Fuller-Kruschke method: description, equations & calcs for 3v icosa

[Gerry Toomey]

Andrew and others,

The attached Excel file presents the spherical trig calculations for a 3v icosa, class I dome, using the Fuller-Kruschke method of subdivision. This gives a level base at the 4/9 and 5/9 truncations.

 

The work is divided into three steps:

Step 1. Calculation of spherical coordinates
Step 2. Calculation of chord factors
Step 3. Conversion of spherical coordinates to Cartesian coordinates & compilation of an OFF report for visual display.

 

The mathematical "route" I followed here differs from that taken by Dave Kruschke in his 1972 booklet "Dome Cookbook of Geodesic Geometry", pp 18-23. But it results in the same chord factors. The only numbers I have taken as "given" (i.e., as the starting point) are the spherical coordinates of the icosahedron.

 

The derivation of coordinates and chord factors in this Excel file is independent of any other subdivision method. In other words, the Fuller-Kruschke approach is not simply the result of tinkering with method 1, 2 or whatever.

 

I've tried to simplify the calculations and make them as transparent as possible by writing out the key equations (highlighted in green) in standard math notation and by including some narrative. The equations can also be viewed in Excel notation in the function bar (fx) at the top of the worksheet.

 

Corrections, comments & queries are welcome.

- Gerry Toomey

 

 

On May 20, 10:32 pm, andrew777 <andrewj...@gmail.com> wrote:
> Hello TaffGoch,
>
> Thanks for the work on the eight frequency picture that I sent in. Nice
> deductive reasoning, just what I asked for, thanks.
>
> How do I get the equations or algorithms for the Fuller-Kruschke method and
> solutions? If there is a mathematical library of all the methods for any
> kind of dome? If not, I think it would be good to catalog these methods
> like you have done in the sketchup 3-D warehouse (which is visual record).
> Just a thought,  a mathematical and algorithmic record should be made too,
> for repeatability and variation for new experimenters and learners. A right
> brain (3-D warehouse) and left brain library. *It* might eliminate
> redundant questions because all the mathematics and algorithms will be laid
> out along with visual examples of domes.
>
> Andrew

[andrew777]

Hello Gerry and all:

Thank you for your work on the mathematical presentation of the three frequency class I dome,using the Fuller-Kruschke method of subdivision.

I am attaching a hand drawn sketchup drawing of a eight frequency dome using the divide command. I would like to develop parametric methods of drawing in AutoCAD or Sketchup all the dome methods.

You asked me how did I get an interest in domes? My interest came when I read many books by Bucky Fuller. I built a lattice wood dome similar to a bamboo geodesic dome, as a result, I was chosen to spend two days with Bucky Fuller, after that I was hooked. I invented a construction method for concrete that could be used for building domes or other structures. The patent really didn't go anywhere, but my passion for domes remained.

Andrew

[Gerry in Quebec]

Andrew,
This excerpt may interest you. It's from the preface to Edward Popko's
recent book, Divided Spheres: Geodesics & the Orderly Subdivision of
the Sphere:

"You will not find a few things in this book. Complete computer
programs are omitted.
Instead, an appendix called “Geodesic Math” collects the basic
algorithms. These
short computer program fragments in Appendix C show how to perform
each subdivision
method described in this book. These are all that is needed for those
interested in writing
their own geodesic computer programs. The resulting geodesic grids for
every technique
explained are also listed in Appendix D You can input this data into
display programs or
compare it with the results of programs you might write."

http://www.crcnetbase.com/doi/pdf/10.1201/b12253-1

>  icosa v8 dome.skp
> 343KViewDownload

[andrew777]

Hello Gerry,

I appreciate the descriptions and and equations for the Fuller-Kruschke method that you sent me. I've been trying to avoid buying Divided Spheres: Geodesics & the Orderly Subdivision of the Sphere, However, you have convinced me that this is what I have to do. The price is a little steep.

Andrew

[Gerry in Quebec]

For the same reason, I haven't bought the book either. One of these
days... if a used copy for comes up for sale around $20.
Gerry

[TaffGoch]

This may be of interest to those readers who are studying Kruschke-Fuller "lesser circle" subdivision.

