operational math

[Dick Fischbeck]

why is no one doing it

[Bryan L]

What do you mean by operational math? 

Gerry helped me get my head around Spherical Trig and Hector helped me with some higher level stuff. I have many spreadsheets (rough around the edges maybe) to work out any type of dome calculations. Over the years Gerry has posted many, with colour formatting showing input areas and results...

What do you want to see?

[Dx G]

Is this what you mean by Operational Math or are you referring to something else?   Why is it important, and how do you see it being used?

https://mathworld.wolfram.com/OperationalMathematics.html

DxG

[Adrian Rossiter]

Hi Dick

On Fri, 15 Sep 2023, Dick Fischbeck wrote:
> why is no one doing it

Because it is for people who base their maths on the properties of paper?

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

[Dick Fischbeck]

Hi Adrian- That's a pretty good definition. I know not everyone here thinks Fuller is worth studying but he thought math needed to come from experience and not from postulates and axioms like dimensionless points and planes without thickness.

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[Bryan L]

If you want to describe a face of a cube, triangle on a dome, a triangle drawn on a piece of paper, any flat surface, a dot on a piece of paper, you don't need to think about thickness. The mathematical concepts that evolved came from the experience of far greater minds than Fuller.

The whole IVM volume thing only applies to regular tetrahedrons / octahedrons. As soon as you deviate from them it is of no value. Trying to describe volume in terms of tetrahedrons only applies to a solid that can be divided into equivalent Regular sized tets. Even then saying something has a volume of x tets, you need to specify an edge length or height, to give it any meaning. 

The 3D cartesian coordinate system is the minimal mathematical model that describes the spatial world we live in. Introducing a 4th dimension may simplify things when dealing with, once again, Regular tets / octahedrons, but it stops there. The cubic packing of tets is NOT the only packing found in nature, and using IVM to describe anything other than that fails miserably.

The postulates and axioms are necessary to provide a solid basis for what follows. They have withstood the test of time and there is no magic bullet that will simplify them. Sure they can be presented in different ways to help one grasp the ideas, but the basis remains.

Fuller was a salesman. He liked to talk as if he had found something new or revolutionary. He hadn't. It was just another view on what was already known. He didn't even build the first dome.

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[Dick Fischbeck]

Hi Bryan- You may be stating the prevailing view in this group (and most groups). As you probably know, it is not one I subscribe to.

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[Dick Fischbeck]

I'd say the paper has some stretch in it, too. Fold the paper and you have a compression element. That's just the way I think about the ivm as opposed to a rigid xyz. So it is more about topology than edge lengths. I see edge lengths as being secondary to angles.

[Dick Fischbeck]

I haven't posted claim 1 in a long time. Some might find it interesting in the context of ivm v xyz.

1. A geodesic structure comprising a plurality of conical elements, each conical element of said plurality of conical elements being a structurally single component having a cone base, a cone wall and a vertex, said cone wall defined by straight lines that extend from said base and intersect each other at said vertex, the length of a straight line from said vertex to said cone base defining a cone-wall length, wherein said plurality of conical elements are arranged in an overlapping arrangement, so as to form a shell that surrounds an inner volume, wherein a portion of said base of a first conical element overlaps with a portion of said cone wall of an adjacent conical element, such that at least one straight line of said cone wall of said first conical element extends substantially parallel to at least one straight line in said cone wall of said adjacent conical element so as to form together a straight strut between said vertex of said first conical element and said vertex of said adjacent conical element, and wherein said plurality of conical elements are arranged such that a distance and a direction of displacement between any two vertexes of adjacently placed conical elements provides an adjustability of said straight strut that is a strut distance that is infinitely variable between a minimum limit and a maximum limit by adjusting an amount of overlap, said maximum limit of said straight strut being slightly less than a sum of cone-wall lengths of any two adjacent conical elements and said minimum limit of said straight strut is being slightly greater than said cone wall length of one of said two adjacent conical elements.

[Gerry in Quebec]

Hi Dick,

Here's how I think of the wonder and power of mathematics, especially geometry:

Human experiences and observations of the natural world are absorbed by the human brain/intellect and transformed into abstract concepts, generalizations, comparisons and associations. Then, with the help of other abstractions we call numbers, those cerebral processes generate rules, tools and instructions for organizing space-time into a multitude of novel patterns, i.e., ones that differ from, but build on, the original human experiences and observations. There is synergy here: the output is richer than the sum of the initial experiences and observations.

So, mathematical axioms, postulates, and conjectures are complements or extensions of human experience, not its competitors. My 2 cents' worth...

- Gerry in Québec

[Dick Fischbeck]

Since we are talking about paper, if you don't know Bradford, it's worth a look.

https://www.wholemovement.com

[Dick Fischbeck]

Hi Gerry- I don't see any competition between the two coordinate systems. They both work. However, ivm is an infant.

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[Hector Alfredo Hernández Hdez.]

I am writting
to Eric Marceau
he lives in Quebec.

about maths in geodesic design
(about the Mexican method in particular)

It will be necessary to write a book, with basic software included
(spreadsheet, visualizations, constructions in Geogebra and programs in C language)?

[charles lasater]

Thank you for this wonderful contribution. What fun!

 

Sent from Mail for Windows

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[Robert Clark]

Dick, could you post your claim 1 from your patent in your own simpler layman speak rather than patent attorney speak?  The legalese is just to hard to follow.  Thanks.

[Eric Marceau]

Hi Dick,

What you posted below is a perfect example why I gave up on trying to harvest any kind of knowledge from Fuller's published works! 

There is a time for words, and there is a time for math.

If the person (who conceives of a concept) has someone else, a collaborator, with whom he can share his ideas, it falls onto them both to define the relevant math that embodies the concept being conveyed.  In that manner, it would ensure that the fullness of that knowledge and wisdom is correctly communicated in unambiguous form to posterity!

To have, instead, deliberately chosen to convey formulaic knowledge thru mostly words,
is either

an obvious lack of foresight ... for someone who saw himself as a futurist,

or

a deliberate act on his part ... to ensure that his words will be debated until the end of time, knowing full well that they are at times open to interpretation.

I prefer to not, for lack of a better word, "waste" my time trying to plumb the depths of the deep, dark waters of Fuller's writings, because I don't have the necessary "bright lights" to illuminate the dark spots in the messages being communicated.

I prefer, and am quite happy to continue doing so, to rely on second hand interpretations by those who, because of their love for the material, or adoration of the person, devote the necessarily huge amount of time to try to absorb, digest, interpret and elucidate what his writings, regretfully, encrypt!

To you, Dick, I commend you for the obvious patience and dedication that you appear willing to devote to such material.  You demonstrate the tenacity that is required to pursue such mysteries until they are revealed.

Unfortunately, I cannot allow myself to follow you into what I refer to as "the rabbit hole".

Eric

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[Eric Marceau]

Hector,

I believe that what Dick is referring (Operational Math) to is unrelated to what we are doing regarding the "Mexican" method.

Operational math appears to be focused on IVM-based referential geometry, which is of no interest to me for reasons better stated by Bryan L. in his earlier response to Dick.

Eric

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[Hector Alfredo Hernández Hdez.]

ok, thanks

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[Curt McNamara]

Greetings Eric!

FYI: what Dick posted was the first claim of a patent. When you create a patent, it is required to describe things in English. If Dick posts the patent number, you could look at the drawings that the language explains.

                Curt

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[Dick Fischbeck]

Hi Robert- I could do that but you might like to read this intro. Let me know if this does or doesn't make the claim clearer. I am happy to answer any question.

https://archive.bridgesmathart.org/2004/bridges2004-347.pdf

[Dick Fischbeck]

Hi Eric- I did give you a spreadsheet recently (which is pretty mathy).

For anyone interested: https://patents.google.com/patent/US7389612B1/en

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[Dx G]

Dick,

 Here is a question for you.  The text of that article on Bridges mentions that frequency is not part of the dimensional calculations.  However, I've seen domes that look a lot like Randomes with configurations that resemble frequency.

For example, if you look at the photos on Intershelter, what you see are cones that rather look like they would cover a full pentagon and/or hexagon.  The cone tip looks like where you would raise the center of a given polygon at the hub connecting the radial chords at or near the surface of the sphere. If you trace the fasteners, they take a path that looks a lot like where polygon perimeter struts would be.

https://intershelter.com/

If we compare that to the photo in your article, you can see that the cones have a much smaller radius, so that the circular overlap lands at the approximate center of a polygon perimeter chord.   

 Perhaps you could explain what it is I'm seeing there with respect to the cone size, or other relevant factors.  It rather resembles frequency, but perhaps just an illusion?

thx

DxG

[Dick Fischbeck]

Not sure I understand your question yet, but Craig Chamberlain's omnisphere, aka intershelters, are made of compound curved elements and not simply curved elements. 

And that has made all the difference. -cite Rober Frost  

On Mon, Sep 18, 2023 at 6:15 PM Dx G <yipp...@gmail.com> wrote:

Dick,

 Here is a question for you.  The text of that article on Bridges mentions that frequency is not part of the dimensional calculations.  However, I've seen domes that look a lot like Randomes with configurations that resemble frequency.

For example, if you look at the photos on Intershelter, what you see are cones that rather look like they would cover a full pentagon and/or hexagon.  The cone tip looks like where you would raise the center of a given polygon at the hub connecting the radial chords at or near the surface of the sphere. If you trace the fasteners, they take a path that looks a lot like where polygon perimeter struts would be.

https://intershelter.com/

If we compare that to the photo in your article, you can see that the cones have a much smaller radius, so that the circular overlap lands at the approximate center of a polygon perimeter chord.   

 Perhaps you could explain what it is I'm seeing there with respect to the cone size, or other relevant factors.  It rather resembles frequency, but perhaps just an illusion?

thx

DxG-- 

[Dick Fischbeck]

https://memorialutah.com/obituaries/craig-chamberlain/

http://www.tedhayes.us/domevillage/JHUSA.html

https://patents.google.com/patent/US3203144A/en

[Dick Fischbeck]

My mistake. Should be:

https://patents.google.com/patent/US4270320

[Dx G]

Ok, let's look at it another way.  For triangle element domes, you can make a dome larger by using larger triangles, or by increasing the number of triangles without increasing their size.

What about randomes?  If you start with a given dome size and cone size, if you want a bigger dome but not such big cones, is there a process for that - making a larger dome from more numerous cones?

thx DxG

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[Dx G]

Ah, very good.  Look at Figure 4.  He shows just what I described. The cone center is the polygon center (raised), and the fasteners fall essentially on the strut lines of the polygon perimeter.    This is different from what is illustrated in your articles, where the cones are much smaller than the polygon, approx half the radius.

  I don't see a compound angle there, it looks like the cone is made by taking out a pie slice, like yours. Or am I missing something?

thx DxG

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[Dick Fischbeck]

Hi dxg- 

On Mon, Sep 18, 2023 at 7:07 PM Dx G <yipp...@gmail.com> wrote:

Ok, let's look at it another way.  For triangle element domes, you can make a dome larger by using larger triangles, or by increasing the number of triangles without increasing their size.

 I don't use triangular elements to build a dome. I use vertex elements. These elements have an angular defect or deficit based on dividing 720 degrees, the curvature of a sphere or tetrahedron, by the number of elements.

What about randomes?  If you start with a given dome size and cone size, if you want a bigger dome but not such big cones, is there a process for that - making a larger dome from more numerous cones?

Oh, yes. 720 degrees / number of elements. 

thx DxGlp+unsu...@googlegroups.com

[Dick Fischbeck]

Like to build an icocaherdon, that is 720/12 or 60 degrees.

[Dx G]

Great, I like the simplicity.  So if you want to estimate the diameter of the cones for a given dome diameter and given number of elements, or, want to estimate the dome diameter from a given number of cones of a given cone diameter, is there an equally simple relationship? 

thx DxG

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[Dick Fischbeck]

Yes, I think so.  I wrote this a while back.

"For a hemisphere-

2.4 A / a = N

Big 'A' is the area of the dome footprint.

Little 'a' is the area of the vertex element, independent of shape (circle, square, etc.)

N is the number of elements.

The constant allows for a 20% overlap of the elements."

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[Ashok Mathur]

Dear Hector

You have stated your case quite nicely.

But I urge you to remember Dick is one person who has undermined/ blasted Fuller quite a lot by showing that geodesic domes do not derive their strength by a precise layout of the vertices on the surface of a sphere .

No maths at all is needed to a strong structure!

Ashok

Sent from my iPhone

On 19-Sep-2023, at 12:27 AM, Hector Alfredo Hernández Hdez. <hecto...@gmail.com> wrote:



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[Adrian Rossiter]

Hi Dick and DxG

On Mon, 18 Sep 2023, Dick Fischbeck wrote:
> Yes, I think so. I wrote this a while back.
>
> "For a hemisphere-
>
> 2.4 A / a = N
>
> Big 'A' is the area of the dome footprint.
>
> Little 'a' is the area of the vertex element, independent of shape (circle,
> square, etc.)
>
> N is the number of elements.
>
> The constant allows for a 20% overlap of the elements."

I had a quick look at a derivation of the formula.

The area, A, of a circle with radius R is: A = pi * R^2

The area, H, of a hemisphere above it is: H = 2 * pi * R^2

H = 2 * A

Divide H into N equal areas to use as vertex elements: b = H / N

H / V = N
2 * A / b = N

Let a be a vertex element 20% bigger than b: a = 1.2 * b

b = a / 1.2

Giving

2.4 * A / a = N

[Dx G]

Adrian,

 Thanks for the calcs, but wouldn't the area of a hemisphere be half, rather than twice the area of the circle?  And for the sake of continuity, how is V defined?  Or did I miss something there...

thx DxG

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[Adrian Rossiter]

Hi DxG

On Tue, 19 Sep 2023, Dx G wrote:
> Thanks for the calcs, but wouldn't the area of a hemisphere be half,
> rather than twice the area of the circle?

The curved surface of a hemisphere is half the area of the
corresponding sphere, and twice the area of the planar circle
whose circumference is the equator of that sphere (i.e. the
"base" of the hemisphere).

> And for the sake of continuity,
> how is V defined? Or did I miss something there...

That is a typo, 'V' should have said 'b' (and magically becomes
'b' in the line after it is used). I changed letters as 'b' is more
like Dick's 'a'.

[Curt McNamara]

Greetings Bryan! 

Numerical analysis software (which includes both finite element and electromagnetic modeling) uses irregular tets as the fundamental unit of analysis. Any object can be modeled with irregular tets.

Check the numerical analysis section here. The article also has a great summary of irregular tet properties.

https://en.m.wikipedia.org/wiki/Tetrahedron

The software most likely uses 3-tuples to specify the location of the tet corners.

Hope this helps! BTW Edison didn't invent the light bulb. An engineer world say he reduced it to practice.

      Curt

On Sun, Sep 17, 2023, 8:47 AM Bryan L <bhla...@gmail.com> wrote:

If you want to describe a face of a cube, triangle on a dome, a triangle drawn on a piece of paper, any flat surface, a dot on a piece of paper, you don't need to think about thickness. The mathematical concepts that evolved came from the experience of far greater minds than Fuller.

The whole IVM volume thing only applies to regular tetrahedrons / octahedrons. As soon as you deviate from them it is of no value. Trying to describe volume in terms of tetrahedrons only applies to a solid that can be divided into equivalent Regular sized tets. Even then saying something has a volume of x tets, you need to specify an edge length or height, to give it any meaning. 

The 3D cartesian coordinate system is the minimal mathematical model that describes the spatial world we live in. Introducing a 4th dimension may simplify things when dealing with, once again, Regular tets / octahedrons, but it stops there. The cubic packing of tets is NOT the only packing found in nature, and using IVM to describe anything other than that fails miserably.

The postulates and axioms are necessary to provide a solid basis for what follows. They have withstood the test of time and there is no magic bullet that will simplify them. Sure they can be presented in different ways to help one grasp the ideas, but the basis remains.

Fuller was a salesman. He liked to talk as if he had found something new or revolutionary. He hadn't. It was just another view on what was already known. He didn't even build the first dome.

On Sun, 17 Sept 2023 at 22:06, Dick Fischbeck <dick.fi...@gmail.com> wrote:

Hi Adrian- That's a pretty good definition. I know not everyone here thinks Fuller is worth studying but he thought math needed to come from experience and not from postulates and axioms like dimensionless points and planes without thickness.

On Sun, Sep 17, 2023 at 3:08 AM Adrian Rossiter <adr...@antiprism.com> wrote:

Hi Dick

On Fri, 15 Sep 2023, Dick Fischbeck wrote:
> why is no one doing it

Because it is for people who base their maths on the properties of paper?

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

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[Curt McNamara]

OK, should have searched a bit more. Some of you may be interested in barycentric coordinates.

     Curt

https://en.m.wikipedia.org/wiki/Barycentric_coordinate_system

[Dx G]

That's pretty interesting, I'll have to look at it in more detail.

Eric and I were discussing TINs, which go way back before the 1990's. Primarily for surfaces rather than volumes.  They may have some assets for certain efforts.  He sent this one.

https://gistbok.ucgis.org/bok-topics/triangular-irregular-network-tin-models

DxG

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[Dick Fischbeck]

Hi Robert- I found an old explanation of the invention in plain English. I might as well add it to this thread so it can be more easily accessed to anyone interested.

On Sun, Sep 17, 2023 at 9:58 PM Robert Clark <clark.rob...@gmail.com> wrote:

Dick, could you post your claim 1 from your patent in your own simpler layman speak rather than patent attorney speak?  The legalese is just to hard to follow.  Thanks.

