Lobel Frames

[Wagyx Xygaw]

Hello there,

Lobel Frames does not seem to be very popular among the geodesic community, I've seen it referenced on this group around ten times only.
However as a polyhedron enthusiast, I think they are a bit underrated and not known well enough.
They may be totally acceptable reasons for this situation and also bad reasons, I won't list them, what would be the point ?

Simply put, Lobel frames are shapes entirely made out of equilateral triangles. They were invented by the late Alain Lobel, a French architect.
He has put up a website that is still running www.equilatere.net with lots of drawings, renders and explanations of his ideas and work.
I suggest you take a look after reading the rest of this message :)

Lobel Frames are classified into 9 families that are parametrized by 3 or 4 numbers each, giving rise to an infinite number of shapes.

To facilitate the understanding and the exploration of these shapes, I have set up a few years ago a series of web pages that simply display them in an interactive viewer.
Here is the page : https://asliceofcuriosity.fr/blog/posts/geometry6.html
The models were kindly given to me by the Lobel's family which I contacted as well as the source code that generates them.

My long term goal is to make a program that would generate any Lobel frame given its family and input parameters.
However, the code is written in Basic and very hard to understand (at least for me). I would like to translate it or redo it in a more modern language, maybe extend it at some point with new families.

So the second main goal for this message is to look for people that have been working on Lobel Frames and have already produced such code for their application and get their help.

If you have any piece of information, please share it in this thread.

Thank you very much for reading this message,
Wagyx

[Dx G]

Wagyx

  Many thanks for posting this. I've been working with domes for a long time, and never heard of these, so I am grateful for the information.

I will study the web sites and see what is there. A lot of the dome literature devoted to reducing the variation of parts is focused on spherical polyhedra, or something close.  However, other shapes can be of equal interest, even if their vertices do not land on a sphere.  People always ask "can't we make a dome out of all the same triangles" and there are always a few options, but usually they are not 100% equilateral. Sounds like Lobels have their calling.   I do see quite a few web sites devoted to "identical elements", so clearly I need to do some study of these. 

One thing I do look for are polyhedra, or tessellations that have some value as structures for actual buildings - anything from a garden shed to a home or more gigantic shells.   Thus, I am particularly interested in studies, simulations, actual tests, etc. that have examined the assets, or lack thereof, of a polyhedon as it is exposed to wind, snow loads, earthquake/seismic stress, impact, etc.  If you already have anything on this that I won't see in the websites, all such leads are welcome.

Also, let me add that in the past I was quite fluent in various versions of BASIC.  Although I had to dust off a lot of brain cobwebs, I have already converted one program into a spreadsheet, which I think is far easier for most people to use.  I am in the process of doing the same for several more programs.  Although a program is far more efficient for iterative routines, some translations can be set up in things like spreadsheets that deliver the exact same results, and in many ways, far easier to work with and edit. So if you want to send me the BASIC code with any available notes, I'd be happy to look it over and see if I think I could translate into a spreadsheet or something more user friendly to people of the 21st Century.

Many thanks again for pitching in on this, you sure found the right group.

Dx G

[Dx G]

Wagyx,

   I've looked at a few items, and even some research papers on Lobel Frames.  One aspect of this is still not clear to me, despite looking for an explanation.  Likely its a common question.  If one starts with the premise of using nothing but equilateral triangles, how would one generate anything but a flat surface?   I see the following possible options:

1) The triangular panels would have to be curved, bent, tapered or something other than flat

2) Different sizes of equilateral triangles would be joined, which would create variation in strut and panel sizes

3) Other triangles would be required in between equilateral triangles.

Since this came from my initial review, maybe I missed something.   Perhaps there is a written explanation already available which discusses this issue. 

thx  Dx G

[Chris Kitrick]

The issue of having curvature with only flat equilateral triangle is easily resolved as long as some vertices have less than 6 surrounding triangles. Take a look at the geometry shown in the website that Wagyx provided.

Example:

[Dx G]

Many thanks Chris.  I should have noticed these were 3-D renditions and those triangles were turned.  I am grateful you pointed this out.  Likewise, that option did not make my list, and certainly should have, since it is, in fact, an important means by which the shape is conferred to the polyhedron.   This does indeed present some interesting options.  

I'm also curious as to whether one could consider some of the other options to improve the shape and structural integrity of the polyhedron.  We know if the strut junctions are too flat or too steep, they can weaken the shell.  If one is willing to depart from purism, different size equilaterals could be included, especially if one only needs a few. Likewise, a few isosceles, although the prospect of using only equilaterals has some attractive advantages in fabrication and reductions (or potential elimination) of material waste.  I also wonder if the addition of some tensegrity elements could enable useful hybrid structures that one could not achieve from equilaterals alone.   I am anxious to see if others have already explored these and either got them worked out, or put them aside having found issues not yet solved.  So will have to dig into this more. As they say, the reach should exceed the grasp.

In addition, I noticed the "universal" connector, which certainly does provide some benefits.  However, it is rather elaborate and doesn't look like an inexpensive option. However, the Wangers may be another means (much simpler, less material) to achieve at least some of that.  The universal connector I'm working on might also be an asset.  It would also enable use of materials like bamboo without cracking, crushing and other downsides of connectors I've seen.

Dx G

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[Wagyx Xygaw]

They are not only 3D renditions but live 3D views that you can turn around to inspect.

[Charles Lasater]

Wagyx Xygaw,Thank you, a real treasure.

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[Dick Fischbeck]

Yes, but all the faces are flat.

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[Wagyx Xygaw]

Thank you for showing so much interest Dx G !

Mathematically speaking, a true hemispherical dome cannot be made out of equilateral triangles unless you use millions or billions of them and thanks to the manufacturing tolerances.
In the same way, you can make any image out of a regular grid of square colored pixels if you have millions of them, so many that your eyes cannot distinguish them anymore.

From his writings, Alain Lobel was more concerned about using a unique shape and see what could be build out of it.
I don't know whether he worked on the mechanical aspect of his structures, if you cannot find it on the website, it either means he did not work on it or never shared his studies publicly.
Personally, I don't have any knowledge about these issues nor know any Lobel frames related work.
If you happen to collect papers, blog entries or any other resources regarding Lobel Frames, please share them here so that this thread can act as a hub.

Finally, let me try to answer your question about how can curved surfaces be made out of perfect flat equilateral triangles.

A quick side note: the lengths of the edges of the Lobel Frames seems to be all equal to 1 up to at least the sixth decimals

for what I have checked but since there is no proper mathematical proof for them, this might still be a concern and also a drive

for recomputing the shapes with more precision and better algorithms.

For starters, a simple fact : neither the tetrahedron, the octahedron nor the icosahedron are flat surfaces despite being made of equilateral triangles only.
Let's look at the number of triangles around a vertex, there are three cases:

- the vertex is locally spherical meaning that the sum of the angles of the triangles is strictly less than 360°. This happens when there are 3, 4 and 5 triangles

  only and necessarily introduces curvature into the shape. The Platonic solids I talked about previously are the simplest examples of shapes that are made out of a single type of vertex.

- the vertex is locally euclidean which happens when there is six triangles only : 6*60°=360°. But, the six triangles do not have to be co-planar, think of a simple crease, with three triangles on each side.
- the vertex is locally hyperbolic, the sum of angles if strictly greater than 360° which happens for any number of triangles strictly larger than 6.
  Lobel shapes do not make use of hyperbolic vertices, only types 3, 4, 5 and 6 with type 6 being use predominantly.
This criterion does not tell you much about why curvature can exist with type 6 vertices but I don't see any reason why it could not exist (which is not a proof either).
It seems more like only with type 6 can all the triangles be co-planar in a very specific and unique configuration where all dihedral angles are 180°.

To convince you even more I have made a solver for the type of faces you can put around a vertex some time ago:
https://asliceofcuriosity.fr/blog/posts/geometry8.html#3D
Put the number of faces to six, choose each face number of sides (default is 3 for the equilateral triangle),
then you can explore the configurations with the sliders that control the dihedral angles between adjacent faces.
Enjoy !

Also, thank you for proposing to help me with the code. I'll share it with you soon.

Have a good day.

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[garry fish]

Hi All,

I am very new to this group and just love the content so much. However I do not confess to the understanding of the math behind the structure/shapes described.

In my opinion Alain Lobel addressed the forming of covered spaces in a way that most builders would welcome with open arms, namely if one shape (an equilateral triangle) is made then can multiples of this be assembled into a free standing structure that is easy and cheap to build but strong and durable......

Knowing that it only takes three points to form planar surface and if we join planer surfaces together, we still get a planar surface that grows as more and more triangles are joined together. I believe that the "hubs" must be flexible enough to allow the individual triangular planar surfaces to deflect/distort to for the intended structural shape BUT what controls the cells to develop into a structure as seen below:

[Wagyx Xygaw]

Hi Gary and welcome here,

An icohedron (or icosphere) is made entirely of equilateral triangle and is not a planar surface unless you have a definition for a planar surface that i don't understand.
In the Lobel Frames, the triangles are equilateral (up to the sixth decimal), there is no distortion of the triangles involved.
For this specific family (C6), there are rotational and reflection symmetries as well as constraints that you can read in the name of the shape.

You put all of these constraints in a solver and voila !

[garry fish]

Hi Wagyx

Your reply is very much appreciated and because it looks like such a nice group, I felt happy to reach out to you all.

My understanding is that a planar surface can be defined by any three points and I looked at the concept of these structures as being made up of hundreds of equilateral triangles, all the same size and identical in every way (easy to construct in a workshop). However, if I wanted to assemble them into an agreed Alain Structure, what would drive the overall assembly to form the desired shape....? 

My guess is that the vertex of each triangle would need to be flexible and the pure fact that they were joined according to an assembly plan, then the desired structure would form almost naturally according to the plan. If this is correct, then the structure could be unstable and would therefore need to be supported/strengthened if a building was to be the final intention of the exercise. Just thought that for earthquake regions, this inherent flexibility of the frame could be beneficial.  

I love the idea of producing lots of triangles (or other shapes) that when assembled together will morph into a beautiful and striking building. Is this possible on a commercial basis?

Thanks in advance,

Garry

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[Wagyx Xygaw]

Hi Garry,

You seem to share the same dream/vision that Alain Lobel had.

I am sorry I did not get your question correctly. I cannot answer it properly because I have zero knowledge and no experience about architecture, material and real life structure engineering.

The people with the answers might be lying around here though. Let's hope some of them are investigating the matter and would like to share their discoveries, that would be awesome !

The only models I have made myself were using Geomag and can fit in one's hands so it's hard for me to tell how it would work for a building and what kind of material, joint mechanism one would need.

My experience with rod magnets and balls is that the structure will snap into shape by fiddling a bit, pushing here and there when putting the last pieces in. No idea how that works on a bigger scale.

Alain Lobel came up with a universal connector : https://www.equilatere.net/frame.php?page=en/mass/connectors.php

Or you could have only hinges linking the triangles, connecting two faces like Jovotoys, no need for a vertex then.

Regarding morphing into shape, there are origami like techniques that are quite impressive. Don't know if they are viable though, both structurally and commercially.

[Dx G]

The universal connector by Lobel is an interesting approach.  Those interested in such things might also want to look at Wangers.
https://www.wangerflange.com/

Let me also reprint the list that was posted some time ago.  It appears to me that the Lobel connector does have some of the features on the list, but misses several.  In being a purist, one would want as many of the features as possible, or preferably all, in order for it to be a truly universal connector several. 

1) Can accept a variable number of struts, often 5 or 6, but others as well.

2) A single design allows different face, axial and dihedral angles without requiring machining operations such as cutting, bending, drilling, etc.  That is, they are interchangeable without compromising strength and not risking structural failure.
  In addition, the part itself does not require precise angle and dimensional machining tolerances and is, to some extent, self adjusting.

2a)  Ideally, in all configurations, the connector successfully operates in compression, tension, torsional and other forces.

3) Can be used with different strut materials (wood, metals, pvc, even bamboo, etc.)
 
4) Can be used with different strut shapes (hollow round pipe or tubing, round solid rod, square or rectangular cross section, etc.)

5) Hub can accept plain cut struts, where the ends do not require machining (drilling, compound angle cuts, etc.) and other customized modifications.  Nor do the struts require specialized ends or caps to be attached in order to join them properly.  This objective can be a real asset for something like bamboo, where the brute force approach of using lag bolts or even machine bolts are not good choices.

  In particular, the intention is to join parts in ways that strengthen the structure, rather than weakening the connections among structural elements.
 
6) The hub design also lends itself to use with panelized domes that require various face, axial and dihedral angles for proper assembly.

7) Does not require specialized materials or parts.
Ok, this one is part of my continuing rant.
 When I look at dome parts in the patent literature, and even in the market place, too many require complicated schemes that have to be made especially for that intended use, and aren't good for anything else. We see this even more now with the use of 3D printers. Often, this approach requires different parts for different domes, or even different hubs, so hubs or strut ends are not interchangeable and have to be custom made. Although the inventors may believe that this complexity protects them from being copied, the down side of that approach is that many such inventions go nowhere and quite a few are abandoned.
  For purposes of expanding the adoption of domes, IMHO it makes more sense to use materials that are commonly available, manufactured in large volumes for something else (PVC pipe, as an example) that will likely be around for a while, whether or not they are used for domes. This also helps keep the price down and availability up, unlike some of the wacky designs I see in some of the patents that would cost a fortune to fabricate and have little chance of ever being made in volumes that would make them more affordable.
  Thus, the motivation for this item 7, here in my list.

Dx G

[Charles Lasater]

I suppose you would have to pre-assemble them as hubs then slip the tubes on.

On Sat, Aug 31, 2024 at 6:14 PM Robert Clark <clark.rob...@gmail.com> wrote:

This is a universal Lobel connector for use with tubular struts.  It's not a complete finished design but is conceptual. Could be molded or Die-cast. Better yet, it could be 3D printed in large quantities using Evolve Additive Solutions' STEP (Selective Thermoplastic Electrophotographic Process).

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[Robert Clark]

Yes, that would be the idea.  There would need to be a method for holding the parts inside the tubes so they do not slip out.  There would be just one massed produced part that could create 6-way, 5-way, and 4-way hubs that would maintain a 60 degree angle between adjacent tubes, but bend like hinges along the triangles formed by the tubes. This is just a quick stab at an idea that passed through my head. It's a bit clunky right now but I think would work.

[Charles Lasater]

Seems to me the 60 hex and 72 pent would need different parts. ?

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[Robert Clark]

No, they are all the same parts.  Remember, a Lobel frame is ONLY made from equilateral triangles. So, any two adjacent tube struts will always be 60 degrees.

[Eric Marceau]

Hey Robert,

Elegant design!

Given the difficulty in trying to insert tubes already assembled at one end into a "joint coupling", it seems to me that you would need to

• in the first instance, assemble a connector component into each tube individually (but defer their fixation until finally assembled), then
  
  
• insert and fix the pin joining adjacent tubes.

Is that how you envisioned the process?

Eric

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[Charles Lasater]

Robert, it may seem too crude but if the tube ends were pressed flat and drilled, one bolt could connect them. A  little bending might be involved but not difficult once bolted. Anyway that's what I am trying.

Love your 3D ideas.

On Mon, Sep 2, 2024 at 9:03 AM Robert Clark <clark.rob...@gmail.com> wrote:

Eric,

I gave it another try on the Lobel Universal Connector design.  This time I've taken another (simpler) approach.  The connectors are attached to the tube struts first before assembling them to other struts of a Lobel frame.  The hinge pin could be secured with a screw through the side of the tube to keep it from slipping out.  Obviously there's more finishing and cleaning up of the design that needs to be done.

Robert

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[Dx G]

I appreciate the, creative energy, time and effort on the connector venture.  Don't want to throw a wet towel on it, but I couldn't help wondering if its circling back around to connectors and hubs that are already available or common.  I'm on the hunt for assets currently lacking in the historic or contemporary options.  So the open question is, what will a new one offer that the existing brain trust doesn't already have on the radar screen?  Don't take that to mean it doesn't have one (or more), I would just like the proposed option description make a point of declaring what those (new) assets are from the inventor's standpoint. 

 Dx G 

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[Robert Clark]

Charles, you are correct that a Lobel frame could be constructed with the method of flattening tube ends and drilling holes.  In a Lobel frame, all the struts should be the same length or distance from theoretical vertex to vertex.  Depending on the Lobel frame design, differenct amounts of bending at the ends of the tube may be required. As the bend angle increases, the working length of the tube strut would shorten. It might not matter, but is something to consider.  The universal Lobel connect I'm trying to design would eliminate any variation in strut length regardless of Lobel frame type.  Also, it would be easier to disassemble and reuse the struts without unbending and rebending the ends of the tubes for other configurations. Thanks for your comment. Questions and comments are what gets me thinking creatively.

[Robert Clark]

Dx G,

Thank you for your comment. Is there already a Lobel frame connector or hub available?  I suppose the Wanger flange meets those requirements. The Lobel designed connector with the 3-way universal joint looks pretty cool (in CGI animation), but extremely expensive.

When we are talking universal connector, I have always assumed it means struts that can be positioned or adjusted to any angle or orientation.  I had not taken it to also include any cross-sectional shape of strut (square, round, hollow, solid) of any material  (wood, metal pipe, plastic tube, bamboo).  I think it would be like trying to go online and find a magical universal fastener that can attach any types of material together such as wood, metal, concrete, plastic, tile or any other material you can think of.

What we never mention for any of these proposed dome structures is the use and strength requirements.  But, it feels like it is assumed that regardless the application, the connectors and dome should be able to handle forces equivalent to supporting the weight of an elephant and surviving a tornado.

I don't know how many of the group members here are actual mechanical designers/engineers or are just dome enthusiasts.  In the engineering world, we are reigned in by realistic, physical limitations of material, cost, and procurement.  Specialized machinery is always more useful than do-all gadgetry that does nothing.  I keep coming back here hoping to see a little bit more sharing of actual design.  