I stumbled across a reference, this morning, that describes the math for lesser-circle (or, "small circle") geodesic tessellation of the icosahedron, specifically, for the production of triangular grids for earth mapping:

http://www.ncgia.ucsb.edu/globalgrids-book/song-kimmerling-sahr/

Equations are clearly described and illustrated, making the reference a good resource for study. I searched further, and found the original Oregon State University thesis, from a few years preceding. This PDF document is downloadable, of course, even though it is a little older. The primary advantage to the thesis document is that it includes "C" code for all of the spherical trig & cartesian-coordinate conversion subroutines:

http://ir.library.oregonstate.edu/xmlui/bitstream/handle/1957/9875/Song_Lian_1998.pdf

While I could read the PDF online, I couldn't get Adobe Acrobat Reader to save the file to my local drive (blocked?) So, I dug it out of my browser cache subdirectory, and saved a copy. Since you may not know how to do this, I've attached the PDF, below.

-Taff

[Gerry in Quebec]

I've just revisited the "Geodesic Math" section my dog-eared copy of Domebook 2 and now realize that Class I, Method 4, described by Joe Clinton, is the Fuller-Kruschke "truncatable" method that we have discussed here in some detail at various times.

Here's Clinton's comparatively brief  (truncated?) description of Method 4, bottom left of p. 107:

"This method is an alteration of methods 1-3 allowing for truncation within the equatorial zone of the spherical form. It is developed with lesser circle as well as great circle arcs so that truncation may be done without requiring special elements. A set of parallel planes, falling in the equatorial region, are provided through the geodesic sphere, perpendicular to any given polar axis. Due to the less symmetrical characteristics of this method, it is used primarily for small frequency structures. The number of relative differences in edge lengths are greater than any of the other methods."

- Gerry in Quebec

P.S. According to one source I read a while back, the person who worked out Fuller's truncatable-dome subdivisions was Bill Wainright.

[TaffGoch]

Regarding the document, "Discrete Global Grids," Chapter 1, "Developing an Equal Area Global Grid by Lesser Circle Subdivision," the link that I posted is "dead." The website managers felt it necessary to move the document.

Doing a little digging, I found the updated URL: http://escholarship.org/uc/item/9492q6sm

Note that this is the 2002 version of the 1998 PDF file that I directly uploaded in my previous post. Since the download of the 2002 version seems to be working (for now,) at the new URL, you might want to grab the later version, while it's still available.

(Thanks for the "heads up," Charlie.)

-Taff

[AzaFran]

​This may be of interest:

http://bookzz.org/book/2202101/32d10f

Sharing is caring,

Thnks for your postes!

Blessings

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[Gebenus Arwid]

[Gebenus Arwid]

Hello Gerry

I had built my first geodesic dome 3v 3/8 dome 6 years ago. This dome is used on our christmas market as a Grünkohl location. In the summer we use my dome as a shell and sell cocktails from there on a big city event. Now I want to manufacture my first greenhouse dome. I would like to use the Kruschke 3v 5/9 method. 

As a connector between the struts, I would like to manufacture exact hubs at the CNC milling machine. I have also used your "3v-icosa-Fuller-Kruschke-derivation.xls" to calculating the correct dimensions and angles. 

Unfortunately, I can not calculate the correct angle inclination of the struts. According to Domerama.com the strut angles are: A 9,49 °, B 11,02 °, C 12,16 °, D 12,74 °. I have been able to calculate all the necessary dimensions and angles. But at the strut angles I do not get any further. 

I calculated for A 9.48871° and for D 12.740055 °. For the strut angles B and C I have too big deviations. Can you please help me? 

To understand how I can calculate the correct strut angles.

Best Regards

Arwid

Am Samstag, 25. Mai 2013 16:42:36 UTC+2 schrieb Gerry in Quebec:

[Gerry in Quebec]

Hello Arwid,

The angles at which to cut the ends of the struts, so they line up with a cylindrical hub, are as follows:

A struts: 9.49 degrees

B struts: 11.02 degrees

C struts: 12.17 degrees

D struts: 12.74 degrees

These angles are derived directly from the four chord factors using the following trigonometry formula:

Strut-end angle = arcsin (chord factor / 2)

For a hub-&-strut dome, you also need to know the "radial" angles that define the distribution of struts around the hubs. At a given hub, these radial angles will add up to 360 degrees. In contrast, the face angles of the five or six triangles converging at a hub will always add up to less than 360 degrees.

Here's a link to a simple Excel spreadsheet that calculates radial and face angles. It may be of use to you. The post is dated Feb. 17, 2017.

https://groups.google.com/d/topic/geodesichelp/JOVMOQLeOj4/discussion

I like your little cocktail bar.

