atantion

[alex illi]

Hello,

I’m trying to understand calculating methodic you use in Scetch Up for geodesic domes. What are the calculations based on? According to the calculator given on Desertdomees elements’ length in your 3D models of geodesic domes correspond only in 4V (class1), while the rest of constructions do not correspond to these calculations. You can check it yourself, for example, model 5V (class 1) http://www.desertdomes.com/dome5calc.html
Is it a mistake or a different calculating algorithm? Could you please clearify this issue? As far as I understand Sketch Up is a accurate programme.

Thank you in advance. My best regards,
Alexander


--
Александр
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web www.space-expo.ru

[TaffGoch]

Alex,

There are several methods to calculate strut lengths. The most common
are conceptually described in "Domebook 2"

Ideally, you want the struts to be as close in length as possible.
This however, typically produces more "unique" strut lengths. If
you're after a more simple design, there are division methods that
produce a minimal number of different (unique) strut lengths, but end
ups increasing the variability in lengths (longer "longs" and shorter
"shorts.") Each division method has it's pros and cons. (These are
also summarized in Domebook 2.)

I prefer a method that produces less-variable lengths, even though one
of the outcomes is additional unique lengths. I use an algorithm from
physics, employing an electric-charge-repulsion concept used to evenly-
distribute points on the surface of a sphere. You can read about the
method by Googling the phrase, "Thomson problem"

http://www.google.com/search?as_epq=thomson+problem&num=100
_________________________

So, you're correct -- SketchUp is accurate, and is not at fault. My
results do not correspond to DesertDomes results (different
calculation method.)
_________________________

Note that you can download a copy of "Domebook 2" here:
http://groups.google.com/group/geodesichelp/browse_thread/thread/51760c3c225ffd66

Regards,
Taff

[alex illi]

2009/12/3 TaffGoch <taff...@gmail.com>

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please can you tell me exactly where in domebook 2 is this argotithm which is used in sketch up (which page)

thanks a lot 

[TaffGoch]

Alex,

The several methods, commonly used, are described in "Domebook 2"
starting on page 106. The methods are depicted and described, but
computer algorithms or formulae are not provided. (The book predates
personal computers.) Using the descriptive geometry, you can apply
planar or spherical trigonometry to derive formulae, if that is your
ultimate goal.

As I indicated earlier, the methodology I use is not typical, and is
NOT in "Domebook 2". I use "electron charge repulsion" calculations,
comparable to those used in the java applet here:
http://thomson.phy.syr.edu/thomsonapplet.htm

The differences in results can be minor, and each method has it's
advanatages and disadvantages. Unless you CLOSELY examine the attached
image (and pdf file,) you may not even notice the differences. The
first (green) is "method 1." The second (orange) is "method 2." The
third (blue) is "repulsion." So, as you can see, there is no ONE
correct method for tessellating an icosahedron, to make a geodesic
sphere.

Taff

[TaffGoch]

By the way, THE major advantage to "method 1" division is that you can
evenly "cut" the sphere in half, along the struts at the equator (as
long as you have used an even-number frequency division.)

Taff

[Ken G. Brown]

Joseph Clinton's did some of the original mathematical work in his 1968 'Structural Design Concepts for Future Space Missions' done for NASA.
Don't know if the document is available online anywhere.
Can purchase it on microfiche from NTIS, product code N6929417, Report NASA-CR-101577:
<http://www.ntis.gov/search/index.aspx>, search for 'Structural Design Concepts for Future Space Missions'

A google for shows up some good links,
eg: <http://mr-fusion.hellblazer.com/pdfs/geodesicmath.pdf>
or <http://www.simplydifferently.org/Geodesic_Dome_Notes?page=1>

Ken G. Brown

>--

[TaffGoch]

Good call, Ken.

It's my understanding that the NASA contract document was updated in
1971, and is, indeed, freely available for download, at the NASA
Technical Reports Server, in two parts:

http://ntrs.nasa.gov/search.jsp?N=0&Ntk=all&Ntx=mode%20matchall&Ntt=%22Advanced%20Structural%20Geometry%20Studies%22

NASA CR-1734
Advanced Structural Geometry Studies
Part I - Polyhedral Subdivision Concepts for Structural Applications
Joseph D. Clinton
September 1971

NASA-CR-1735
Advanced Structural Geometry Studies
Part II - A Geometric Transformation Concept for Expanding Rigid
Structures
Joseph D. Clinton
September 1971
_______________________

As it turns out, I had already downloaded the first part, and have had
it in my "digital library" since 2007. I didn't have the second part,
though. It, too, looks like and interesting read.

Alex, the subdivision concepts are covered in these two documents,
along with a description of the math (no computer code, though.)

[Again, please note that these methods differ from the "repulsion"
technique I used for my 3D Warehouse models.]

Taff

[Ken G. Brown]

Yay!

Ken

[TaffGoch]

The NASA document, to which Ken referred, is available for free PDF
download from the NASA Technical Reports Server:

http://ntrs.nasa.gov/search.jsp?N=0&Ntk=all&Ntx=mode%20matchall&Ntt=NASA-CR-101577

Report Number: NASA-CR-101577
Title: Structural Design Concepts for Future Space Missions
Abstract: Mathematical model for subdividing and transforming
polyhedron into structural sphere
Publication Year: 1968
(Recently to the NTRS database; March 2009)

This document does include computer code (FORTRAN).

Taff

[TaffGoch]

Jay Salsburg updated NASA-CR-101577, to clean-up the text and replace
graphics with computer-generated 3D color images.

You can download a copy of the PDF at his website:
http://www.salsburg.com/geod/geod.html

(Jay's own "GeodesicMath" PDF document is also available there.)

Taff

[TaffGoch]

Alex,

You might also benefit from Tom Davis' explanations of geodesic math:

http://www.geometer.org/mathcircles/geodesic.pdf

[Gerry in Quebec]

Hi Taff and Alexander,
As Taff mentioned, there are various ways to manipulate an icosahedron
to create geodesic spheres or domes. The so-called class I, method 1
described in Dome Book 2 by Joe Clinton and company, and also by Hugh
Kenner in his book, Geodesic Math and How to Use It, is very useful as
a starting point of dome design, as are other methods. But because
each builder, each building method, and each building application has
its own requirements and peculiarities, it seems to me like a good
idea to adapt any given geodesic geometry to fit the actual need.

Even-frequency class I, method 1 domes sit flat at the equator. But
this is not the case for other potentially useful truncations such as
those just above and below the equator. For example, the 4/9ths and
5/9ths truncations (also called 3/8ths and 5/8ths) of the 3-frequency
icosahedral sphere do not sit flat. The same is true of the 5/12ths
and 7/12ths truncations of the 4-frequency icosa. Buckminster Fuller
produced 3- and 4-frequency dome designs that do indeed sit flat at
the truncations immediately above and below the equator, as well as at
the equator itself. But he didn’t publicly describe the math behind
it.

In the early 1970s, a mathematics teacher from Wisconsin, USA, named
David Kruschke, used spherical trigonometry to demonstrate Fuller’s
level-base method for both 3 and 4 frequency domes. The chord factors
in his little publication, called Dome Cookbook of Geodesic Geometry
(first published in 1972, with another edition in 1975, but now out of
print) are still used today by various dome manufacturers because of
the obvious advantages of having all base-level vertices lie exactly
in the same plane. Kruschke mentions his reasons for publishing this
work. In the preface of his booklet he writes: “Unlike Domebook One
and Domebook Two, the chord factor results here are in close agreement
with Buckminster Fuller’s (a fact that is important when one is
working with three frequency domes....). Oddly enough, this confusion
and this book would have been unnecessary if Fuller would have
published his derivations.”

