Re: 12' plywood hex dome with flat base (4V class II) made from 6 sheets of underlayment. Photos and pattern. What do you think of hex domes?

[Dick Fischbeck]

Robert - I like this a lot. Good work. Well done.

Dick

On Thursday, August 13, 2015 at 4:15:23 PM UTC-4, Robert Clark wrote:

This spring I built this 12 foot diameter dome, 8 foot high in the center.

[Ken G. Brown]

Nice!

What would the end result look like if you filled in all the spaces in the nesting pattern? ie, only one cut between pieces?

Ken G. Brown

On Aug 13, 2015, at 14:15, Robert Clark <clark.rob...@gmail.com> wrote:

This spring I built this 12 foot diameter dome, 8 foot high in the center.

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<plywood dome 4V class II - COLOR.JPG><dome parts nesting pattern.JPG><CAM00137 (2).jpg><CAM00141 (1).jpg><CAM00145 (1).jpg><CAM00138 (1).jpg>

[TaffGoch]

Robert,

You are to be commended.

I like the style. I played with the concept, back in 2012, to produce this Class-I, 3v model (never progressing further):

I wanted to produce a usable, good-size sphere composed of only two unique parts. Yes, there is some distortion of hexagons, immediately adjacent to the pentagons, but I think the design retains an attractive appearance.

(The "circles" aren't components, but are the rounded tips of each "Y" component. In this model, you can "see through" the top-most piece, so the rounded tip of the underlying piece is visible. Two overlapped semi-clrcle tips produce each circle.) 

The concept deserves more of my attention. 

"Where does the time go?"

-Taff

[TaffGoch]

Rob,

I meant to ask about the cutting. Did you employ a laser cutter? (I don't see burned edges inherent to laser cutting.)

-Taff

[Ken G. Brown]

What I mean is not having all filled in, but modify the shapes as nested so that there is only one common cut line between pieces. Then see what it looks like on the dome. I envision the holes in the structure would be a bit partially filled and perhaps be a somewhat different shape. 

Ken,

from my iPhone

On Aug 15, 2015, at 17:00, Robert Clark <clark.rob...@gmail.com> wrote:

I think you're asking what it would look like with all the holes filled in? The shape is based on a 4V class II geodesic sphere which I've attached here. I essentially removed the corners of all the triangular panels. Then, I played with the lengths of the red and blue pieces so that it would have a flat base.

I also made a modified version that only uses 2 unique panels (3 if you don't count flipping or mirroring). It's the pink and grey model I've attached. This would also make a pretty cool flyeye dome.

-Rob in Massachusetts

<4V class II - 240 sided geodesic.jpg>

<plywood dome - 4V class II - 2 panel.JPG>

<4v class II dome.JPG>

<new flyeye dome 4V class II - 02.JPG>

[Ken G. Brown]

Yes, that is what I suppose I was getting at, no waste and less cutting. 

Ken,

from my iPhone

On Aug 15, 2015, at 21:01, Robert Clark <clark.rob...@gmail.com> wrote:

I see what you're saying now. It would be a lot of really odd quirky looking pieces. But, as long as the bolt holes are in the proper places and the pieces can still flex without any buckling, then I guess it would still work. And, almost no waste plywood would be left over.

- Rob

[Scott Ihrig]

Robert - I recently built a 10ft diameter 3v geo dome from ripped 2x4's. Been nuts over domes for a long time and I finally built one. I absolutely love yours. Wish I could build one like it. Awesome job!

[Scott Ihrig]

Hi Rob - Love your dome and would like to build one. Can you share your dxf files for this dome pattern or sell them perhaps?

On Saturday, August 15, 2015 at 8:59:16 PM UTC-5, Robert Clark wrote:

No, it's cut with a CNC router using my supplied dxf files. I used an online nesting program to lay the pieces out on the plywood as compactly as possible. I had to really search on Craigslist to find someone that does CNC routering. I live in Massachusetts and the CNC guy was in Connecticut - a 2 hour drive.  He charged $50 per plywood sheet and it required 6 sheets. I ordered the bolts from McMaster-Carr. All in all the dome cost about $500 labor and materials.