- Gerry in Québec

[Gerry in Quebec]

Hi Arwid,

I looked at your spreadsheet and noticed that you have a heading titled "Radial Struts". This section contains four values: 975.59 mm, 1125.7 mm, 1236.1 mm and 1290.6 mm. I don't understand what these "Radial Strut" lengths refers to; and the equation used to calculate those lengths doesn't make any sense to me. The two variables in your equation are strut length (A, B, C and D values in column B) and what you call "Central Angles".  (The values you give for central angles are exactly one half of the actual central angles.)

You then use those "Radial Strut" lengths as variables to calculate "Radial Angles" using the law of cosines, the same equation used to calculate the "Face Angles". This is not the appropriate equation for calculating the radial angles.

So, I would suggest that, if you want to understand the relationships among the various angles and lengths, we first define all the terms you need to build a hub-&-strut dome. This includes face angle, radial angle, axial angle and central angle. All the key angles can be derived from the chord factors.

The key equations are in the "dome angles" spreadsheet I posted. In that spreadsheet, the "central angles" of struts are twice the angles at which you need to mitre the ends of dimensional-lumber struts for a hub-&-strut dome. I've attached a diagram to help illustrate some of the following dome-related definitions of terms:

Chord: The line or length between two points on the surface of a sphere -- in this case, the imaginary sphere that exactly surrounds a geodesic dome.

Spherical radius: The distance between the centre of the imaginary sphere around the dome and a point on the sphere's surface, such as the end of a chord.

Chord factor: Length of a chord when the dome's spherical radius = 1 unit.

Central angle: The angle between the two radii that lead from the spherical centre to the two end points of a chord.

Axial angle: The angle between a chord and a radius that leads to either of the chord's end points. The mitre angle (in degrees) for cutting the end of a strut in a hub-&-strut dome is 90 - axial angle. This mitre angle is half the central angle.

Face angle: The angle between two chords. The three face angles of a triangular face of the dome add up to 180 degrees. The five or six face angles converging at a hub add up to less than 360 degrees.

Radial angle: The angle between two lines connecting the centre point of a hub's outer face to the points where two adjacent struts connect to the rim of the hub. Radial angles divide up the outer surface of the hub into 5 or 6 sectors and therefore add up to 360 degrees.

Dihedral angle: The angle formed by the intersection of 2 faces of a dome, normally 2 triangles. A dihedral angle in this context is less than 180 degrees. (Not included in illustraiton.)

The terminology described above is what I and many others use to talk about domes. But, of course, not everyone uses exactly the same language, so you may find the same concepts have different names in other sources.

- Gerry in Québec

On Monday, February 27, 2017 at 6:57:41 PM UTC-5, Gebenus Arwid wrote:

Hello Gerry

thank you so much for your very fast feedback. 

Until now I had calculated all necessary lengths of the A-B-C-D struts, the face angles for the triangles 

and matched them with the 3D drawing of Taff Goch in Sketchup.  

At this time I try to understand your Excel Calculation for the Dome Angles.

I am not able to convert your Angle Calculation to the Kruschke Method.

Please look to my attached Excel Kruschke Dome Calculation.

I'm very sure my face and central angles are correct. 

The result of my radial angles can not be right.

Maybe you will be able to teach me about it???

Best Regards

Arwid

Am Samstag, 25. Mai 2013 16:42:36 UTC+2 schrieb Gerry in Quebec:

[Gerry in Quebec]

Arwid,

In my previous post, there was a text error in the diagram. In the middle illustration, it should be angle BAC, not CAC. The corrected diagram is attached.

- Gerry

[Gerry in Quebec]

Hi Arvid,

I spot-checked data on sheet 2 of your Excel file and the numbers look fine. Good luck with the project.

- Gerry 

On Wednesday, March 1, 2017 at 1:04:09 PM UTC-5, Gebenus Arwid wrote:

Hello Gerry,

Now I have proceeded according to your guide line and I believe that my result is satisfactory.

Best Regards

Am Samstag, 25. Mai 2013 16:42:36 UTC+2 schrieb Gerry in Quebec:

[Gebenus Arwid]

Hello Gerry

Thank you so much for your support.

Best Regards

Arwid

Am Samstag, 25. Mai 2013 16:42:36 UTC+2 schrieb Gerry in Quebec: