Lesser Circle Domes?

[andrew777]

Hello TaffGoch and all:

TaffGoch, you drew a geometric depiction of a lesser circle sphere for a bamboo dome, I am interested in seeing the underlying polyhedron and how you chose all of the lesser circles. I would like to draw the lesser circles for different frequency domes. If you could please place the underlying frequency dome under your model and depict where the circles cross the surface of the dome to obtain the uninterrupted circle paths it would be greatly appreciated. Do lesser circle domes favor certain frequencies and methods?

Always trying to learn new things,

Andrew

[TaffGoch]

The discussion, from the past, where this was introduced:

https://groups.google.com/group/geodesichelp/browse_thread/thread/ef07ec874d36e217

Andrew,

There really isn't an underlying polyhedron, other than what is most often employed (the icosahedron.)

The reason that I stated that there was only ONE solution to this presentation, and the reason other readers could establish identical numbers, is because a particular "intersection" characteristic must be satisfied.

In the attached image, the plane intersections around the equator must lie precisely on the equator. I've colored, yellow and green, a couple of the planes. Note that their crossing junction, on the circumference, is on the equator.

If the parallel-plane pairs were spaced any closer, or farther apart, those intersections would move off the equator -- so, only ONE solution, for this particular example.

______________________________

Other parallel-plane separations are used to produce different frequencies, but they too rely on a comparable "rule" for proper positioning of intersections. This is the basic underlying premise for Kruschke's subdivision method, depicted graphically, without mathematical calculations.

-Taff

[TaffGoch]

(I should have pointed out that the GIF image is animated, so download it, or view it full-size, to see the equatorial "bits")

[andrew777]

TaffGoch,

Thank you for the animated illustration and explanation.

 I know that the group has already discussed the subject I am talking about, however, with all the geodesic methods talked about, I can't ferret out a program from the group, that produces the Kruschke method mathematically for many frequencies.  Do the Temcor domes use the Kruschke method?

Has anyone come up with a generalized mathematical solution for the above problem supplying chord factors, dihedral and axial angles, if so, could you please point me to them. 

I haven't seen David Kruschke's book but I surmise it breaks down the lesser circles problem into equations and algorithms for a generalized solution or that a generalized solution could be derived from the book. With the advent of all-purpose printer scanners why doesn't Kruschke supply copies of his book to anyone who wants it? I'm willing to pay $45 for a 45 page book.

Any help will be appreciated,

Andrew

[Gerry in Quebec]

Hello Andrew,
Dave Kruschke posted a few messages here late last year (November?),
including an offer to one member (Jim F.) to loan him a copy of his
1972 book on geodesic math ( "Cookbook of Geodesic Geometry"). So
maybe you can get a copy that way -- just "reply to author".

A couple of points about the Fuller-Kruschke truncations, as applied
to class I icosa domes....

My own work with this "method" suggests their approach works fine up
to frequency 5. At frequency 6, not all the strut pathways can be made
into lesser circles or great circles. The 7/18th and 11/18ths cut-off
points refuse to cooperate! But this really isn't important for dome
construction purposes because, from a single 6v design, you can have
domes that sit flat at the 8/18, 9/18 (hemisphere) and 10/18
truncations.

Dave's booklet shows the derivation of the chord factors for the 2v,
3v and 4v icosa using spherical trigonometry. The 3v and 4v have been
the ones of interest to builders over the years because they provide a
flat base at various cut-off points, as compared with method 1 layouts
which are problematic because they undulate at non-hemispheric
truncations.

In a separate message, I'll post a jpg of Antiview images of the 5v
icosa lesser-circle dome (solution 9 in my numbering scheme), which
sits flat at all key truncations, plus a photo of a 5v, low-profile
greenhouse skeleton (solution 2a, flat at 7/15 & 8/15). So, the notion
that odd-frequency icosa domes don't sit flat can be chucked out the
window.