If I do another dome, I want to find a place that laser cuts the plywood.

- Rob

[Dick Fischbeck]

Hey Rob - I've shared your photos with others and several people are taken with your dome. They are asking for dimensions as well.

Dick

[Jamie Meredith]

Thank you very much for sharing this information!  I have been looking for just this kind of dome to use for a project that I recently started here in Nashville - https://www.facebook.com/nashvilleurbangreenhouseinitiative - Our goal is too build 300 greenhouses in Nashville over the next 5 years, with each greenhouse linked to a centralized network which reports environmental metrics as well as providing each greenhouse operator to report yields, pests, and post questions or photos to the global greenhouse network.  

I have a couple of quick questions:

• Can the width of the pieces be made less and still maintain structural integrity?  I am thinking about how to maximize light to the inside of the dome.
• How would I go about making the dome larger in diameter?  I need three dome sizes - 12', 18' and 24'
• Would you be interested in contracting to supply the cut files for the larger versions?
• Could one bolt be used for the connections rather than two?

Thank you again for sharing this!  Looking forward to picking up plywood today and giving this a spin!  I have someone locally that can CNC the materials today for me!  Woot!

On Thursday, August 20, 2015 at 9:52:19 AM UTC-5, Robert Clark wrote:

Scott,

Your dome picture is awesome. It's exactly what I'm hoping to do with my plywood dome.

I've attached the dxf file and an image of the parts list for hardware that I ordered from McMaster-Carr. The plywood I used was 7/32" birch plywood underlayment from Home Depot. You'll need 6 sheets to complete the dome (with a few extra pieces left over).  There are notches in some of the pieces to identify which piece is which. I think you can use slightly thicker plywood for a stiffer dome. If you can't get someone to CNC the parts I guess you can use a bandsaw.  The important thing is getting the holes in the exact locations.

The pieces with 2 notches are the red pieces in the cad image and the pieces with just one notch are the grey. The notch in one piece will always mate with the notch in another piece. Good luck and post pictures of your progress.

- Rob

[homespun]

Rob,

 

    I have begun making a little paper model using the first template you posted.  I am going to make the entire sphere of 240 pieces will work on it this weekend and send pictures as soon as it is done.

 

                Dan

 

https://www.facebook.com/pages/The-Octa-Tetra-Museum/250284981705042

 

 

From: Robert Clark

Sent: Thursday, August 20, 2015 10:23 AM

To: Geodesic Help Group

Subject: Re: 12' plywood hex dome with flat base (4V class II) made from 6 sheets of underlayment. Photos and pattern. What do you think of hex domes?

[Jamie Meredith]

Thank you for your answers Robert!  Feel free to email me at jlmeredith(at)gmail.com or contact me through Google+ 

I am curious if you have come up with a plan for addressing it being flat along the bottom?  Or is it flat and the photos just don't represent the final state?  

I will need to figure out how to tie this into a 3' tall base.  Would love to use the same routed piece design some how for visual continuity.  The raised wall is required for the install of beds within the greenhouse.  

I saw it mentioned previously a question about how to preserve the wood.  Assuming this is not directly exposed to the elements (i.e. covered in greenhouse plastic), I don't think there would be a huge issue with just leaving it as it is.  There maybe a small issue with mold growth during wet seasons, but nothing that would be of significant concern I don't think.  I also think a quick shot of ply on the wood would act as a perfectly acceptable protective coat for many years.  Just make sure to get a high UV acrylic poly like this Olympic 55263a/01 Waterproofing Sealant Wood Toner.

I like the other hub design with the sandwiched circle, but your initial design here is the right mix of mass producible and lightweight engineering.  For my purposes, I want to be able to have a palette of 1/2" plywood delivered to a CNC shop and they deliver me 10 complete units in short order.  This design allows for that and with some jigging, and entire dome could be packed in a crate and shipped with no issues.  Assembly is easy, and not a lot of mixed parts.  It really is perfect is so many ways.  

