How to make any shape out of triangles?

[FLWQ]

Hi,

I'd like to understand how you can construct any shape (not just
domes) out of triangles, with the triangles being as similar (and big)
as possible.

Does anyone know of some search terms / sources that can help me on my
way?

[\Constructor Structural\]

To you, kind time!
How to make any shape out of triangles?
Very simply
See picture!
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**********************************************
О©╫О©╫О©╫О©╫О©╫О©╫О©╫лёО©╫О©╫О©╫О©╫ О©╫О©╫О©╫О©╫О©╫О©╫, О©╫О©╫О©╫О©╫О©╫О©╫О©╫ О©╫О©╫О©╫О©╫!
See picture!

Fri, 15 Oct 2010 09:59:34 -0700 (PDT) О©╫О©╫О©╫О©╫О©╫О©╫ О©╫О©╫ FLWQ <alfreds...@gmail.com>:

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[Alfred van Dijk]

Thanks for your reply :)

I think I mean something like this:

http://www.google.com/images?um=1&hl=en&safe=off&q=triangulated%20surfaces&ie=UTF-8&source=og&sa=N&tab=wi&biw=1439&bih=725
http://www.cgal.org/Manual/3.3/doc_html/cgal_manual/Surface_mesh_simplification/Chapter_main.html
http://mathworld.wolfram.com/Triangulation.html

My geometry will be simpler.

After thinking about it a bit, my question I think becomes: are there programs or methods that let me choose the regularity of the triangles that result?

(For me it is better if all the sides of the triangles have about the same length, and all the triangles are about the same size.)

[Ken G. Brown]

Computer graphics, scientific visualization, finite element analysis, Computational Fluid Dynamics might be some areas to look for mesh generation techniques for arbitrary forms.
Possible search terms: triangular mesh generation

Ken G.Brown

[TaffGoch]

Alfred,

 

With spheres, it's easy to take advantage of symmetries (rotational & mirror) because a sphere is a surface that can be described mathematically.

 

With irregular shapes, there is no simple (mathematical) way to describe the surface. It is done, however, with all 3D renderings (think Avatar.)

 

You can search for "3D" and "mesh"

http://images.google.com/images?as_q=3d+mesh

 

Finding the most-efficient triangular mesh network is the subject of current scholarly research, patents and "proprietary information" advances -- lots of competition to find the fastest, most-efficient computer algorithms. (Way over my experience & pay-grade!)

 

-Taff

[TaffGoch]

 

I do know that a plugin, for SketchUp, is available, that will simplify an existing mesh, similar to the "hand" images you attached. The plugin, however, does not provide for comparably-sized and shaped triangles. (With the plugin, you still have to start with a more-complex mesh, upon which to work.)

 

The only mesh methodology, of which I am familiar, that produces comparable triangles, are for reasonably "horizontal" meshes, such as a landscape or terrain surface (survey results.)

 

-Taff

[TaffGoch]

Alfred,

 

None of the above are to say that such a program/algorithm is not available. I'm merely writing of my personal familiarity.

 

For example, many meshes start out by laser-scanning an object (the hand, for example.) The laser-scanning routine surely has an "interval" sub-routine that determines the size and spacing of vertices.

 

-Taff

[FLWQ]

Thanks for your replies Taff and Ken :)

I think this should give me a useful start.

[Gerald de Jong]

It's an optimization problem, and it will depend a lot on what
constraints you impose. Is it a definable surface that you want to
triangulate, for example? The stuff I've done with elastic intervals
does a kind of optimization like this, where everything individually
"strives" for a length but adjacent ones have to compromise if it
can't be reached.

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skype: beautifulcode
ph: +31629339805

[Alfred van Dijk]

I'm still not clear on the exact shapes I want, but they will be much more simple than the hand I posted earlier.

They will probably be "rounded" buildings. The sides of the building then can be thought of as sections of a sphere, on which the triangles can be drawn. The difficulty lies, I think, in the corners.

Possible building blocks (which might then be combined) attached.

In the Eden Project / rounded cube example, you can use the normal method for the surfaces. But what about the corners?

And how to construct the Russian dome?

[Alfred van Dijk]

(Sorry if I wasn't very clear before. I'm still thinking ;)   )

[TaffGoch]

Alfred,

 

You can see what others have done, with specific shapes, by searching within the 3D Warehouse.

 

For example, "onion dome"

http://sketchup.google.com/3dwarehouse/search?q=onion+dome&styp=m&num=50

 

...and "rounded cube"

http://sketchup.google.com/3dwarehouse/search?q=rounded+cube&styp=m&num=50

 

The Eden domes, are, of course, spherical geodesic constructs.