 The RanDome

Buckminster Fuller is well known for his determined effort to address the global shelter shortage by using his concept of Comprehensive Anticipatory Design Science. Design science strategy focuses on using the world's resources to the greatest advantage possible. One measure of a dwelling's success is its performance per kilogram of material. By increasing a material's performance through design, and thus the efficacy and efficiency of both the cost and labor of construction, the possibility for sheltering all the world's population will also increase.
Fuller made great progress in his efforts to address the problems of shelter. The geodesic dome is a milestone in the history of shelter. The RanDome is an advancement in building design that will take Fuller's geodesic shells to new levels of functionality and versatility. The RanDome is a new kind of geodesic shell which is far simpler than existing space enclosing building strategies. RanDomes are mass-produced and assembled from a variety of materials. RanDomes are inexpensive, strong, long lasting and weatherproof - exactly the type of structure to address the global problem of inadequate shelter.
The RanDome achieves these benefits by changing, fundamentally, how we view the basic building element. The RanDome structure is not an assemblage of edges and faces, as are traditional shell structures. It is assembled from a number of overlapping cones called vertex elements. A RanDome has all the structural benefits of a traditional geodesic shell structure but without the complexity of design, manufacture and construction. The RanDome construction strategy is intuitive. It is a Trimtab improvement on mass shelter construction by virtue of elegant simplicity. This building strategy needs no advanced educational degrees or construction experience. It transcends literacy, language and cultural barriers. Everyone, even children, can now build a sturdy, weatherproof and healthy shelter.
A RanDome intended for human shelter is assembled from numerous identical vertex elements. Vertex elements are cones. A cone is fabricated from a sheet of material. We use a typical sheet in the shape of a circle, square or rectangle, whose maximum dimension is usually one meter. The semi-rigid sheet material can be any kind available. The preferred material is weather-, insect-, rot- and fire- resistant. Examples of suitable materials are metal, corrugated plastic, plywood, cardboard, fiberboard, fiberglass, or similar products.
That all vertex elements are identical leads directly to manufacturing efficiency. When one is fabricating a large number of identical elements, the economies of scale and ease of process are best taken advantage of.
This is a construction method previously unknown and unexploited. Some basic mathematics can be used to explain our unique approach. Until now builders have assembled their structures using only two of the three topological features. Leonard Euler discovered 257 years ago that all polyhedrons can be reduced no further than into edges, faces and vertexes. For example, a cube has 8 vertexes, 6 faces and 12 edges. People build shelters using face elements or edge elements. People do not yet build shelters using vertex elements. The basic building component of a RanDome is a vertex element and this makes all the difference.
Cones are simple to calculate and easy to fabricate. The fabrication begins with a flat sheet of building material. An angle is made in the shape of a narrow triangle from the center to the perimeter of the sheet. The sheet is then cut and lapped, or folded without cutting, according to this angle, to create a vertex, thus producing a cone element. The angle of this lap or fold is determined using simple mathematics. The relationship of the individual angle in degrees to the whole structure is 720/n, where n is the number of cone elements employed. For example, if 100 sheets are used for a half-sphere structure, the required angle is 360 degrees divided by 100 pieces, or 3.6 degrees per angle.
The cones can be mass produced and stacked efficiently for shipping. They are assembled on-site, or pre-assembled before deployment, by anyone using a technique similar to shingling a roof. The elements are overlapped just as one would overlap shingles on a traditional roof. The upper element extends outside and over top of the lower element. Rain water is forced to shed onto the ground. Leaks are virtually impossible. Elements are attached to each other with common fasteners such as nuts and bolts or rivets. Double adhesive tape, magnets or clamps hold the elements in place during this fastening process. Each fastener is made weather tight. Erection of the shell structure can proceed from the top to the bottom or the reverse. Top down erection is preferred, raising the structure during the process of erection, to allow all work to take place at ground level. Doors, windows and vents are installed in the traditional manner, adding frames where needed.
One of the fascinating and beautiful aspects of the RanDome is that the placement or arrangement of the structure's elements during erection is done in an approximately random fashion. No precise measurements are required. Elements are overlapped and fastened together with a predetermined average distance between cone vertexes. A 50% overlap of the cone elements is typical. In climates with heavy snow and wind loads expected, or when the available sheet material is relatively thin, overlap will be maximized to strengthen the shell. Double and triple layers of elements are desirable depending on the expected function of the shell structure. This layering makes possible very strong domes using very thin sheet material. In extreme climates, either very hot or very cold, two or three concentric shells with an air space between the shells will be useful. For added strength, these shells can be bridged with short connectors such that one shell supports the other. Finally, with slightly more complexity, any shape RanDome is possible just by varying the cone angle of the elements of a particular structure.

[Dick Fischbeck]

Hi Robert- One more explanation, this time in pictures and very few words. Joe Clinton put this together. Thanks, Joe.

https://vimeo.com/412523832

[Robert Clark]

Very nice.  Are you currently manufacturing the Randomes?

[Dick Fischbeck]

Oh, no. I know nothing about business! I am a special ed tech at the local high school resource room. I am retired but love what I do so I'm going to do it for as long as I can.

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[Ashok Mathur]

Dear DxG

Tins are new and fascinating to me.

Are there open source software for playing tins?

Regards 

Ashok

Sent from my iPhone

On 20-Sep-2023, at 4:56 AM, Robert Clark <clark.rob...@gmail.com> wrote:

Very nice.  Are you currently manufacturing the Randomes?

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[Dx G]

Ashok,

 I haven't made much use of TINs myself, but if you do a search on [software for tins surface modeling]  you will see quite a few options and related info. 

DxG

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[Adrian Rossiter]

Hi Curt

On Tue, 19 Sep 2023, Curt McNamara wrote:
> OK, should have searched a bit more. Some of you may be interested in
> barycentric coordinates.

I use a structure similar to the paper for the Antiprism wythoff
program, where points are specified as barycentric coordinates
and placed in triangles of a mesh (derived from a polyhedron).

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

Here is an example animation where the single point of the snub pattern
is moved around the perimeter of thw reference triangles (of a mesh
derived from an icosahedron)

https://antiprism.com/misc/anim_wyt_snub2.gif

[Adrian Rossiter]

On Wed, 20 Sep 2023, Adrian Rossiter wrote:
> Here is an example animation where the single point of the snub pattern
> is moved around the perimeter of thw reference triangles (of a mesh
> derived from an icosahedron)
>
> https://antiprism.com/misc/anim_wyt_snub2.gif

Also, the tetrahedron version is fun

https://antiprism.com/misc/anim_wyt_snub_tet2.gif

[Dx G]

Adrian

Quite nice.  Would be interesting if one could control the animation speed or step through it.  

Have you done any that illustrate various tessellations?  Those would be useful for explaining this for Class 1, 2 and 3 domes and related polyhedra. 

DxG

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[Gerry in Quebec]

Small Stella, by Rob Webb, is commercial software promoted as a "simple program for viewing polyhedra and printing nets for their physical construction". Like Adrian's dynamic images, Small Stella can morph a polyhedron into its dual. Small Stella has a pretty good library of polyhedra. 

 

https://www.software3d.com/

- Gerry in Québec

[Dx G]

Good lead Gerry.  Do you know if Small Stella does tessellations?

DxG

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[Adrian Rossiter]

Hi DxG

On Wed, 20 Sep 2023, Dx G wrote:
> Quite nice. Would be interesting if one could control the animation speed
> or step through it.

I think there are programs that will let you step through the frames
of a GIF.

> Have you done any that illustrate various tessellations? Those would be
> useful for explaining this for Class 1, 2 and 3 domes and related
> polyhedra.

The Antiprism wythoff/conway programs include many Conway operators,
including the Class III geodesic pattern and the "Class III" ortho
patterns, and all the operators can be applied to any polyhedron.

conway u3_5 ico -f w | antiview
conway o3_5 cube -f w | antiview

or, matching tiles to the colours of the original faces

off_color -f U ico | wythoff -c u_3_5 -C A,f | antiview
off_color -f U cube | wythoff -c o_3_5 -C A,f | antiview -t no_tri

I posted a dome example before here

https://groups.google.com/g/geodesichelp/c/Hr_QJbKx2CI/m/Nu36-UBdBAAJ

[Gerry in Quebec]

Hi DxG,

There are 22 tessellations of geodesic spheres in the Small Stella library. These vary from 1v to 6v, and include tetrahedral, octahedral and icosahedral polyhedra. But it seems to be more of a potpourri than a systematic presentation of geospheres.

Small Stella's big sister program, Great Stella (which I don't have), is more powerful as it is a polyhedral design tool rather than a simple display/educational tool. A year or so before his death in 2017, the well known American mathematician and polyhedron modeler, Father Magnus Wenninger, told me he had adopted Great Stella as his preferred modeling tool because he was getting tired of doing a lot of heavy lifting with simpler math tools.

But I digress!

Cheers,

- Gerry in QC

[Adrian Rossiter]

On Wed, 20 Sep 2023, Adrian Rossiter wrote:
> including the Class III geodesic pattern and the "Class III" ortho
> patterns, and all the operators can be applied to any polyhedron.
>
> conway u3_5 ico -f w | antiview
> conway o3_5 cube -f w | antiview
>
> or, matching tiles to the colours of the original faces
>
> off_color -f U ico | wythoff -c u_3_5 -C A,f | antiview
> off_color -f U cube | wythoff -c o_3_5 -C A,f | antiview -t no_tri

In case it isn't clear, the second models are connected like the
first, here is the canonical vesion of the planar geodesic icosahedron

off_color -f U ico | wythoff -c u_3_5 -C A,f | canonical | antiview

[Eric Marceau]

Just making this point, in case anyone has insights about this ...

Interesting that the vertex coordinate system is assigned in clockwise fashion (each side being a vector pointing towards the origin of the adjacent edge's vector).

I that out because of the distinction between that and the system used in many reference grids that have been shared in the group when discussing the "Mexican Method", where the numbering is only 2 values assigned from a common edge origin point at one of the base face's vertices.