- Gerry Toomey in sunny Quebec

On May 4, 10:38 pm, andrew777 <andrewj...@gmail.com> wrote:
> TaffGoch,
>
> Thank you for the animated illustration and explanation.
>
>  I know that the group has already discussed the subject I am talking
> about, however, with all the geodesic methods talked about, I can't ferret
> out a program from the group, that produces the Kruschke method
> mathematically for many frequencies.  Do the Temcor domes use the Kruschke
> method?
>
> Has anyone come up with a generalized mathematical solution for the above
> problem supplying chord factors, dihedral and axial angles, if so, could
> you please point me to them.
>
> I haven't seen David Kruschke's book but I surmise it breaks down the
> lesser circles problem into equations and algorithms for a generalized
> solution or that a generalized solution could be derived from the book.
> With the advent of all-purpose printer scanners why doesn't Kruschke supply
> copies of his book to anyone who wants it? I'm willing to pay $45 for a 45
> page book.
>
> Any help will be appreciated,
>
> Andrew
>
>
>
>
>
>
>
> On Friday, May 3, 2013 10:48:43 PM UTC-4, TaffGoch wrote:
>
> > (I should have pointed out that the GIF image is animated, so download it,
> > or view it full-size, to see the equatorial "bits")
>

> > On Fri, May 3, 2013 at 9:46 PM, TaffGoch <taff...@gmail.com <javascript:>>wrote:
>
> >> The discussion, from the past, where this was introduced:
>

> >>https://groups.google.com/group/geodesichelp/browse_thread/thread/ef0...

[Gerry Toomey]

- Antiview images of 5v icosa dome (solution 9). Lesser circles make for a flat base at the 6/15, 7/15, 815 & 9/15 truncations.

- Photo of a low-profile 5v dome framework (hub & strut, 2x6) for a greenhouse, solution 2a, 7/15 truncation.

 

Gerry in Quebec

[andrew777]

Hello Gerry,

Great information.

You hit the essence of what I wanted to understand about class I icosa structures. What other classes can derive lesser circle domes - what about class II method 3 structures? Are there ways to obtain higher frequency lesser circle domes? I'm interested not only in the flat bases but also in the uninterrupted strut/circle pathways of the overall surface of the domes

Please tell me about software, what programs can I use to parametrically create lesser circle domes.

I have used antiview and antiprism before, how do you obtain lesser circle domes from antiprism. I saw  something about c2m3icosa.exe and antiview on the site can that be used for the lesser circle domes?

Is that the only software you use to  formulate  lesser circles structures?

Thanks for the information and and pictures,

Andrew 

 

[Gerry in Quebec]

Andrew,

You may want to check out CADRE Geo, version 6. This software has a
"spheric" dome option that lets you create horizontal lesser circles,
with quite a bit of design flexibility. However, I don't think you can
design the oblique strut pathways to be lesser circles. You can
probably download a trial copy of the software free of charge.

www.cadreanalytic.com

I don't think it's possible to obtain high-frequency class I domes in
which all pathways (not counting the partial great circles connecting
pentagon centres) are lesser circles. As I mentioned, I couldn't make
the 6v icosa completely "lesser circle". But you can certainly level
the base of higher-frequency domes, Fuller-Kruschke style, that is,
maintaining icosa symmetry, with no need to tinker with strut lengths
in the bottom row of triangles. I've been able to apply the concept to
8 and 9 frequency domes in addition to 5v and 6v. I haven't tried it
on a 7v.

Off-hand I don't know whether you can make lesser-circle domes for
classes II and III. I suspect you can for class II. I'd have to fumble
around with some CAD models for an hour or so to get a clear idea as
to how much "pretzel twisting" is possible without breaking symmetry.

The approach I've been using for modeling domes and other shapes is
pretty old-fashioned, I guess. I do the math in Excel using a mix of
planar and spherical trigonometry to establish the vertex coordinates.
Then I convert those into cartesian coordinates and make a list of all
the polyhedron's faces (triangle 1-2-3, 2-4-5, and so on). Often I use
the Solver function either to improve the design or for error-
checking. The coordinates and list of faces are compiled into an OFF
file (simple text file with an OFF extension) for display in Antiview.
I'm not aware of the c2m3icosa.exe utility of Antiprism you mentioned.

A while back I posted some images of great-circle domes based on the
patent by John D. Jacoby. This geometry may be of interest to you.
Just search for the word Jacoby within the GeodesicHelp group. There
are two threads with identical subject names. Here's a link to
Jacoby's patent:

http://www.google.com/patents?id=DREuAQAAEBAJ

Lastly, the strut pathways of Temcor-type domes created by Don Richter
are lesser circles as long as the geometry respects certain
restrictions on the underlying parent polyhedron. If you stray from
those restrictions (e.g., by choosing an inappropriate ratio of dome
height to footprint diameter), then many of the strut paths will
deviate from the lesser-circle path where they cross from one sector
of the dome to another.

I hope this helps.

- Gerry in Quebec

On May 7, 3:08 pm, andrew777 <andrewj...@gmail.com> wrote:
> On Tuesday, May 7, 2013 12:33:39 PM UTC-4, Gerry in Quebec wrote:
>
> > - Antiview images of 5v icosa dome (solution 9). Lesser circles make for a
> > flat base at the 6/15, 7/15, 815 & 9/15 truncations.
> > - Photo of a low-profile 5v dome framework (hub & strut, 2x6) for a
> > greenhouse, solution 2a, 7/15 truncation.
>

> > *Gerry in Quebec*

[Dick Fischbeck]

Katrina is into small circle domes called Charter-Sphere domes, invented by her dad, TC Howard.

https://www.facebook.com/pages/Synergetics-Inc-Architecture-and-Engineering/191074047574927

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[Gerry in Quebec]

Andrew,
A correction.... I said in the previous post that the strut paths in
Temcor domes could all be lesser circles if certain restrictions on
the underlying polyhedron are met. Now, in looking at a few models, I
have to conclude that, with the exceptions of the dome footprint and
the edges of the polygon at the dome apex, there's probably no across-
the-dome or around-the-dome path that follows a purely lesser-circle
path. At sectoral boundaries, all strut paths deviate from
circularity. Take a look at one of Taff's SketchUp models (e.g., the
Amundsen-Scott South Pole dome) and you'll see what I mean.

- Gerry in Quebec

[Gerry in Quebec]

I've always liked this 5v, lesser-circle dome designed by Buckminster
Fuller. It's the religious center at Southern Illinois University in
the USA. Pretty sure "solution 9" shown in illustrations I posted
earlier today is the same geometry.

- Gerry



http://www.cardcow.com/99614/religious-center-southern-illinois-university-edwardsville/

> *Gerry in Quebec*
>
>  5v-icosa-Toomey-solution-9.jpg
> 216KViewDownload
>
>  Frame-after-snowfall.jpg
> 548KViewDownload

[andrew777]

Hello Gerry,

All the bamboo dome and rigid tensegrity domes, I believe, have interrupted circle paths as shown in the pic attached.  If I remember correctly this is a 12 frequency dome.

I just wonder what method was used to obtain  the interrupted circle pathways mathematically, similar to the one TaffGoch sent me in the gif of the bamboo dome.

Since you do your calculations with Excel you probably don't remember this e-mail:

"Dondalah

10/31/11

Hi Gerry,

This is the procedure that Adrian gave us.  The example is for a 12v sphere.

c2m3icosa  -f  12   >tri.off

off_trans -R 0,0,1,1,0,0,0,1,1.61803398874989484,1,1,1 tri.off | poly_kscope -s I > sph.off

antiview sph.off -e 0.005

Cheers,

Dondalah

   

download

 c2m3icosa-exe.b64

I will study the attached pic to see if I can draw the structure in sketchup.

Thanks for your help.