 

-Taff

[Alfred van Dijk]

Thanks, I'll continue searching later, on the theory, programs, and examples. For now I haven't found these kinds of shapes triangulated yet.

[Richard Fischbeck]

More ideas on irregular surfaces.

http://www.fishrock.com/conics/default.html

http://eat-a-bug.blogspot.com/2009/05/more-geodesic-shells.html

http://eat-a-bug.blogspot.com/2009/05/geodesic-shell.html

See google images "Shigeru Ban".

[Camilla Fox]

One (inefficient, but conceptually easy) approach is to generate the
Delaunay triangulation on a set of starter points, then use the
longest/shortest edges to iteratively massage the data. The package
'qhull' is what I used, and I needed some dummy points inside and
outside the shape to make it work out nicely.

Starting with random, and massaging it in the direction you want works
well for shapes that are mathematically definable (I did it for a
sphere and torus) and where the look you want does not emphasize the
underlying symmetry.

-Camilla

[Richard Fischbeck]

In this picture, I don't see pentagon but I do see stretched hexagons around the edges. No one ever said the pentagons could not be in the same hemisphere.

http://myrmecos.files.wordpress.com/2008/09/hethead1.jpg

[Richard Fischbeck]

Here's one with pentagons.

http://biology.touchspin.com/images/2006.12.23.19%20Dragonfly%20EyePOI1.100x.LF.7.jpg

[Hector Alfredo Hernández Hdez.]

  I do see in the image pentagons, search for "circles" with 5 "circles" around, pay close attencion

On Sat, Oct 16, 2010 at 6:32 AM, Richard Fischbeck <dick.fi...@gmail.com> wrote:

In this picture, I don't see pentagon but I do see stretched hexagons around the edges. No one ever said the pentagons could not be in the same hemisphere.

http://myrmecos.files.wordpress.com/2008/09/hethead1.jpg

--

[homespun]

Dick,

    Please state what this is a picture of, or post a link that will lead to it's description.

         Thanks,

         Dan Suttin

[Richard Fischbeck]

Sorry, Dan. I've talking to Dave D. about insect eyes, compound eyes and biology, and I forgot I hadn't brought up the topic here lately I guess. I have a bunch of fantastic images of dragonfly eyes that I'll put together and share.

Dick

[Richard Fischbeck]

Oops. It is a wasp.

http://myrmecos.wordpress.com/2008/09/29/a-wasp-in-intricate-detail/

[Richard Fischbeck]

If you google microscopy compound eye, you get a great selection of images.

I know this thread is titled,

How to make any shape out of triangles?

but I think it is relevant to that. Except these are the duals of the triangles.

[Alfred van Dijk]

>> I know this thread is titled,
>>
>> How to make any shape out of triangles?
>>
>> but I think it is relevant to that. Except these are the duals of the triangles.

Agreed. And thanks for the posts! I am reading the posts, but I
haven't that much to add because I'm a newbie, still.

Hector said this picture had pentagons, but I can't find any! (not
with "5 circles around" them anyway)

http://myrmecos.files.wordpress.com/2008/09/hethead1.jpg

I do see the hexagons becoming rounder toward the middle, and becoming
more irregular towards the sides, becoming sometimes almost squares.

[Richard Fischbeck]

Yes. Think of an elastic 3-way mesh being stretched over a ball. The irregularities are around the edges of the hemisphere. There may sometimes be advantages in this structural assembly technique. It is certainly an advantage in the curved bamboo shells I've seen.

[Biagio Di Carlo]

"Benoit Mandelbrot, who developed ground-breaking mathematical theories about fractals and complexity, passed away today as announced in the New York Times. His theories have deeply influenced knowledge in many disciplines, and he received 15 honorary doctorates during his lifetime."

[Richard Fischbeck]

http://www.nytimes.com/2010/10/17/us/17mandelbrot.html

[Richard Fischbeck]

We are very familiar with the same strategy frequently used with a 2-way wire weave in microphones covers for a long time.

http://blackwallpaper.org/wp-content/uploads/2009/11/microphone.jpg

[Hector Alfredo Hernández Hdez.]

I hope you can see

[Gerry in Quebec]

Alfred,
The late French architect Alain Lobel, who invented the Lobel Frame,
did a lot of research into the used of joined equilateral triangles to
create three dimensional shapes. He put the question like this: What
volumes are possible based on a single module: The equilateral
triangle?" He ended up describing a whole family of forms. His
website, in French and English, is still operational:

www.equilatere.net

Gerry in Quebec

[Alfred van Dijk]

Again, thanks for the replies!