For comparison, here are the chord factors, to 5 decimal places, for
class I, method 1 (i.e., Clinton, Domebook 2, Kenner, Rick Bono’s
Windome program, and the Desert Domes website) and for the Fuller-
Kruschke layouts.

Class I, method 1:
3v icosa: 0.34862, 0.40355, 0.41241
4v icosa: 0.25318, 0.29524, 0.29453, 0.31287, 0.32492, 0.29859

Fuller & Kruschke:
3v icosa: 0.32971, 0.38229, 0.44106, 0.42149
4v icosa: 0.22219, 0.25958, 0.30906, 0.31287, 0.32492, 0.32942

In the case of the 3v icosa, the level base of the Fuller-Kruschke
design comes at a price: there is an extra chord factor (strut length)
to deal with and three triangle types instead of two.

As for the 4v icosa, the Fuller-Kruschke design has the same number of
chord factors (6) and the same number of triangle types (6). Two of
the triangle types are mirror-image scalene triangles (but otherwise
identical), which can slow down work in the shop. As I said, the main
advantage of this design is that the base is level at three useful
truncations: 5/12, 6/12 and 7/12.

But why stop there? It's possible to further simplify the design (and
fabrication) by (1) getting rid of ALL the scalene triangles, which
reduces the number of triangle types from 6 to 5, and (2) reducing the
number of unique edge lengths from 6 to 4 – at the same time
maintaining a level base at the 5/12ths and 7/12ths truncations (but
not at the equator).

Here are the new chord factors (4 instead of 6):
0.23766, 0.27741, 0.32611, 0.31405.

With this set of chord factors, Taff, there is indeed an increase in
the ratio of longest strut length to shortest. But it’s pretty small.
The ratio is 1.372 compared with 1.283 for the Class I, method 1 dome.
But it’s less than the ratio for the Fuller-Kruschke design, namely
1.483.

Most of the triangles in this alternative layout are quite close to
equilateral and I think the resulting dome would be aesthetically
pleasing. But so far I’ve only modeled it mathematically, in Excel
using the spherical coordinate system, not in any CAD program like
SketchUp. I’ve uploaded a jpg image to the files section. It shows
which strut goes where. File: 4v-icosa-solution3c-level-base.jpg.

Solution 3c is just one of several potentially practical triangle
layouts that stray from the conventional Clinton-Kenner geometry, the
Fuller-Kruschke geometry, and the electric-charge-repulsion
distribution of points that Taff uses. This “pretzel-twisting”, with
fabrication efficiency in mind, can be applied to other parent
polyhedra (e.g., octahedon) and other frequencies (I’ve done it with
5v, 6v and 9v), as well as class II dome layouts.

Any feedback on this will be most welcome.

Cheers,
Gerry in Quebec

> Note that you can download a copy of "Domebook 2" here:http://groups.google.com/group/geodesichelp/browse_thread/thread/5176...
>
> Regards,
> Taff

[TaffGoch]

On Mon, Dec 7, 2009 at 6:57 PM, Gerry in Quebec wrote:
> ...But so far I’ve only modeled it mathematically, in Excel using the

> spherical coordinate system, not in any CAD program like SketchUp.

Gerry, here's your geodesic sphere, in 3D.

Taff

[Gerry in Quebec]

Hi Taff,
Thanks for that. It's wonderful to see it in 3D -- and visually
confirm that the base is indeed level at the 5/12ths and 7/12ths
cutoff points.
Cheers,
Gerry in Quebec

>  Toomey.skp
> 137KViewDownload
>
>  Toomey.png
> 350KViewDownload

[ashokch...@gmail.com]

Dear Taff and Gerry

Would 5/12 in Tommey.png be equivalent to a shallow cap division of a 4 v icosahedron?
Regards

Ashok

[Gerry in Quebec]

Hi Ashok,

The 5/12 truncation of the 4v icosahedral sphere is taller than the 4 icosa-cap, the latter being a 4/12 truncation with the base doctored to sit flat.

- Gerry