Andrew

[andrew777]

Hello Gerry,

Correction, bamboo domes and original rigid tensegrity domes of this type (non nexorade)  all have  un-interrupted circle paths/struts on their surface. The 12 frequency tensegrity dome has un-interrupted circle paths/struts on its surface

Andrew

[TaffGoch]

Hmm, I count an 8-frequency subdivision (Class-I)

-Taff

[andrew777]

  Hello TaffGoch,

I didn't count the frequency in the picture, the picture is from 1962 ,University of Southern Illinois, of the 72 foot tensegrity dome that Bucky's students built according to his specifications:

this is a direct quote from Bucky's book "Tensegrity":

795.01   We can take advantage of the fact that lumber cut at the "two-by-four" size represents the lumber industry's most frequently used and lowest-cost structural lumber. The average length of the two-by-fours is 12 feet. We can take the approximately two- inch dimension as the mid-girth size of a strut, and we can use an average of 10-foot lengths of the tensilely strongest two-by-four wood worked by the trade (and pay the premium to have it selected and free of knotholes). We can then calculate what size of the spherical dome__and what frequency__will produce the condition of "just-kissing" contact of the two-by-four ends of the islanded two-by-four chordal struts with the mid-girth contact points of one another. This calculates out to a 12-frequency, 72-foot-diameter sphere that, if truncated as a three-quarter sphere, has 20 hexagonal openings around its base, each high enough and wide enough to allow the passage of a closed body truck.

795.02   We calculated and produced such a 72-foot, three-quarter-sphere geodesic dome at the Edwardsville campus of Southern Illinois University in 1962. The static load testing of all the parts as well as the final assembly found it performing exactly as described in the above paragraphs. The static load testing demonstrated performance on the basis of the load-distributing capabilities of pneumatics and hydraulics and exceeded those that would have been predicted solely on the basis of continuous compression.

You are the expert, I am just trying to learn. Any help will be appreciated.

Whether it is 12 or 8 frequency it a higher freguency than 5.

Andrew

[Gerry Toomey]

Andrew,
Thanks for the photo of the 72-foot dome. I've attached a line illustration that makes it a little easier to see strut lines in the context of a fully triangulated class I dome (8v, as Taff pointed out, not 12 v). For this diagram I leveled the base ring into a lesser circle. Fuller and/or his colleagues used a different geometry to get the various lesser circles seen in the photo, though the base does seem to undulate.

 

When I said earlier that the maximum frequency for a lesser-circle class I icosa dome was 5, that was based on the restriction that the dome be fully triangulated (5-way and 6-way vertices with 4-way vertices around the footprint). Removing that restriction, perhaps you can have lesser-circle domes at higher frequencies such as 8v, in which you have only 4-way vertices in the main part of the dome and 3-way vertices around the footprint.

 

By the way, for a fully triangulated 6v icosa, class I sphere, only one of the repeating non-great-circle pathways strays from lesser circle status. So it's a "near miss", falling a wee bit short of the high-level symmetry seen in the Fuller-Kruschke division of the 3v and 4v icosa, class I, and in the 5v Fuller-Sadao religious center dome in Edwardsville, USA. In the attached diagram (6v, solution 8e), the thick green line is an example of one of the strut paths that doesn't quite make the grade of lesser circle.

- Gerry 

 

 

 

 

 

 

On May 8, 7:15 pm, andrew777 <andrewj...@gmail.com> wrote:
> Hello Gerry,
>
> Correction, bamboo domes and original rigid tensegrity domes of this type

> (non nexorade)  all have * un-interrupted* circle paths/struts on their
> surface. The 12 frequency tensegrity dome has *un-interrupted* circle

[Gerry in Quebec]

Someone has reminded me that the SIU-Edwardsville religious center
dome was a joint effort by architect Shoji Sadao and Buckminster
Fuller. The two also collaborated on the giant Expo '67 dome in
Montreal. Here's a nice black-&-white pic of the two in front of the
Edwardsville dome.
- Gerry in Quebec

On May 7, 7:06 pm, Gerry in Quebec <toomey.ge...@gmail.com> wrote:
> I've always liked this 5v, lesser-circle dome designed by Buckminster
> Fuller. It's the religious center at Southern Illinois University in
> the USA. Pretty sure "solution 9" shown in illustrations I posted
> earlier today is the same geometry.
>
> - Gerry
>

> http://www.cardcow.com/99614/religious-center-southern-illinois-unive...

[Gerry in Quebec]

This time with the link!
http://www-sul.stanford.edu/depts/spc/exhibits/fullersadao.html

[andrew777]

Hello Gerry,

Based on the the picture I sent of the 8 Frequency Lesser Circle dome, and Hugh Kenner's illustrating a 12 frequency rigid tensegrity dome and Bucky Fuller's statement in Synergetic I of a 12 frequency rigid tensegrity dome, I believe some knowledge has been lost. I hope we can find it.

Your quote: "Fuller and/or his colleagues used a different geometry to get the various lesser circles seen in the photo, though the base does seem to undulate."

I believe the undulation is due to the fact that it is a basket weave dome. What do you surmise is that the geometry that Bucky  or his colleagues is using?

Thanks,

Andrew

[TaffGoch]

I've realized that I have only 3v Kruschke domes posted in the 3D Warehouse, and that I should produce a 3D model that includes as many Kruschke frequencies as possible, for upload.

I've done 3v and 4v, fairly confidently, and just added a 5v. Since Kruschke's book only goes up to 4v, I have no reference to check my work.

Gerry, are you willing to share your chord-factor results for > 4v Kruschke breakdowns? Attached below are my 5v results.

-Taff

[Gerry Toomey]

 

Here are the chord factors I get for the 5v icosa, class I, Fuller-Kruschke style, using Excel. Very close to yours, Taff. For some numbers, there are small differences at the 5th decimal place.

 

Yes, I can post chord factors for some other level-base layouts at higher frequencies. But as I mentioned earlier in this thread, beyond 5v I don't think it's possible to make class I icosa domes sit flat at all major truncations in the Fuller-Kruschke style, namely those truncations lying within the icosahedron's central band of 10 parent equilateral triangles. But, of course, there are partial solutions that are potentially quite useful for dome construction.

 

I'll dig out solution 9a for the 6v icosa and post it soon. Blair at Dome Inc. is the only person who has so far seen details of that one.

 

I haven't looked into 7v, but can do so and compare results with any SketchUp model you might produce.

 

Andrew.... did you want specifics on how to calculate dihedral angles between triangular faces of a dome?

 

- Gerry Toomey in Quebec

 

 

 

 

 

 

>  Kruschke spheres.png
> 189KViewDownload
>
>  Kruschke sphere; 5v (1).png
> 142KViewDownload
>
>  Kruschke sphere; 5v (2).png
> 107KViewDownload
>
>  Kruschke sphere; 5v (3).png
> 66KViewDownload

[Hector Alfredo Hernández Hdez.]

Very good work Gerry.

2013/5/15 Gerry Toomey <toomey...@gmail.com>

--

[Gerry Toomey]

On May 11, my post on lesser circle domes got disconnected from the main thread ("Lesser Circle Domes?") because I didn't type the subject line correctly. Here's the message and attachment.

- Gerry

 - - - -

 

Andrew,
The 5v and 6v illustrations I posted earlier depict domes whose struts come
together at a single vertex, rather than rigid tensegrities where struts
are interleaved, under and over, as in the 8v Edwardsville dome. I don't
know what numbers Fuller used for the Edwardsville rigid tensegrity...

I took a quick look at the last chapter, Rigid Tensegrities, of Part I of
Hugh Kenner's book. The illustrations contain very little detail, but the
text makes it clear that the "dips and gaps" of tensegrity structures are
reduced (as frequency increases) to the point where you can slide one 2x4
over another and bolt them together, with the tightened bolts serving as
the tensegrity tension members. This is the "just-kissing" contact Fuller
referred to in the passage you quoted. But there didn't seem to be enough
information in Kenner's book, at least for me, to be able to apply the
tensegrity equations to make an 8v rigid tensegrity. Taff is your best bet
for learning more about rigid tensegrities, nexorades, rotegrities.
.
On the topic of lesser-circle domes, where all vertices lie on the
spherical surface, I'm attaching a jpg of an 8v icosa dome layout (solution
5) that follows the general lines of the 8v Edwardsville dome, but not
exactly. Like that dome, the strut patterns are not fully triangulated --
it defines a set of hexagons, pentagons and triangles, rather than
triangles only. Removing the "omni-triangulation" restriction allows for
ALL pathways to be lesser circles, as I suspected.

In this dome, there's no ring of struts along the equator. If you "connect
the dots" along the equator,  though, you'll see that the resulting pathway
undulates. However, the base of this dome, unlike that of the Edwardsville
rigid tensegrity, does not undulate. Ditto for the strut paths that mark
the perimeters of pentagons higher up.

- Gerry in Quebec

On May 9, 11:44 pm, andrew777 <andrewj...@gmail.com> wrote:
> Hello Gerry,

[Gerry Toomey]

Here is solution 8e for the 6v icosa, class I. In this subdivision, a dome will sit flat at the 8/18, 9/18 (equator), and 10/18 truncations, but not at the 7/18 and 11/18 truncations. It is like the Fuller-Krushcke subdivision of the 4v icosa because the strut pathways one row of triangles above the equator and one row below form lesser circles, and the central pathway of struts exactly follows the equator (a great circle). It also has something in common with the Fuller-Kruschke 3v icosa: the angular distance between the parallel strut pathways just above and below the equator is 21.6246 degrees.

Taff, I've also attached a comma-delimited Excel file containing the Cartesian coordinates of the 28 vertices of the principal parent triangle.
- Gerry in Quebec

[TaffGoch]

As you state, Gerry, the central three planes of this 6v solution are level, for application as ground planes, and produce the most useful of the disk intersections.

The top- and bottom-most disks do not produce intersections leading to a level ground plane, but I think this calculation provides the three central ground planes that folks are most likely to employ.

(The "csv" file is appreciated, as it made modeling much quicker, and less-complicated.)

_____________________________

If higher frequencies do not produce solutions that resolve into handy, level ground-plane intersections, perhaps, we can stop at 6v. I question how many people will build a Kruschke dome over 6v. I am, however, willing to model, and would welcome alternative opinions.

-Taff

[Gerry Toomey]

Thanks for the images. Here's a summary of 3v to 6v icosa, Fuller-Kruschke, to complement yours.

- Gerry in Quebec where spring is being really stubborn

[andrew777]

Hello Gerry,

Thanks for the work on the 8v-icosa-lesser-circle-dome-Tri-Pent-Hex (8v icosa dome layout (solution 

5) ) please send a sketchup model of this particular solution, I want the lesser circle one.  Equations and algorithms are what

I prefer, so that I could repeat the groups results and experiment further. I'm going to attach some pics of my nexorades, sorry they are not artistic renderings like TaffGoch. 

Andrew

[TaffGoch]

Andrew,

I see that your 8v reciprocal frame is a Class-II (triacontahedron) subdivision. I had not tried Class-II, yet, but had planned to do so.

Spurred into action by your 8v version, I worked up a 6v Class-II this afternoon.

-Taff

[TaffGoch]

Andrew,

This SketchUp model is a close approximation of Gerry's 8v "tri-hex" geodesic dome/sphere; solution 5. This is not the only possible arrangement of the disks (as "solution 5" implies.) Other "solutions" can be produced, which are also valid, and have slightly different appearance, due to using different intersection criteria for calculations/solving.

Note that this model is, upon opening, in the "View" mode of "Edges" only. You can change the display mode, to show the disks, rather than just their outlines.

I've also employed a "Section plane" to display a dome, rather than the full sphere. You can turn off the section plane with just a click. Alternatively, you can move the section plane to one of the other "levels" of dome truncation, to change the appearance.

The displayed appearance is also influenced by SketchUp's "Fog" feature, to diminish the appearance of the portions of the sphere that are farther away from the viewpoint. (Open the "Fog" window to display controls; to turn off.)