Alain Lobel's work seems even more useful than the others because his
structures seem relatively easy to build.

The concepts are all ridiculously complex, so it'll take me quite some
time to take it in ;)

[homespun]

On the Lobel site, the first FAQ has the following question and answer with
the picture I have attached.
Can someone explain this better?
Dan
__________________________________________________________________________________________
Q: 6 equilateral triangles flat form a hexagon which is a figure that one
cannot make convex. How do you procede ?

A: Positive and negative angles (mount and valley).

[Gerry in Quebec]

Hi Dan,
I have in front of me a cardboard hexagon. In this instance it's a
regular hexagon, that is, with all edges of equal length and all
interior angles 120 degrees. I made it by taping together six
equilateral triangles. If you were to play with such a "hinged"
hexagon, you'd see it's possible to twist it into various
configurations. For example, you could end up with four dihedral
angles (between triangular faces) that are less than 180 degrees
(convex) and two that are more than 180 (concave). And of course you
could flip this "floppy" hex on its back, and you'd have the reverse
configuration. In either case, on its back or not, each triangle lies
in a different plane. Various combinations of these traits (concavity,
convexity, and dihedral greater than, less than or equal to 180
degrees) are possible, but I haven't tried to count or catalog them.
In any case, Lobel's key insight, as I understand it, was that by
combining equilateral triangles in different ways, you can make 3D
building blocks to create more complex and highly varied 3D shapes --
which is what Alfred seems to be interested in. This is just my quick
"layman's" reading of what Alain Lobel was up to.
Cheers,
Gerry in Quebec


On Oct 17, 4:29 pm, "homespun" <uncle...@homespun4homeschoolers.com>
wrote:

> On the Lobel site, the first FAQ has the following question and answer with
> the picture I have attached.
> Can someone explain this better?
>         Dan

> ___________________________________________________________________________­_______________

> Q: 6 equilateral triangles flat form a hexagon which is a figure that one
> cannot make convex. How do you procede ?
>
> A: Positive and negative angles (mount and valley).
>
>
>
> ----- Original Message -----
> From: "Gerry in Quebec" <toomey.ge...@gmail.com>
> To: "Geodesic Help Group" <geodes...@googlegroups.com>
> Sent: Sunday, October 17, 2010 3:59 AM
> Subject: Re: How to make any shape out of triangles?
>
> Alfred,
> The late French architect Alain Lobel, who invented the Lobel Frame,
> did a lot of research into the used of joined equilateral triangles to
> create three dimensional shapes. He put the question like this: What
> volumes are possible based on a single module: The equilateral
> triangle?" He ended up describing a whole family of forms. His
> website, in French and English, is still operational:
>
> www.equilatere.net
>
> Gerry in Quebec
>
>
>

>  hexagone.jpg
> 13KViewDownload- Hide quoted text -
>
> - Show quoted text -

[homespun]

Gerry,
Thanks so much,
Dan

----- Original Message -----
From: "Gerry in Quebec" <toomey...@gmail.com>
To: "Geodesic Help Group" <geodes...@googlegroups.com>
Sent: Sunday, October 17, 2010 6:07 PM
Subject: Re: How to make any shape out of triangles?

[Ken G. Brown]

Ronald D. Resch, recently deceased, had several interesting patents related to 3D forms:

Construction-element Ronald D. Resch
<http://www.google.com/patents?id=PTs0AAAAEBAJ&printsec=abstract&zoom=4#v=onepage&q&f=false>

GEOMETRICAL DEVICE HAVING ARTICULATED RELATIVELY MOVABLE SECTIONS
<http://www.google.com/patents?id=rPtXAAAAEBAJ&printsec=abstract&zoom=4#v=onepage&q&f=false>

SELF-SUPPORTING STRUCTURAL UNIT HAVING A SERIES OF REPETITIOUS GEOMETRICAL
<http://www.google.com/patents?id=rPtXAAAAEBAJ&printsec=abstract&zoom=4#v=onepage&q&f=false>

Self-supporting structural unit having a three-dimensional surface
<http://www.google.com/patents?id=K781AAAAEBAJ&printsec=abstract&zoom=4&source=gbs_overview_r&cad=0#v=onepage&q&f=false>

Ken G. Brown

[Alfred van Dijk]

Thinking out loud...

