Class I Icosahedron 7V Truncations {3,5+}(7,0)

[Chris Kitrick]

The only fully symmetric truncatable Class I geometries are 3V(3,0), 4V(4,0), and 5V(5,0). For 7V(7,0) there are 3 possible variations, each allowing 4 different truncation levels. The attached diagram illustrates all three variations. All require the same number of different (unique) faces and edges.

Cheers,

Chris

[Eric Marceau]

Hello Chris,

That is an impressive piece of investigative work!  Well done!

Can you identify the algorithm/approach used for generating the grid geometry for each of the 3 scenarios presented?  I am guessing one is "traditional" (a.k.a. Clinton), one is Hernandez (a.k.a Mexican), and I'm not able to guess what the 3rd scenario might be (Kruschke?).

I am not asking for the logic for evaluating various scenarios, to identify those with the clean "slicing" planes, although, if you were willing to share, I am sure there are those within the Community who would be keen to see that piece of masterwork as well.

Thank you,

Eric

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[Levente Likhanecz]

hi chris,

3-4 years ago we've discussed these kruschke balls.

the 7v variants have some "ugly" snaking out of the flat lesser circles.

the 5v fully harmonic.
off_color -e J kruschke5v.off | antiview

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[Chris Kitrick]

Eric,

The method involved in finding the truncations is a combination of using spherical trigonometry, spherical coordinates, and Cartesian coordinates. Symmetry is maintained through geometric transformations and all computations are done on only three LCD sections of a single icosahedral face where the icosahedron is positioned vertex up on the z axis and the face used for computation is aligned on the x axis (see diagram). Determining truncations for 2V(2,0), 3V(3,0), and 4V(4,0) configurations are actually quite simple given the symmetrical constraints. The 5V(5,0) truncation requires some additional iteration to find the single valid configuration. Beyond 5V it is no longer possible to have all the lesser circles within the middle section of the icosahedron to lie on flat planes. The attached diagram illustrates why the 6V(6,0) and 7V(7,0) have no single solution. In both these cases there is one vertex that is required to lie on 3 lesser circles. It is possible to have a single vertex be co-planar on 2 lesser circles simultaneously, but not three. Of course as my original post illustrated there are sub-configurations possible where a subsets of lesser circles are planar.

From a efficiency perspective the truncatable class I configurations are sub-optimal, with relatively high edge and face counts.

Regards,

Chris

[Gerry in Quebec]

Eric & others,

Here's an old thread on class I truncatable domes, including an illustrated spreadsheet showing the derivation of the 3v icosa Fuller-Kruschke subdivision, using a procedure similar to the description posted by Chris.

https://groups.google.com/g/geodesichelp/c/zG4Mm__cVHI/m/5WbjH5gTLgIJ

That subdivision method is popular among U.S. and other dome building companies:

https://groups.google.com/g/geodesichelp/c/CRMbxEszbKo/m/VFyj5c-iKAAJ

 

Here's more on truncatable class I icosahedral domes including TaffGoch's SketchUp models:

https://groups.google.com/g/geodesichelp/c/UmbvayfzGGU/m/DIoi0I8oHQAJ

 

- Gerry in Québec

[Eric Marceau]

Thank you, Gerry,

I guess I have my "homework" laid out for me!  🙂

Eric

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[Gerry in Quebec]

Hi Lev,

I'd love to know how you convert an OFF file into an mp4 movie. Is it easy to do for those of us who don't have much experience with software and IT?

And by the way, Adrian Rossiter's collaborator on Antiprism, Roger Kaufman, has produced a great little OFF file viewer, separate from Antiprism & Antiview. It's all done on line in your browser. 

https://asliceofcuriosity.fr/blog/extra/polyhedra-viewer-antiprism.html?

Open the link above and then click on the "Choose files" option in control panel on the right side of the screen. That lets you navigate to an OFF file stored on your device. Just open the OFF file, and "OFF View" displays the polyhedron. It's not as powerful as Antiview, but it allows those who don't use Antiprism/Antiview to nevertheless view and manipulate a dynamic image of the polyhedron on screen very easily (with basic mouse moves).