-Taff

[TaffGoch]

Andrew,

I played with photometrics, a bit, last night, to get an impression of the relative scales of the components in the b&w photo. I noted that the lesser circles appeared to be, approximately, separated by the same measure (the green lines, and the central gap between the green-line endpoints.) Also, the individual "strut" lengths between intersections (yellow segments) of the topmost horizontal circle look to be the same approximate length, except for the pentagon base (blue.)

Knowing the golden-ratio relationship between the pentagon base and the length of a pent-star "arm," I could establish the circumference of the top horizontal circle, relative to the pentagon center (and subsequently, relative to the vertex of an icosahedron, which shares the pentagon center point.)

I 3D modeled these relationships to provide a "solution" of my own. The circles are more-nearly equivalently-spaced than Gerry's "solution 5," but he could have a solution (of a different number designation) that matches. 

This just goes to show that there is no one solution. What you get, in the end, is established by the starting criteria.

-Taff

[andrew777]

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

[Gerry in Quebec]

Andrew and others interested in dome math,

I'll try to put something together, namely an algorithm (equations and
their sequence of use) with line drawings and text. This will show you
how to find vertex coordinates and chord factors for Fuller-Kruschke-
style domes. However, I think the largest number of lesser circles
(usable truncations) I can include in a single algorithm is 3. In the
case of the frequency 5, class I icosa, I had to use Excel's Solver
function for the final step. You'll recall that the high-profile 5v
dome has four parallel, horizontal lesser circles. If I use the 4v as
an example, that at least will show you the procedure, using spherical
trig, to level the base of any dome at pairs of truncations (5/12 and
7/12 in the case of the 4v).

This will be a bit time-consuming as I need to "translate" bit and
pieces from Excel math notation into English to make it transparent.
So, before I put any effort into this, please let me know whether this
is the type of presentation you're looking for.

Some time ago I posted a "bare bones" algorithm for the Temcor method.
If you can work your way through that file, you should be able to
design various types of Temcor domes, by building on the algorithm.
The variables are the number of rows of triangles, the ratio of dome
height to base diameter, and the number of triangles converging at the
dome apex. I also posted a procedure for deriving vertex coordinates
and chord factors for class II icosa, method 3 domes, as outlined in
Appendix 1 of Kenner's book.

You should be able to find these old threads using the "Search this
group" function. The attachments are Excel files. I'd welcome feedback
on that stuff from group members. A note of caution..... I'm not a
professional mathematician, just a dome enthusiast who enjoys
geometry, carpentry, and reading about architecture.

One advantage of SketchUp is that it does all the math for you in the
background. So it really does seem an excellent way to design and
model geodesic and other structures, as compared with the step-by-step
math I've been using. Also, SketchUp isn't prone to the human error
typical of complex and/or repetitive mathematical computation. I am,
still, only a passive user of SketchUp. I use it to view, inspect and
measure components of other people's models, mainly Taff's; I don't
know how to create my own models. The main problem I've had with
SketchUp so far is inconsistent values when using the tape measure.

Cheers,
- Gerry in Quebec

> 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

[Gerry Toomey]

Variation on a theme.... Here's the fully triangulated version of the 8v icosa lesser-circle dome we've been discussing. As with Taff's "solution", the planes of the four horizontal, parallel lesser circles are evenly spaced.

- Gerry in Quebec

 

On May 19, 6:30 pm, TaffGoch <taffg...@gmail.com> wrote:
> Andrew,
>
> I played with photometrics, a bit, last night, to get an impression of the
> relative scales of the components in the b&w photo. I noted that the lesser
> circles appeared to be, approximately, separated by the same measure (the
> green lines, and the central gap between the green-line endpoints.) Also,
> the individual "strut" lengths between intersections (yellow segments) of
> the topmost horizontal circle look to be the same approximate length,
> except for the pentagon base (blue.)
>
> Knowing the golden-ratio relationship between the pentagon base and the
> length of a pent-star "arm," I could establish the circumference of the top
> horizontal circle, relative to the pentagon center (and subsequently,
> relative to the vertex of an icosahedron, which shares the pentagon center
> point.)
>
> I 3D modeled these relationships to provide a "solution" of my own. The
> circles are more-nearly equivalently-spaced than Gerry's "solution 5," but
> he could have a solution (of a different number designation) that matches.
>
> This just goes to show that there is no one solution. What you get, in the
> end, is established by the starting criteria.
>
> -Taff
>

