is "GHT" an abbreviation of?

It's not "GerryHectorTaff", is it?

-Taff

[Bazil]

Hi Hector,

I am not sure what you mean there.

I meant dividing with n > 2

Method I is what i called chord division and it appears Method II is
what i called angular division

This thread should probably die because i thought it had vanished and
posted my subsequent thread.

thx,

Bryan

On Feb 4, 6:30 am, Hector Alfredo Hernández Hdez.

<hectorh...@gmail.com> wrote:
> HI,
>
> Angular division and chord divition are the same thing.
> ie exista a function biyective between them
>

> this means that t1=t2  < ==>   chord1=chord2   for angles less than pi/2.
>
> the math problem to solve for "mexican" method its reduced to solve a
> no-linear equation system.
>
> and for frecuency >= 6  is a system with more equations than freedom degre,
> for this reason we need add fredom degres to have a equalt quatity.
> A way to add freedom degres is qiben freedom a one or more r.
> this is:, some r dont need r=1 necesarily.
>
> "mexican method" is one improvement to Clinton method , with out
> incertidumbre in radios ...
>

> BY THE WAY, I WILL PROPOSE TO CALL     *GHT* method insteat of
> "mexican"  *WHAT DO YOU THINK ABOUT IT GUYS?
> *

[TaffGoch]

Bryon,

Equal-angle subdivision of the PPT (principal polyhedral triangle) edge is, indeed, method-2.

See method descriptions in Domebook 2, pages 106 & 107.

-Taff

[Hector Alfredo Hernández Hdez.]

You are RIGHT Taff :)

Also
Gerry: Equations
Hector: Equations & Cad Maniputations
Taff: Cad Maniputations

I think so.

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