Re: Class I, method 2 derivation

[Gerry Toomey]

Attached to this message is an illustrated Excel file showing the derivation of spherical coordinates and chord factors for the 4v icosahedral sphere, class I, using Clinton's method 2. It's a step-by-step approach based on analytical geometry/trig.

 

Cheers,

Gerry in Quebec

 

 

 

On Mon, Feb 8, 2010 at 4:36 PM, <geodesiche...@googlegroups.com> wrote:

  Today's Topic Summary

Group: http://groups.google.com/group/geodesichelp/topics

• V6 Class 1 truncated [7 Updates]

 Topic: V6 Class 1 truncated

Gerry in Quebec <toomey...@gmail.com> Feb 08 08:04AM -0800 ^

 
I used class I, method 2 (page 107 in Domebook 2) to calculate the
chord factors for the 4v icosa. I end up with 5 chord factors:
0.275904; 0.321244; 0.296068; 0.313122; 0.302931.
 
The Class I 4v layout created by Mexican mathematician Hector
Hernandez has only 4 chord factors (number of chord lengths equals
dome frequency):
0.275904; 0.321243; 0.298311; 0.313239.
 
So, it now looks to me like his method is NOT method 2 as I suggested,
but something rather different from the Domebook/Clinton methods.
 
Can anyone verify that the first set of (5) chord factors listed above
are correct for method 2? I don't think I've come across a list of
class 1, method 2 chord factors.
 
Thanks,
Gerry in Quebec
 
 
 
 


 

Blair Wolfram <thedo...@gmail.com> Feb 08 11:15AM -0600 ^

 
Gerry;
 
Joe Clinton himself told me not to trust the chord factors in Domebook 2. He
said he was able to correct a number of errors and misprints for the final
printing in 1974 , but still the 1974 DB2 has errors.
 
I assumed you would be the person to confirm which chord factors are
correct!
 
Blair
 


 

TaffGoch <taff...@gmail.com> Feb 08 12:33PM -0600 ^

 
Gerry,
 
I modeled, in SketchUp, a 4v-class1-method2 icosa, and have attached the
chord results. They are identical to those found in DB2. Since I used the
method described in DB2, that's no surprise.
 
You apparently used a different method for finding the three vertices in the
center of the icosa face?
_______________________
 
BTW, I'm trying to "back-engineer" the Hernandez results. So far, my model
is in agreement, to only 4 decimal places, using successive approximation.
(Not a satisfactory technique.) How did you derive the "mexican"
vertices/chords?
 
Taff


 

Adrian Rossiter <adr...@antiprism.com> Feb 08 07:38PM +0100 ^

 
Hi Gerry
 
On Mon, 8 Feb 2010, Gerry in Quebec wrote:
 
> Can anyone verify that the first set of (5) chord factors listed above
> are correct for method 2? I don't think I've come across a list of
> class 1, method 2 chord factors.
 
If method 2 is equal central angle on the icosahedron edge
divisions then I get
 
geodesic 4 | off_report -C E
 
[edge_lengths_cnts]
0.27590448425526748 = 120
0.297790443995401 = 120
0.29938145926502907 = 60
0.31321026301214749 = 120
0.32124407128004651 = 60
 
Here is a model which colours the edges by their symmetry orbits
 
http://www.antiprism.com/misc/geo_i_4_edges.jpg
 
The brown edges have length 0.299381...
The green edges have length 0.297790...
The yellow edges have length 0.313210...
 
Keeping vertices on the sphere, if you shrink the brown edges a
little it will lengthen the green edges a little and they can be
the same length. Doing this looks like it would lengthen the
yellow edges. It wouldn't change the other two lengths (purple and
red, and blue).
 
I would guess then that this was the construction for the four
strut-length model.
 
Adrian.
--
Adrian Rossiter
adr...@antiprism.com
http://antiprism.com/adrian


 

Gerry in Quebec <toomey...@gmail.com> Feb 08 11:48AM -0800 ^

 
Hi Blair,
Thanks for the heads-up on Domebook 2. The reason I was looking to
confirm the 5 chord factors for class I, method 2, is that I’ve never
seen these numbers in any book, website or computer program. Domebook
2 and Kenner’s Geodesic Math describe method 2 in cursory fashion, but
neither publication actually lists either the spherical coordinates or
chord factors (actually.... there’s one octahedral example in Kenner).
I derived the 4v icosa chord factors using trigonometry following the
brief instructions at the top of page 107 of Domebook 2. I thought
maybe someone else had already crunched the numbers for various
frequencies and that these might be available somewhere for
comparison.
 
It looks to me like Hector Hernandez in Mexico has created a new class
I breakdown method. It has the advantage of minimizing the total
number of chord lengths (unique lengths = dome frequency).
Unfortunately, domes based of this method don’t sit flat at useful
truncations. Hence the need to tinker with the strut lengths in the
bottom row of triangles, which of course increases the number of
unique strut lengths. But the "Mexican" method does seem to produce
domes with very little variation in triangle shape.
 
I don’t know the math behind Hector’s method. I arrived at the chord
factors using iteration in Excel – lazy person’s method :-). In case
you or others are interested, here are the numbers for frequencies 3,
4 and 6. (The 2v solution is identical to class 1, method 1. I need to
double-check 5v.)
 
3v :
A = 0.36695882,
B = 0.42406256
C = 0.40419443
 
4v:
A = 0.27590384
B = 0.32124334
C = 0.29831132
D = 0.31323941
 
6v (This is the one Taff modeled in SketchUp.)
A = 0.18426311
B = 0.21569298
C = 0.19484588
D = 0.21345118
E = 0.20318958
F = 0.20934982
 
Cheers,
Gerry
 
 
 


 

"Ken G. Brown" <kbr...@mac.com> Feb 08 12:59PM -0700 ^

 
See the Dec 4, 2009, geodesichelp thread titled "NASA-CR-101577 - Structural Design Concepts for Future Space Missions" for a link to download Clinton's early NASA report detailing how to calculate chord factors.
 
I'm wondering if the 'new' breakdown method has previously been covered in the report.
 
Ken G. Brown



 

TaffGoch <taff...@gmail.com> Feb 08 02:05PM -0600 ^

 
Gerry,
 
The chord factors for 4v,class1,method2, in Domebook2, are provided on page
112 (bottom-left corner.)
"...
0.2759044
0.2995157
0.2977157
0.3132066
0.3212449"
_____________
 
I did not derive mine mathematically, but with "physical" construction in
SketchUp, followed by measuring.
...
0.275904
0.299516
0.297725
0.313207
0.321244
 
Not precisely the same, but good to 4 decimals. (Maybe I should double-check
my construction technique....)
________________
 
I agree that the Hernandez subdivision method appears to be "new." That's
why I am so interested in your method for derivation. I am, basically, using
iteration in my successive-approximation technique (which is a pain, since
I'm doing the modeling work, rather than letting Excel do the heavy
lifting.)
 
Taff


 

--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en