Eric

On 2023-09-19 10:14, Curt McNamara wrote:

OK, should have searched a bit more. Some of you may be interested in barycentric coordinates.

     Curt

https://en.m.wikipedia.org/wiki/Barycentric_coordinate_system

On Tue, Sep 19, 2023, 9:01 AM Curt McNamara <cur...@gmail.com> wrote:

Greetings Bryan! 

Numerical analysis software (which includes both finite element and electromagnetic modeling) uses irregular tets as the fundamental unit of analysis. Any object can be modeled with irregular tets.

Check the numerical analysis section here. The article also has a great summary of irregular tet properties.

https://en.m.wikipedia.org/wiki/Tetrahedron

The software most likely uses 3-tuples to specify the location of the tet corners.

Hope this helps! BTW Edison didn't invent the light bulb. An engineer world say he reduced it to practice.

      Curt

On Sun, Sep 17, 2023, 8:47 AM Bryan L <bhla...@gmail.com> wrote:

If you want to describe a face of a cube, triangle on a dome, a triangle drawn on a piece of paper, any flat surface, a dot on a piece of paper, you don't need to think about thickness. The mathematical concepts that evolved came from the experience of far greater minds than Fuller.

The whole IVM volume thing only applies to regular tetrahedrons / octahedrons. As soon as you deviate from them it is of no value. Trying to describe volume in terms of tetrahedrons only applies to a solid that can be divided into equivalent Regular sized tets. Even then saying something has a volume of x tets, you need to specify an edge length or height, to give it any meaning. 

The 3D cartesian coordinate system is the minimal mathematical model that describes the spatial world we live in. Introducing a 4th dimension may simplify things when dealing with, once again, Regular tets / octahedrons, but it stops there. The cubic packing of tets is NOT the only packing found in nature, and using IVM to describe anything other than that fails miserably.

The postulates and axioms are necessary to provide a solid basis for what follows. They have withstood the test of time and there is no magic bullet that will simplify them. Sure they can be presented in different ways to help one grasp the ideas, but the basis remains.

Fuller was a salesman. He liked to talk as if he had found something new or revolutionary. He hadn't. It was just another view on what was already known. He didn't even build the first dome.

On Sun, 17 Sept 2023 at 22:06, Dick Fischbeck <dick.fi...@gmail.com> wrote:

Hi Adrian- That's a pretty good definition. I know not everyone here thinks Fuller is worth studying but he thought math needed to come from experience and not from postulates and axioms like dimensionless points and planes without thickness.

On Sun, Sep 17, 2023 at 3:08 AM Adrian Rossiter <adr...@antiprism.com> wrote:

Hi Dick

On Fri, 15 Sep 2023, Dick Fischbeck wrote:
> why is no one doing it

Because it is for people who base their maths on the properties of paper?

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

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[Bryan L]

Hi Eric,

fair observation.

Because most of the work of the group was on the subdivision of a Principal Polyhedral Triangle (PPT), it was enough to describe a subdivision on just one triangle. All of the other polyhedron triangles being reflections, rotations or translations of the original.

In the interests of simplicity, an x, y coordinate system was then used to define vertices within the PPT. I remember Hector putting it forward in the Mexican threads but it was used for other subdivision methods as well.

Hope that helps,

Bryan

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[Adrian Rossiter]

Hi DxG

On Wed, 20 Sep 2023, Dx G wrote:
> Have you done any that illustrate various tessellations? Those would be
> useful for explaining this for Class 1, 2 and 3 domes and related
> polyhedra.

I remembered, depending on your interests, there is also the unitile2d
program, that will lay out uniform tilings of the plane on a number of
well-known surfaces

https://www.antiprism.com/programs/unitile2d.html

Here is an example of tiling a torus with unitile2d and then applying
a geodesic pattern to it

https://groups.google.com/g/geodesichelp/c/rixxwHF02iQ/m/WLdQG3uxEA0J

[Hector Alfredo Hernández Hdez.]

@Bryan share all designs of geodesics of a 5 different lengths struts, there are a lot of 😃

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