More questions LOL,

• Any chance you can show us a video of this with some flex or push on the sides?  
  
• Have you tried pulling down weight from the center?  How much deflection do you get with say 50lbs?

On Thursday, August 20, 2015 at 4:35:58 PM UTC-5, Robert Clark wrote:

Jamie,

I'm sorry, I didn't finish answering your other questions. 

Making the dome larger would require larger pieces (less pieces per plywood sheet and therefore more plywood sheets) and thicker plywood that still is capable of a certain amount of flexing.

You must use 2 bolts because you do not want it to act like a hinge or the whole dome would collapse.

We could discuss contracting to supply larger cut files later.

- Rob

[Scott Ihrig]

Wow Rob, thank you so much! This is fantastic! I can't wait to try it!

On Thursday, August 20, 2015 at 9:52:19 AM UTC-5, Robert Clark wrote:

Scott,

Your dome picture is awesome. It's exactly what I'm hoping to do with my plywood dome.

I've attached the dxf file and an image of the parts list for hardware that I ordered from McMaster-Carr. The plywood I used was 7/32" birch plywood underlayment from Home Depot. You'll need 6 sheets to complete the dome (with a few extra pieces left over).  There are notches in some of the pieces to identify which piece is which. I think you can use slightly thicker plywood for a stiffer dome. If you can't get someone to CNC the parts I guess you can use a bandsaw.  The important thing is getting the holes in the exact locations.

The pieces with 2 notches are the red pieces in the cad image and the pieces with just one notch are the grey. The notch in one piece will always mate with the notch in another piece. Good luck and post pictures of your progress.

- Rob

On Tuesday, August 18, 2015 at 11:20:28 PM UTC-4, Scott Ihrig wrote:

[Scott Ihrig]

I found what seems like a cost effect wood sealer product. I saw a YouTube video of someone building a dome who treated all of the components prior to assembly. He mentioned this product in his video. Sorry I cannot find the video but I did save the info on the preservative. The company is in the UK with distributors/retailers here in the US. I plan on trying this product soon. Hope this info helps you guys.

http://www.valhalco.com/index.php

On Saturday, August 15, 2015 at 1:34:36 PM UTC-5, Robert Clark wrote:

I was asked, "How did you protect the plywood components from the weather?"

Well, I didn't - not yet.  I've already disassembled it and the pieces are stacked in the corner of my dining room. They form two stacks about 18 inches high and the wood pieces without bolts weigh about 50 lbs. I'm going to take it to another location and assemble it on top of a round platform a couple feet off the ground. I'll probably paint it and then cover the outside with a single layer of fiberglassed mat cloth to water proof it and still let the light in. It will be in the shade of trees to protect it from the heat of the sun. I want to design a round hobbit style door for it. The dome will just be a place to hang out and relax. There won't be any insulation so it wouldn't be very comfortable in cold weather unless I ran a heater or small stove.

Thanks.

- Rob in Massachusetts

[Gerry in Quebec]

Rob & Scott,

The Valhalla literature says the company is based in British Columbia, Canada. I had a dome client in the US a few years ago who treated all the Douglas Fir 2x4 & 2x6 struts with Vallhalla wood preservative. It comes as a powder that you mix with water. They used a sprayer. I heard somewhere that some Canadian parks have used Valhalla to treat their picnic tables, with good results.

- Gerry in Quebec 

 

 

On Friday, August 21, 2015 at 9:09:05 AM UTC-4, Robert Clark wrote:

Scott,

I like that it is non-toxic and can be used in and around a garden. Says one application lasts a lifetime and it doesn't seem very expensive. I'll definitely look into this. Thanks.

