Nexorades/Rotegrities - Frequencies 1 through 3

[TaffGoch]

Here is the extent of what I've modeled, in order of increasing parts "density":

1v {0,1} (icosahedron) (original "Rotegrity" by Richard Boyt, 1970)

2v {1,1} (i.e., class-II)

2v {0,2} (i.e., class-I)

3v {2,1} (i.e., class-III)

3v {0,3} (i.e., class-I)

I don't expect to fine-tune any higher frequencies, to achieve equal 1/3rd band subdivision (as it's a real pain to iteratively-produce manually, in SketchUp.) Also, the density of parts has reached a maximum practical count for actual construction of physical models.

Now, I just have to find an inexpensive, readily-available source of semi-flexible material to use for the bands (one to one-and-a-quarter inch width.) Any suggestions?

-Taff

[TaffGoch]

By request, I have posted dimensioned images to my deviatART webpage gallery:

These dimensions will produce spheres of unit (1.000000) radius. As with geodesic chord factors, multiply by your objective radius, to make bands that will assemble desired-size spheres.

-Taff

[TaffGoch]

If the band from the first rotegrity (icosahedron) is used as the pentagon bands of the other four "nexo-rotegrities," the scale of sphere complexity becomes more apparent:

(Essentially, all pentagons are, now, the same size.)

Radius ratio, smallest:largest = 1:3.44

-Taff

[biagiodicarlo]

Thank you very much, Taff !!

A very interesting work.

- Biagio

Il giorno 03 ago 2016, alle ore 07:00, TaffGoch <taff...@gmail.com> ha scritto:

If the band from the first rotegrity (icosahedron) is used as the pentagon bands of the other four "nexo-rotegrities," the scale of sphere complexity becomes more apparent:

<Nexorades; pentagon scale.png>

(Essentially, all pentagons are, now, the same size.)

Radius ratio, smallest:largest = 1:3.44

-Taff

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[TaffGoch]

Final rotegrity "catalog"

I can't imagine anyone having the fortitude to construct any of the frequency-4 rotegrities - they're composed of a LOT of bands

-Taff

[TaffGoch]

Some physical examples -- then and now....

Richard Boyt's rotegrity:

2v{0,2} geometry -- Drawing from "Rotegrity" leaflet, Parity Products, simple science toys, Eugene, OR (apparently, now defunct)

Father Magnus Wenninger's "Order in Chaos" has a rotegrity quality to it:

Other than these, I can't find any examples, from past decades.
___________________________

Recently, rotegrities are being studied, as "nexorades." I found an art project that employs 2v{0,2} geometry -- that of the Terraform "Plug-In Ecology" project:

Have you come across any other online examples (other than the group) ???

-Taff

[TaffGoch]

Spotted online: Kortrijk, Belgium -- May 2016 street festival

-Taff

[TaffGoch]

​Hiroshi Murata's "Da Vinci Dome" has a 3v{0,3} spherical rotegrity basis​:

-Taff

[TaffGoch]

For those who want the actual 3D geometry (rather than just pretty pictures,) I've posted the rotegrity "catalog" SketchUp model to the 3D Warehouse:  Rotegrities - Pentagon Scale


___________________________________

I've also modeled the Terraform "Plug-In Ecology," in two versions:

Without the large pentagon "entrance":  Terraform "Plug-In Ecology"

With the pentagon "entrance":  Terraform "Plug-In Ecology" - Pentagon

-Taff

[TaffGoch]

Note:  My references to "Terraform" should have been "Terreform"

-Taff

[Rikus Keyser]

Hi Taff

Your work is so fascinating. I am interested to build a 2v{1,1} rotegrity of 1 meter in diameter. (That is to say if I you kindly give me permission to do so.)
I just have a question regarding the formula as per your diagram attached. If radius is 0.5m, I take it the distance between the attachment points will be 0.5(0.252981) for the red bands. Or is it more complex than that to scale it up or down?

Many thanks for taking the time to answer me.

Kind regards

Rikus

[TaffGoch]

Rikus,

Your formula interpretation (calculation) is correct.

-Taff

[Rikus Keyser]

Thank you very much!

R.

[Rikus Keyser]

Hi

The 1m diameter scale model worked out perfectly! 

Now for the real one. 3m diameter. :-) I'll send a picture of that one too in a few weeks (we only need it constructed end of April).

Thank you again so much. Without you this would not have been possible.

Warmest regards

Rikus

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[Ashok Mathur]

Dear Rikus,

Congratulations on the successful building of 1m dome.

Fuller had found that materials do not scale up linearly, that is to say, if a material of one and quarter inch width worked perfectly for a length of 1 meter, then it might become soft/putty like when its length reaches say 2 meters.