As I've said earlier, I do enjoy the discussions, interaction, questions and comments because it forces me to try and think outside the box.

[Dx G]

Robert,

  Well, it does occur to me that perhaps what you are after, for this application, is a subset of the features on the earlier "universal connector".  Most prominently, all your struts are the same length, and there may be little or no immediate need accommodate different materials and/or diameters, although that would be a plus.   So perhaps all that is left is a "variable angle connector" (VAC), which readily accommodates different axial and face angles, along with different strut counts at the junction/hub.  Depending on your vision for the innovation, that name may be more reflective of the target, but if not, additions/corrections/deletions to using that name would help hone our thinking.

 The strength of the part(s) is certainly not a trivial issue. My guess is a part would need to be designed and evaluated, as-installed, before the structural software could crunch out the loads and failure modes.  One can always estimate something like a total snow or wind load on a general shell structure, but just where the compression/extenstion/torsional forces occur, and their magnitude, would have to be done on the as-built structure.   So might we say the part before the force?   :-)

Dx G

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[Dick Fischbeck]

Hi Robert

As a non-engineer, I wonder if there is an advantage in building like this when it seems to me that flat and cylindrical sheets should achieve a similar result and forms. Geodesic domes have the structural benefit of creating compound curvature at each vertex.

Lobel puts it like this. "Since the forties,a time when we witnessed various attempts to divide the sphere,(Bauersfeld and Fuller),a great deal of research has been concentrated on the equal division of space, with the obvious objective of industrialization (Peter Pearce).

Of course, dividing a flat surface or of single curvature offers little difficulty. The problem begins when it is a question of creating surfaces described as having a double curvature which have a rigid form. It should be noted however that the term double curvature loses something of its meaning when it is applied to surfaces with facets which have no tangents. 
This is the problem on which I have concentrated for several years and to which I have found several solutions which I shall present here."

Just two cents.

On Mon, Sep 2, 2024 at 12:03 PM Robert Clark <clark.rob...@gmail.com> wrote:

Eric,

I gave it another try on the Lobel Universal Connector design.  This time I've taken another (simpler) approach.  The connectors are attached to the tube struts first before assembling them to other struts of a Lobel frame.  The hinge pin could be secured with a screw through the side of the tube to keep it from slipping out.  Obviously there's more finishing and cleaning up of the design that needs to be done.

Robert

On Sunday, September 1, 2024 at 5:28:25 PM UTC-4 Eric Marceau wrote:

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[Robert Clark]

Not sure what you're talking about "flat and cylindrical sheets should achieve a similar result and forms".  Do you think Lobel frames only form flat or cylydrical shapes?

Frankly, I have no great love of Lobel frames.  I find them an interesting concept. The topic went to the connector that Lobel designed. It looked incredibly expensive to manufacture.  As a personal design challenge for myself, I came up with a concept that might be a bit simpler and cheaper.  I have yet to see others offer their design ideas. 

The advantage I saw to Lobel frames when used in conjunction with the Injection molded universal connector I just designed is that only one strut length and one universal connector can create any of the Lobel frame forms listed on the Lobel website.  I realize now how stupid that was.  I am deleting my design post so as not to cause more irritation.

[Dick Fischbeck]

On Mon, Sep 2, 2024 at 5:55 PM Robert Clark <clark.rob...@gmail.com> wrote:

Not sure what you're talking about "flat and cylindrical sheets should achieve a similar result and forms".  Do you think Lobel frames only form flat or cylydrical shapes?

Yes, except if there is a 3-way (tetrahedral), 4- way (octahedral), or 5-way (pentagonal) vertex.

Here are good images of these types of corners.

https://www.equilatere.net/frame.php?page=en/light/air.php

Frankly, I have no great love of Lobel frames.  I find them an interesting concept. The topic went to the connector that Lobel designed. It looked incredibly expensive to manufacture.  As a personal design challenge for myself, I came up with a concept that might be a bit simpler and cheaper.  I have yet to see others offer their design ideas. 

I like your models so thanks. 

[Robert Clark]

So, you didn't realize that the connector parts I designed can in fact be used with 3, 4, and 5-way vertex?  Why would a Lobel frame not include these type of vertex.  This is why I said the connector could be used to create any of those Lobel frames you just showed.  Anyway, my design challenge is done and we don't need to talk about the connector again. Thanks.

[Dick Fischbeck]

Yes, of course, but the point I'm making is that the surfaces between those 3 types of vertexes can only be flat, cylindrical or conical. No compound curvature, not very strong relatively speaking.

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