- Rob

[Scott Ihrig]

Hey Jamie - I have a thought for a flat bottom on Rob's dome. I built a 3V 10ft geo dome with a flat base using 2x4's. It has 15 pieces, or sides, along the bottom. Each base piece is cut to length with a 12 degree angle on each end. It may be possible to do the same thing with Rob's 10ft dome by bolting the ends of the ply to each 2x4 segment. Since the plywood dome is more spherical than a geo dome I cannot be sure how the ply would react to being straightened out along the bottom? Although you could cut vertical kerfs, say 0.75 inches deep, every couple of inches on each segment, making sure the kerfs face inside the dome, to allow the 2x4 to flex. A flat steel tension band could then be installed around the bottom circumference to tighten it all up. Just a thought anyway.

~ Scott

[Scott Ihrig]

Hey Gerry - Thanks for the correction. I'm just happy it can be purchased here in the States and my neck of the woods! And it's nice to hear about someone who has experience with the product and its effectiveness. Thanks!

[Scott Ihrig]

Hey Rob - It is precisely the non toxic aspect of the product, well the reasonable cost too, that compelled me to save the info.

On Friday, August 21, 2015 at 8:09:05 AM UTC-5, Robert Clark wrote:

Scott,

I like that it is non-toxic and can be used in and around a garden. Says one application lasts a lifetime and it doesn't seem very expensive. I'll definitely look into this. Thanks.

- Rob

On Friday, August 21, 2015 at 8:28:51 AM UTC-4, Scott Ihrig wrote:

[norm...@gmail.com]

Nice work Robert!  Great to see a fellow dome builder doing his thing.  Robert's advice on mocking up smaller models out of paper/cardboard is spot on.

couple general thoughts

-does the plywood being used have exterior glue?  Some underlayment doesn't use exterior glue which can lead to delamination when used outside, particularly if it is left bare.

-for sealer application I'd dip and soak individual pieces then assemble when dry so it gets all the surfaces good.  another natural product used for hundreds of years in wood preservation is pine tar and linseed oil combo, although that can end up darker then some people want.  See Norwegian stave churches.  If the dome is going to be used there will be rain to deal with on the exterior and moisture from the inside.

-to me covering something like this in green house plastic doesn't seem entirely practical (not that all choices should be made from that view)...when you can build a 12' stardome with about $60 in pvc conduit that won't rot, isn't effected by UV and won't block as much light: https://simplydifferently.org/Star_Dome

[homespun]

Hi Rob and all,

   Here is my little paper model using Rob Clark’s pattern:

 

 

 

More picture in two files attached.

        

                                Dan

 

From: Robert Clark

Sent: Thursday, August 20, 2015 4:26 PM

To: Geodesic Help Group

Subject: Re: 12' plywood hex dome with flat base (4V class II) made from 6 sheets of underlayment. Photos and pattern. What do you think of hex domes?

 

Very good!  I'm looking forward to seeing it.......

Rob

[homespun]

Robert Clark and all,

     I like your dome so much that I decided to make a larger model – this time thicker (standard corrugated cardboard with cardstock on both faces), and held together with machine screws and nuts.  This time I’ll just make the dome and attach it to a base.  It’s going to take a few weeks, - will keep me busy and out of mischief.

 

                                   Dan

[homespun]

Robert Clark and all,

     Here is the dome model I have been working on, now about two-thirds completed.  ---  Dan

[homespun]

  Before I put it up high.

 

  Dome in the Sky.

 

  View from the floor looking up.

 

   You only see the “tip of the iceberg” when you enter the room.

 

Best wishes,

Dan

[norm...@gmail.com]

nice model work everyone, I envy that model workspace in the pictures.  You should try a light in the upsidedown mounted model.

So am I correct in thinking you could build even higher freq models with only one unit template if you based it on the clinton/goldberg equal strut length design?

[Dick Fischbeck]

Hey Charlie

Would  the Goldberg require a tiny fudge factor in the attachment angles?

Dick

[TaffGoch]

normalson wrote:

"So am I correct in thinking you could build even higher freq models with only one unit template if you based it on the clinton/goldberg equal strut length design?"

____________________________________

Edges would be the same, yes, but not the angles. Recall discussion:

Clinton's Equal Central Angle Conjecture

3D Warehouse • Icosahedral System • Equal Central Angle

With fixed angles built-into the "Y" components, I can't foresee how they could be successfully employed (not with 2-bolt "rigid" connections, anyway.)