Hugh Kenners refers to this as slenderness ratio.

You might consider making a part model of the hexagon and its 6 adjustsant triangles with the material of new lengths of same width as the 1 m dome.
By the way, can you describe in some detail what the bands are composed of and how did you drill holes in them?

Regards

Ashok

Regards

Ashok

[Rikus Keyser]

Hi Ashok

Thank you very much for the response and advice.

The 1 m scale model was made of 3mm plain plywood. Lasercut. The holes were also drilled by laser. The plywood sections (50mm wide) are kept together with 3mm bolt, washer and nut.

For the full scale 3 meter structure we intend to use 9mm thick marine grade plywood (150mm) wide with 8mm bolts, nuts and washers. A central steelpole (20mm diameter) and steel base plates over the two podal hexagons will add structural integrity. Radial rods can also be added if necessary. This whole structure will then be bolted on a 5m high pyramidal base structure. [See attached conceptual image to see where we are going with this :-)]

Any suggestions will be most welcome. (We'll also run it by a structural enigineer before we proceed to the full scale structure.)

Kind regards
Rikus

Regards

Ashok

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[Ashok Mathur]

Dear Rikus,

Thanks for the details.

They show that you are going about in a good manner.

As far as your going to a structural engineer is concerned, before you give him any money, ask him if he has heard of Rene Motro of France?

Regards

Ashok

Regards

Ashok

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[Ashok Mathur]

 Dear Rikus,

My point of view is that the standard structural engineering software (FEA/FEMA) is absolutely incapable of handling tensegrity structures.

There is a long thread in the group where I have explained my point of view.

Than I have been silent on the subject trying to research as to who has done anything to make the proper software for analyzing tensgrities.

The answer to that question is Rene Motro of France.

The tensegrity wiki page has a good write up on him http://tensegritywiki.com/Motro%2C+Ren%C3%A9

Unfortunately, his English work, which costs more than $ 1,000, is completely out of print.
https://www.amazon.com/Tensegrity-Structural-Systems-Ren%C3%A9-Motro/dp/1903996376

I was in USA last summer at Mountain View and I initiated a California wide search for a copy of his book in any public library.

It simply was not present.

 

To my knowledge, his software was last updated in year 2,000. There might be later editions, but I am not aware of them.

From the internet, you can download this MSc thesis of a student of his

Tensegrity Structures  and their Application to Architecture
Valentín Gómez Jáuregui

You can download one of his English paper from here
https://www.amazon.com/Tensegrity-Structural-Systems-Ren%C3%A9-Motro/dp/1903996376

If there are French speaking readers who know more of his work, I would welcome their contributions.

Regards

Ashok

Regards

Ashok

[Rikus Keyser]

Dear Ashok

Thank you so much for all the information. I'll look into the links you have shared. I'll speak to the engineer, but I am sure he would not have heard of René Motro...

I'll keep you posted!

Kind regards

Rikus 

[AzaFran]

Hi folks! thanks for share knowledge, expreriences, proyects and resources. i find a research/study copy online of tensegrity systems for the future you can download at: http://b-ok.org/dl/871005/c716ea

Kind Regards

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------------------------------------------------------------------------------------

D.A.L.E.!

~   "Despertando a la Ecosofía"   ~

-----------------------------------------------------------------------------------

[TaffGoch]

An important book, hard to find a safe internet link for it.

I safely downloaded from AzaFran's link, with no kerfuffle.

-Taff

[Yves Rolland]

Dear TaffGosh,

I have read your communications about nexorades in the same time I was interested in Bamboo last year, and I have proposed a greenhouse "active Bamboubulle" in a FAB14 event that happened in Toulouse last week. Thank you very much (and to Olivier Baverel to). This spherical shape seems optimal because of its sphericity and there is no natural material like bamboo.
I send you a notice and here are a few pictures https://photos.app.goo.gl/RWvYGkLMg0RapKQ22 .
A big thank you again.

Yves Rolland

yesro...@gmail.com

[Olga]

Hi Taff,

 

Do you perhaps have a calculation for rotegrity dome as excel? I found one that is unfortunately only available as a pdf.

 

kind regrds

Olga

[ZelDome]

Dear Mrs Olga,

Your pdf reminded me of a picture I saw earlier on mr. Hiroshi Murata Da Vinci dome page. I made the Excel table, found that it was the same dome (12x magnification). Soma facts are still unknown for me: What are a,b? Two legs (Hex) are similar, whitch is left of blue leg, which is right?
Warning: This is not a dome calculation. No geometric analysis was carried out nor the dome was physically fabricated. My table is just my impression of two pictures of the web with the aim to help you. For the correct information and procedure, please order Mr. Murata manual.