>  Fuller-Kruschke; 8v (C).jpg
> 407KViewDownload
>
>  Fuller-Kruschke; 8v (D).png
> 334KViewDownload

[norm...@gmail.com]

this is an older thread that I found during some searches but could someone relate bucky's statement:

will produce the condition of "just-kissing" contact of the two-by-four ends of the islanded two-by-four chordal struts with the mid-girth contact points of one another. 

and how one would procure the strut lengths to make this kissing/mid-girth contact point rigid Tensegrities?

I really like this design but I'm having a hard time understanding how to replicate it.

thanks

[Bryan]

Here's a variation on your variation Gerry...

This is an 8V Kruschke with 5 truncation planes. 10, 11, 12, 13 and 14/24

[Gerry in Quebec]

Nice work, Byran. For the 8v, that's the maximum number of parallel, lesser circles between the non-polar pentagons. So your model can be considered another member of the collection of class I icosa "Kruschke spheres" assembled by Taff many moons ago. The only one now missing between frequencies 3 and 8 is the 7v, which I guess has 4 parallel truncation planes.... (I don't recall ever seeing a model of a truncatable 7v.)

For anyone interested in these geospheres, which allow for flat-based domes at cut-off points other than the equator, here's a link to the SketchUp models:

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

Topic: Econodome See Taff's post & attachment dated 10/30/15. Gerry in Québec

[Ashok Mathur]

Dear Bryan,

Forgive me if my curiosity has got better of my manners, but I do want to know the context in which you made this variation of 8V dome.

Can you share the background and what alternates did you discard before venturing to make this variation?

What structure did you have in mind while deriving this variation/

Please do share what you feel comfortable sharing.

Pure curiosity as I can not imagine me making a 8v dome ever in future.

Regards

Ashok

Regar

Regards

Ashok

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[Bryan L]

Hi Ashok, I have no plan on building a dome at that frequency.
The context was purely my love of modelling the different dome geometries in sketchup.
I only recently bothered to learn the kruschke method after coming across this thread. So then I modelled the lower frequencies up to what Gerry and Taff had done, and wanted to see what an 8v would give.
I will say I like "eveness" of the  kruschke geometry. Would make it easier to insert doors, windows in a real world structure...

[Ashok Mathur]

DearBryan,

Its a pleasure to learn that there is now a second SketchUp modeler in the group.

Best of luck and carry on.

Regards

Ashok

Regards

Ashok

--

[Bryan L]

Thanks Gerry,
Yes the 8v can't have a plane including the b struts on the pentagon edges.
I am guessing same for the 7v so as you say only 4 planes.
I will get to the 7v. I am just getting my head around solving the 5v with trig because before, to get going,  I just copied a strut from what you & Taff had done. I can't imagine doing it iteratively via construction - so much work - but, it wouldn't surprise me if Taff saw a shortcut...

--

[Bryan L]

Ashok, there are many here before me use sketchup....

[Bryan]

On Saturday, 25 February 2017 22:48:01 UTC+11, Gerry in Quebec wrote:

The only one now missing between frequencies 3 and 8 is the 7v, which I guess has 4 parallel truncation planes....

Here it is... :-)

[TaffGoch]

...and, scaled-down, to unit (1.0) radius, so that chord-factors can be measured (with SketchUp <Tape Measure> tool.)

-Taff

[Bryan]

If you knew how much effort went into solving that in excel, I hope a small oversight could be forgiven... ;-)

[Gerry in Quebec]

Thanks, Bryan. Simple, elegant solutions or designs often require a huge effort.

As with the 8v, good job. 

- Gerry