-Taff

[norm...@gmail.com]

thanks Dick and Taff for clearing that up.   I'll try a part of the model of what I was thinking and prove this to myself as well...

[Dick Fischbeck]

Nailed it. But we are very talking minor angular fudge. Still, Steve Miller makes the main point. He said the "depth of truss, or you could say the wall thickness, counts big time.

-- 

[Dick Fischbeck]

Let us know, Charles. I love your models if I didn't say that before.

[TaffGoch]

Charlie,

Angular "fudging," as Dick calls it, can be seen in the model image that I posted, earlier:

Note the angular "compensation" produced in the pentagon "edges." (That's why I mentioned the "two-bolt" connections. Only single-bolt, "pivoting" connections will automatically provide for fudging.)

You may find that you can get by with one unique Y-component, if you allow for such fudging. Secondarily, the vertices may not all share the same sphere radius, but that, itself, might also be similar enough to be unnoticeable.

-Taff

[Dick Fischbeck]

Taff - The yellow/blue angular adjustments shown seem exaggerated. I expected them to be 2 or 3 degrees. The image shows much larger variations. I mean eyeballing it.

Dick

[TaffGoch]

If I recall, correctly, it was the length of the "arms" that needed adjustment, more-so than the angles.

-Taff

[Dick Fischbeck]

Equal central angle means lengths are the same.

On Wed, Sep 9, 2015 at 9:07 PM, TaffGoch <taff...@gmail.com> wrote:

If I recall, correctly, it was the length of the "arms" that needed adjustment, more-so than the angles.

-Taff

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[TaffGoch]

Yeah, I did that model before any of my Clinton/Goldberg models....

-Taff

[norm...@gmail.com]

thanks for the kind words my friend.  Sometimes my model making is seasonal, a winter hobby, but I've been inspired by the model makers in this thread to get back at it.

so thanks to Robert and 'homespun' and everyone at the discussion board as well.

[Mason Cooley]

Any chance you could post the nested pattern file with dimensions? Thinking of trying with a jigsaw etc.

Thanks

[Mason Cooley]

I can probably send it to a friend and have them overall the measurements.

Thanks

On Saturday, November 28, 2015 at 10:22:29 PM UTC-5, Robert Clark wrote:

Can you import the DXF into a cad program?  There are a lot of angles and curves to the three parts; not easy to create a fully dimensioned drawing.  For reference, the rectangle enclosing the parts represents a 48" x 96" sheet of plywood.

[Mason Cooley]

Also, maybe this was covered above but is there a reason you didn't go with 1/4 plywood? The holes look like they are in the CAD file as well as the notches.

On Saturday, November 28, 2015 at 10:22:29 PM UTC-5, Robert Clark wrote:

Can you import the DXF into a cad program?  There are a lot of angles and curves to the three parts; not easy to create a fully dimensioned drawing.  For reference, the rectangle enclosing the parts represents a 48" x 96" sheet of plywood.

On Saturday, November 28, 2015 at 9:44:45 PM UTC-5, Mason Cooley wrote:

[Mason Cooley]

Thanks for the replies. Good to know. I am going to see if the college where I want to do this for class will be able to do the CNC cuts. I assume the CAD file is good to go and the other files showing the order of parts also worked well for you? Here is our previous dome not far from where Black Mountain College was located and hence Buckminster Fuller's early experiments.

m

On Sunday, November 29, 2015 at 5:22:17 PM UTC-5, Robert Clark wrote:

In hindsight, it might have been better to go with 1/4" plywood for stiffness, but this was more of a prototype and I went with what was cheapest being 5.5 mm underlayment.  Also, the underlayment seemed less prone to having hidden voids.