Best regards,

ZelDome

 

[Olga M]

Thank you ZelDome.

a and b are distances between legs on the floor. I have build a small dome of 50 cm diameter  - it works.

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[Ashok Mathur]

Dear Taff

I have not thanked you for pointing out the link to Rene's book.

I am looking at it keeping in view Hector's requirements.

Thanks

Ashok

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[Bill F]

Hello TaffGogh,

I built a rotegrity art piece based on your work and wanted to share the results with you.

It is built out of aluminum and I made some modifications.

I flipped it inside out because I liked the weave-like look. I also added another internal layer and a central pillar for strength. The two layers sandwiches CNC cut and engraved acrylic panels.

Thank you for your inspiration and posts on how to make one. 

http://www.frymire-art.com/work/reflect-and-connect/

Bill Frymire

[lemondealc .]

Bravo! Great work Bill, and cheers to TaffGogh!

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[Levente Likhanecz]

hi Bill,

there are some higher frequency at this address

https://3dwarehouse.sketchup.com/collection/ac2598a2-575d-445a-9135-a4a690825187/Nexorades-by-Lev?sortBy=createTime%20DESC&searchTab=model&hl=en

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[Levente Likhanecz]

hi Bill here is some compiled rotegrities in excel (macros are for sheet navigation)

https://drive.google.com/open?id=1WTXfiio-VzCcvlwbnXgjmRWjHfR6x7gf

On Sat, May 16, 2020 at 9:04 AM Bill F <bill.f...@gmail.com> wrote:

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[TaffGoch]

Hi, Bill,

Commendations on your work's style and beauty

-Taff

(aka, David Price)

[Edward Popko]

TaffGoch

We are preparing the second edition of a college textbook on geodesics and spherical subdivision and would like to include a vignette about your work.

Can you contact me?

ESP2

[Adam Winter]

What's the first edition called?

Adam Winter

The Bootstrap Music Co.

206-430-4425

ad...@bootstrapmusic.com

www.bootstrapmusic.com

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[Dx G]

I've been looking at Rotegrities and reviewing some related  older discussions.  I noticed something that escaped my attention before. So far, every one I've looked at use 3-strut points of connection at every "hub" in the sphere.  This is unlike the more common hub-and-strut domes that typically have unequal hub counts of 3, 4, 5, 6 or more strut connectors.   This observation came up as part of my continuing work on the Universal Connector. One of the assets listed for the connector is that it lends itself to different strut counts, perhaps as high as 10 or 12 struts.  However, it appears that a 3-strut connector could be of universal use, for at least the Rotegrities I've looked at.  Clearly Rotegrities made of strap type materials that use a bolt, pin or fastener of some kind for connections do not need hubs in the traditional sense. However, having a 3-strut connector that can be used with different, or even mixed strut materials and for various face and axial angles does expand the range of what is easily possible.

   I'm thinking I could be missing something important there, so if anyone is aware of exceptions to such a trend, I'd appreciate any leads on that one. 

Also, saw this and thought it might give you all a good start to 2025

 You only need two tools, WD40 and Duct Tape.

    If it moves, and it shouldn't, use Duct tape

    If it doesn't move, and it should, use WD40

Dx G

[Robert Clark]

Could the 3-strut connectors you're thinking of be used in a structure like this:

[Dx G]

Yes, Robert, they would.  

If one is already using flat profile strut material,  has a means to punch holes (or already making use of perforated material) and access to inexpensive pins/bolts etc., I don't think the universal connector would be quite as simple or as inexpensive.   In particular, I was struck by the fact that I had totally overlooked how nicely Rotegrities of many kinds all use a 3-strut connector, which has advantages. 

However, what the universal connector would offer:

1) One could use round (tubing, pipe, etc) or other strut materials, even mixed materials on a given structure, which can be difficult to join for a routine Rotegrity.  In particular, I'm quite interested in enabling the use of bamboo, which does not accommodate machining well on the ends.  People keep trying, but I've been unimpressed with the results. So, I figure, if you think you can do better, and it has value, then do it.

2) The universal connector can be used for a variety of settings, or even easily taken off a structure and reused for another.  So the connector that lends itself to use on a Rogegrity can also be used for domes and spaceframes...and moved from one to the other if desired, even if you need to go from a 3-strut junction to a 5...and back to a 3 again later. 

The holy grail is this one thing that can be used for anything.  When I review the current asset list developed earlier, I remind myself that even if a connector does not check virtually all the boxes, let not perfection eliminate candidate connectors that might still be highly useful.  