Hats off to Roger!

- Gerry

[Levente Likhanecz]

hi gerry,

not that much magic behind off to mp4.

i just enter the command to display the object in antiview, then i record the spin with the windows 11 built in "snipping tool" / camera.

the antivew windows has a "spin" option. so i can spin the ball either by cursor arrows from keyboard, or startup a slow spin, then i start a recording session, which later saves the recorded movie screenshot in mp4 format.

from sketchup i exported the sphere into collada (dae), and some random on-line "dae to off" converter i converted the dae to off.

in case of antiprism, coordinates better fall between 0 and 1 (so unit radius dome), otherwise it will not handle itself strut coloring, and will not create/display edges.

so, just utilizing windows features.

cheers, lev 

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[Levente Likhanecz]

the screen recorder will work with that online antiview as well

On Sun, Dec 3, 2023 at 3:15 AM Gerry in Quebec <toomey...@gmail.com> wrote:

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[Chris Kitrick]

While the Kruschke method is useful it leads to the assumption that is a general solution paradigm for fully symmetric truncatable class one configurations. His cookbook only solves up to frequency 4. The solutions for frequencies 3 & 4 are determinate and can be found with multiple approaches. The 5 frequency is more complex requiring some iteration to arrive at the solution. Beyond 5 frequency there are no single solutions where every equatorial section is planar. Even Taff's earlier post (https://groups.google.com/g/geodesichelp/c/UmbvayfzGGU/m/EYz-kCNxHwAJ) shows a single solution for frequency 6, which does not exist. 

Chris

[Levente Likhanecz]

yep, taff's 6v has little cheat

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[Gerry in Quebec]

Lev, 

Thanks for the tips on the Windows snipping tool.

[Chris Kitrick]

If anyone is interested in the code that produces the results for all the class I truncations (2 thru 7) I will be happy to provide. The entire program is encapsulated in a single 'c' code file. There is one notation image available as well. The same technique is used for all solutions.

Cheers,

Chris

[Eric Marceau]

Thank you, Chris, for the generous offer.

My thinking is that it might be better to share it with the group directly, rather than individually.

When doing so, you might wish to specifically request that members perform an informal "peer review" to confirm its stated functionality and post the observations from their own usage of the tool (along with suggestions?).

Just a thought.  🙂

Eric

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[Chris Kitrick]

Since google groups are not efficient as repositories of information, I am releasing the aforementioned code to be openly available on github. The code is fully contained in a single file that includes all the necessary utility functions needed for trignometry, spherical coordinates, vector, and matrix operations. A single diagram is included that details the vertex numbering notation used within the program to derive all the solutions. Compiling and running the code should be easy. Understanding the methodology is a bit more work.

Essentially vertices are derived in sequence order. Many vertices for higher order configurations are dependent on prior derived vertices. Some vertices have no dependency on other vertices. For frequency 5 and above one or more iteration processes are needed to determine unknown positions. Questions welcome.

Here is the link: https://github.com/ckitrick/icosa_classI_truncation/releases/tag/v1.0

Good luck!

Chris

[David haughwout]

I'm in, thanks... 

   I just finished a 3/4 3V frame and screwed it together with pocket screws. I'm playing with cnc hubs and cnc plotted strut trace templates for sawing non compound angles. (Super easy) . It went up as planned ..The door was a tough design choice.The dome is made from all recycled 2*4 flooring for a deck deck ....I got 35 dollars invested (screws and platform nails) plus the cost of a  damaged 25%off a 3/8 pvc  sheet ...forgot the cost...roofing is another department (not done)...I designed the dome  from scratch. Started with an icosahedron using Rhino V4 ...building domes virtually allowed me to come up with assembly logistics. and the first one I built is under helping me with roofing ideas. A funny thing about this project, the amount of labor would be nearly the same even if it's scaled it way up... at lest frame wise. 

Dave

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[Eric Marceau]

Thank you again, Chris!  I won't argue with your choice.  GitHub is even better!

BTW, beautiful work on that Dymaxion map!  To me, it exemplifies a labour of love.

Eric

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