- Rob Clark

[Peter Schwarzel]

Hi Robert - If your using solidworks how are you modelling the individual parts as if you model them flat they will not assemble correctly? So are you modelling them as a circular part then flattening (developing) them?  Peter

On Monday, November 30, 2015 at 8:22:17 AM UTC+10, Robert Clark wrote:

In hindsight, it might have been better to go with 1/4" plywood for stiffness, but this was more of a prototype and I went with what was cheapest being 5.5 mm underlayment.  Also, the underlayment seemed less prone to having hidden voids.

- Rob Clark

On Sunday, November 29, 2015 at 9:21:32 AM UTC-5, Mason Cooley wrote:

[etiennem...@gmail.com]

Hey Robert! Really nice design you've made. I'm currently looking at dome design to build a stage in a festival I'm part of and I feel like your design would be the best for our needs. I'd like to know if you still have the assembly file for solidworks (2013) and if you would be OK with sending it to me? I'd have to do some modification to the dome to turn it into a stage (opening in the back to enter, window in the front so we can see the performer and find a way to incorporate a table to the dome) and would prefer to work on this with SW if possible. 

Etienne M.

Le jeudi 9 février 2017 08:23:56 UTC-5, Robert Clark a écrit :

Peter,

I model the curved pieces and assembly in SolidWorks and then I measure the arc lengths along the curved center-lines of the hub legs.  Also, I take measurements of the radial angles between legs.  Then, I take the angles and the arc lengths and manually draw out the flattened shapes in AutoCAD.  To layout all the dxf shapes as efficiently as possible on a 4' x 8' rectangle, I use a free online program called MyNesting.

Robert

[etiennem...@gmail.com]

Sent you a pm with my email address thanks again!

Le mardi 14 février 2017 22:17:13 UTC-5, Robert Clark a écrit :

Etienne,

How big are you going to make this?  The model I built was only 12 feet in diameter using 3/16" plywood (took 6 sheets of plywood).  It was plenty strong enough to support its own weight but would not have been strong enough to climb on.  I'd of had to use 3/8" or 1/2" plywood for that.  It sounds like it will be quite a construction project.  Are you very skilled with SolidWorks?  I am currently using SolidWorks 2016 student edition, but I have a saved 2012 version of the model and assembly.  Are you going to have the parts CNC cut?  I'm guessing you'll use 1/2" thick plywood.  I'd test cut strips of plywood to see which thickness has just enough flex for the diameter dome you are making, but without being TOO flimsy.  The bolt sizes might also need to be increased from the 1/4" hex bolts that I used.  Let me have an email to send you the files.  I would love to see this stage get built!

Robert

[Peter Schwarzel]

Hi Robert - If you get a copy of Rhino3D it will flatten your shapes for you automatically. Peter

On Thursday, February 9, 2017 at 11:23:56 PM UTC+10, Robert Clark wrote:

Peter,

I model the curved pieces and assembly in SolidWorks and then I measure the arc lengths along the curved center-lines of the hub legs.  Also, I take measurements of the radial angles between legs.  Then, I take the angles and the arc lengths and manually draw out the flattened shapes in AutoCAD.  To layout all the dxf shapes as efficiently as possible on a 4' x 8' rectangle, I use a free online program called MyNesting.

Robert

On Thursday, February 9, 2017 at 3:36:15 AM UTC-5, Peter Schwarzel wrote:

[fran...@gmail.com]

Hi all, looking the pictures I think this dome is not 4V class II, It is 2V Class I,  whit fullerene "inscribed In". See my picture atached.

[Ashok Mathur]

Dear Franopio,

To some extent I understand the frequency division  classification among domes based on polyhedra.

But I am completely ignorant of the system of frequency division among fullerene molecules.

Searching on the net for relationship between the radius and the strut lengths of fullerenes I came across this article that does a good job of explaining 1V fullerene.

http://web.eng.fiu.edu/~vlassov/EEE-5425/Ulloa-Fullerenes.pdf

Can you suggest other  links?