I have this theory to test, that the availability of a universal connector could be a significant factor in boosting the acceptance and proliferation of domes where they are actually a better choice, but are currently being overlooked or refused.  If this is partly, or greatly, for want of a universal connector in particular, once an affordable, simple connector is easily available that anyone can use, there are people out there who are not engineers or architects in the formal sense, but may use that connector as a means to discover and invent new building options/configurations  that nobody ever thought of before.   Even if it is not the best, or even adequate for a large permanent structure, it would be an enabling element for some serious testing and prototyping by people who normally couldn't do it. 

 Sure, I could be wrong, and it won't pan out, but often learning what does not work leads us to what will.  If my theory is correct, and there is more luck here than I would imagine, it could also very well be a fuse that sets off a much larger explosion.   I'm workin' on it and invite others to do the same.

Dx G

[Levente Likhanecz]

hi robert,
as i remember, adrian's (rossiter) antiprism calculates middle points with arbitrary distance from end points.
in the rotegrities have some examples:

https://www.antiprism.com/programs/rotegrity.html

cheers, lev

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[Levente Likhanecz]

this switch/parameter does it:

-f <frac> fraction of length for end sections (default: 1/3)

On Mon, Jan 13, 2025 at 5:26 PM Robert Clark <clark.rob...@gmail.com> wrote:

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[Dx G]

Levente,

 Thanks for posting. Adrian's material is great stuff.  In looking though the images, I still see mostly(?) 3 strut junctions.  

So what is the formal (?) difference between a Nexorade and Rotegrity?  Likely this has already been stated somewhere but I have not seen that.

thx

Dx G

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[Adrian Rossiter]

Hi Dx G

On Mon, 13 Jan 2025, Dx G wrote:
> So what is the formal (?) difference between a Nexorade and Rotegrity?
> Likely this has already been stated somewhere but I have not seen that.

For my rotegrity program:

a rotegrity has great circle arc units, with contact points lying on the
arcs (and hence all on a sphere)

a nexorade has cylinder units, with contact points lying on the outside
of the cylinders (and no consideration of a common surface that all the
contact points lie on)

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

[TaffGoch]

Uhmm,...

I don't want to discourage folks, but I'm pretty sure that the integrity of a rotegrity depends, solely, on the rigidity of long "struts."  

If you allow any potential rotation at the hubs, the whole thing degrades.

However, there may be a bright spot. If hub rotation is allowed, the rotegrity may "shrink" into a smaller ball, providing a means for reduced payload size.

"Inflating" it would be a process of restoring long-strut straight configuration.

So far, a "mind experiment." Maybe, I'm wrong. 

Food for thought and discussion?

Taff

(aka; TaffGoch, David Price)

[Dx G]

Taff,

  I'm not sure why you raised the issue, but it is certainly an interesting topic to discuss.  The change in size feature rather reminds me of Hoberman Spheres, although I realize his work is not the only "spin" on that business (ok, pun...) 

  As far as rotation of struts at vertices (hubs), I can't think of anything that is more likely to yield to induced rotation as a pin, bolt, screw, etc. which looks to me like precisely how most Rotegrity struts are joined.  So they seem inherently susceptible as they are. I always thought it was the logistics of the framework that prevented rotation at the point of strut connection, largely relying on triangles, like so many other structures count on for stability.  I can certainly understand how a square or rectangle with no diagonal brace could be deformed into a parallelogram, which would require rotation at the vertices, but that becomes less likely with a triangle.

Perhaps I misunderstand the issue...?

With respect to the universal connector, the design should accommodate variable axial, face and dihedral angles, but like most domes, does not anticipate rotation of struts around a vertex, rather the way a simple flattened-end conduit dome uses bolt holes.   However, the extent to which the connector alone will restrict or allow that rotation is worthy of consideration as I don't recall it coming up before.

Dx G

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[Dx G]

Taff,

 A follow up to my previous message.  I had assumed you were talking about rotation in the plane tangent to the sphere, or where face angles of sorts would be.  It occurred to me, given your mention of strut stiffness, perhaps you were actually referring to rotation at the (3-way) hub in the plane of the axial angle...(?).  If so, my understanding is that many domes, especially those of lower frequency, derive load capacity from the strut and axial angle logistics, although those with a hub that is rigid is better at resisting collapse from vertex inversion than those that would flex or allow axial rotation.  As we go to higher frequencies and hubs take on a flatter disposition, it becomes more likely that a hub can be pushed/pulled down out of the preferred position.  

  In any case, the universal connector I'm working on would be designed to be inflexible in the axial plane, although it could also be configured as a lock-and-release arrangement if that facilitated folding.  Its an interesting feature that could be a useful addition.

Dx G