Regards

Ashok

Regards

Ashok

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[Ashok Mathur]

Dear Franopio

In brief this is what the article says:

S No	Shorter strut Vertex to Vertex length	Longer strut Vertex to Vertex length
For diameter 7.09	1.4	1.46
For diameter 2	0.394922426	0.4118476728
NB	Three alternate struts of hexagonal ring have this length.	All the struts of pentagonal ring have this length

Regards

Ashok

[fran...@gmail.com]

Hi Ashok and Robert, thank you for your reply.
ok, I could look for the mathematical theory that explains this.
But for now you can use this geodesic dome calculator, very complete and in several languages.
.
If you use these parameters you can get your hexa dome:
- Level of detail (frecuency): 2v
- Subdivision class: I
- Rotational symmetry: Pentad
- Fullerene: Inscribed in
- Part of full spher: 1/2
- Sphere radius: your choise
- Connection type: Piped
- Pipe diameter: 0

Now you should look at the hexagons and where they are located. This 2V dome has two hexagons, one regular and one irregular, such as your dome.

A 4V class II dome has six different hexagons and all are irregular. Check it in the calculator. If you use fullerene: described around, you get three hexes all
irregular ones too.
See two images of 4V calse II.

Regards
Franopio

[fran...@gmail.com]

This is the calculator mentioned: 

http://www.acidome.ru/lab/calc/#1/2_Inscribed_Fullerene_on_Piped_D0_2V_R2.5_beams_5x50

 

[fran...@gmail.com]

I correct my previus post: 

2V, class I, fullerene inscribed has three hexagons, one regular and two irregular. 

The important thing is the position of the regular hexagon.

Franopio

[thedomeguy]

I would identify this as a tessellated dodecahedron, the dual of a Six Frequency geodesic sphere.

Sent from my Verizon, Samsung Galaxy smartphone

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[Gerry in Quebec]

A few ideas on the geometry of the 4v icosa class II geodesic sphere, the starting point of Robert's dome....

If you generate its geometric dual, you end up with something that looks very much like the full-sphere version of his dome. The dual is a true polyhedron, so all the pent and hex faces are flat. But, unlike the original class II geodesic sphere, not all vertices are the same distance from the spherical centre.

Examples of the class II geodesic sphere and its dual, generated by Antiprism, are attached (images A & B). In the terminology used by Joseph Clinton in his paper about Goldberg spheres with equal central angles, the dual of a 4v icosa class II geodesic sphere would be an I{2,2}, where "I" stands for icosahedron, as opposed to octahedron or tetrahedron. If you were to then triangulate the dual -- making six triangles per hex and five per pent -- it would end up looking like a 6v icosa class I geodesic sphere (image C).

There are all kinds of breakdown methods for geodesic spheres, of course. In the case of the 4v icosa class II, nine of them have only 4 chord factors. (The one Bucky Fuller started out with in the 1940s, if I'm not mistaken, had 5 chord factors.) One of the 4-chord layouts, image C, has a very interesting property: its dual sits flat at the equator when you slice symmetrically through the equatorial hexagons. (None of the others does this.) So, it's an attractive candidate for a physical building, similar to the Eden domes, but with a lower frequency and a higher profile.

Apologies to anyone who's already seen this stuff in a thread a few years ago.

- Gerry in Québec

On Wednesday, December 6, 2017 at 11:16:18 AM UTC-5, Robert Clark wrote:

Well, now I'm confused!  When I created the dome, it was definitely a 4V class-II geodesic dome made of triangular panels.  I modified the panels by cutting off the corners of each triangle creating a "Y" shape.  I assumed it was still a 4V class-II, but with slightly modified panels.  Now I found TaffGoch's SketchUp diagram for Goldberg polyhedron and it would appear that my dome is now a 2V class-I Goldberg polyhedron.

I'm curious, at what point, as I nibble away the corners of the triangles, does the dome cross a line and go from a 4V class-II to a 2V class-I?

Does anyone have an idea about this or any thoughts?  I really am suddenly baffled.

best regards,

Robert

Regards

Ashok

Regards

Ashok

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[Adrian Rossiter]

Hi Robert

On Wed, 6 Dec 2017, Robert Clark wrote:
> it was still a 4V class-II, but with slightly modified panels. Now I found
> TaffGoch's SketchUp diagram for Goldberg polyhedron and it would appear
> that my dome is now a 2V class-I Goldberg polyhedron.

The Wikipedia page for Goldberg polyhedra describes your dome as
Class II

https://en.wikipedia.org/wiki/Goldberg_polyhedron#Examples

But, the row it appears in on Taff's diagram is row 4 (there
is no model placed on row one), and you can see that on that
row the minimum number of steps (over edges) to travel from one
pentagon to another is 4.

Adrian.
--
Adrian Rossiter
adr...@antiprism.com
http://antiprism.com/adrian

[Adrian Rossiter]

Hi

On Wed, 6 Dec 2017, fran...@gmail.com wrote:
> Hi all, looking the pictures I think this dome is not 4V class II, It is 2V
> Class I, whit fullerene "inscribed In". See my picture atached.

If you choose "described around", then the model produced is what
is normally associated with the other parameters. I.e. if you choose
4V, Class II, "described around", then you get the pattern of
Robert's dome.

"inscribed in" appears to correspond to: produce the usual model
(the "described around" model) and then apply the Conway notation
'zip' operation to it

https://en.wikipedia.org/wiki/Conway_polyhedron_notation#Operations_on_polyhedra

You can make the pattern of Robert's model with the Antiprism command

conway z geo_2_d | antiview

The F4 Class II "inscribed in" model, which would normally be called
a F6 Class I model, is produced with

conway z geo_2_2_d | antiview

[Images attached]

[Adrian Rossiter]

On Thu, 7 Dec 2017, Adrian Rossiter wrote:
> "inscribed in" appears to correspond to: produce the usual model
> (the "described around" model) and then apply the Conway notation
> 'zip' operation to it
>
> https://en.wikipedia.org/wiki/Conway_polyhedron_notation#Operations_on_polyhedra
>
> You can make the pattern of Robert's model with the Antiprism command
>
> conway z geo_2_d | antiview
>
> The F4 Class II "inscribed in" model, which would normally be called
> a F6 Class I model, is produced with
>
> conway z geo_2_2_d | antiview

I have just noticed that the description box for the zip operator
on the Wikipedia page specifically mentions its application to
Goldberg polyhedra, and gives a formula for the transformation.

G(a,b) --z--> G(a+2b,a-b)

G(2,0) --z--> G(2,2) equiv. F4 Class II
G(2,2) --z--> G(2+2*2, 2-2) = G(6,0) equiv. F6 Class I

[Francisco]

Hi Adrian

 

The Robert’s dome has three hexagons, one regular and two irregular. See picture 1.

You say “. if you choose 4V, Class II, "described around", then you get the pattern of

Robert's dome”. But this is not correct because 4v, class II, described around has three hexagons too, but all irregular (very irregular). See picture 2.

 

Look again the Robert`s dome where are the regular hexagons. Now see my third picture attached, you can see the same pattern of the Robert`s dome, This is 2V, class I, inscribed in.

 

I`m not an expert but this is that I can see according to the software from acidome.ru.

 

Regards

Franopio    

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[Adrian Rossiter]

Hi Franpio

On Thu, 7 Dec 2017, Francisco wrote:
> The Robert’s dome has three hexagons, one regular and two irregular. See
> picture 1. You say “. if you choose 4V, Class II, "described around",
> then you get the pattern of Robert's dome”. But this is not correct
> because 4v, class II, described around has three hexagons too, but all
> irregular (very irregular). See picture 2.
>
> Look again the Robert`s dome where are the regular hexagons. Now see my
> third picture attached, you can see the same pattern of the Robert`s
> dome, This is 2V, class I, inscribed in.
>
> I`m not an expert but this is that I can see according to the software
> from acidome.ru.

The Class and Frequency describe a pattern - the connectivity of the
model, what polygons are connected to what polygons - but not the
coordinates of the vertices. Gerry mentioned breakdown methods, which
are used to assign coordinates, and these control geometric properties
like which hexagons are regular.