I notice that in the wasp's eye, the transition between the different
shapes is much more gradual. In many or all man-made structures I've
seen there are just some regular hexagons and pentagons. In the wasps
eye you can follow the transition: in the middle you have regular
hexagons, and toward the edges they become more and more irregular...
You don't have 1 pentagon like in a man-made structure, but a region
of irregular shapes, that mimic that function. (I think I see)

I suppose for this shape (the wasp's eye) this arrangement of
(ir)regular hexagons is pretty much optimized over millions of year of
evolution. So if you could try to mimic that, and do the same for
other shapes, you're probably not too far off the mark.

You could try to mimic this with programs that triangulate any shape,
or with the suggested "Delaunay triangulation" and I guess you could
do a pretty good job with that. But then you get the problem that all
shapes are a bit different, therefore difficult to produce. So you're
probably still better of by combining geometrically symmetrical
objects. Which then only leaves the corners between the objects to be
worked out by the "programs that triangulate any shape" or the
Delaunay triangulation.

Am I on the right track?

http://myrmecos.files.wordpress.com/2008/09/hethead1.jpg

[Camilla Fox]

How you're making it will figure into how hard the irregular shapes
are; I went this route because I liked the construction process, and
cutting out pieces all the same was boring to me. (The little
scrapbooker cutters might've made it more attractive to do regular
shapes, had they been available to me in 2001.)

I made some out of paper and matlab, a number of years ago...:
http://web.mit.edu/cfox/www/spherical-models/2002-05-10/index.html

These have the same underlying random process, but a different way of
spanning between the points:
http://web.mit.edu/cfox/www/plaited-models/
(aie, the web and photography work looks so dated, I should do
something about that)

As far as nomenclature goes, "voronoi region" and "Delaunay
triangulation" are a dual pair.

-Camilla

[homespun]

Hi Gerry and whoever,
Attached is my favorite of the Lobel pictures, and my attempt, just
fooling around, to begin to recreate it.
It is very flimsy, but perhaps if I build the entire structure, it would
be a little more rigid.
Dan

----- Original Message -----
From: "Gerry in Quebec" <toomey...@gmail.com>
To: "Geodesic Help Group" <geodes...@googlegroups.com>
Sent: Sunday, October 17, 2010 6:07 PM
Subject: Re: How to make any shape out of triangles?

Hi Dan,
I have in front of me a cardboard hexagon. In this instance it's a
regular hexagon, that is, with all edges of equal length and all
interior angles 120 degrees. I made it by taping together six
equilateral triangles. If you were to play with such a "hinged"
hexagon, you'd see it's possible to twist it into various

configurations.............................

[Hector Alfredo Hernández Hdez.]

A friend mine, says that hiperbolic structures a so strong.

:)

[TaffGoch]

 

I find several of the Lobel frames intriguing. I like the radial fans of catenary arches, and the "domes" that appear to be based on a triacontahedron.

 

I've experimented modeling the latter, in SketchUp, and it's not easy. (It would likely be easier with a cardboard model.) Since angle inclinations/declinations affect adjacent coincident edges, adjustments in SketchUp requires rotating several triangles at one time, which SketchUp can't do. I can see why the website mentions dedicated software, written to design the different arrangements.

 

Nevertheless, I've come pretty close. (See attached.)

 

-Taff

[TaffGoch]

 

This is the particular Lobel frame that I was attempting to model.

 

~Taff

[Gerry in Quebec]

Very cool model, Dan. Did you take a look at the short video of the
Lobel node and struts, on the Lobel website? That invention was
intended as a way to make the Lobel frames rigid and easy to assemble.
The hardware, of course, would be quite expensive.

I found this mesmerizing:

http://www.equilatere.net/frame.php?page=en/mass/connectors.php

Gerry in Quebec

On Oct 18, 6:45 pm, "homespun" <uncle...@homespun4homeschoolers.com>
wrote:

[Adrian Rossiter]

Hi Taff

On Tue, 19 Oct 2010, TaffGoch wrote:
> I've experimented modeling the latter, in SketchUp, and it's not easy. (It
> would likely be easier with a cardboard model.) Since angle
> inclinations/declinations affect adjacent coincident edges, adjustments in
> SketchUp requires rotating several triangles at one time, which SketchUp
> can't do. I can see why the website mentions dedicated software, written to
> design the different arrangements.

I mentioned the Antiprism minmax program before, which tries to
make edge lengths equal with vertices constrained to a sphere.
It has an undocumented (unfinished) -a u option to make edges unit
length without constraining to a sphere. It should work for these
sort of models so long as the base model is close enough to the
required final model, otherwise vertices may be indented when they
should protrude, etc.

An example command which makes your model is

off_trans -s e geo_2_2 | minmax -a u -s 50 -l 40 -n 20000 | antiview

It takes about 5 seconds and the edges are unit to around 14 decimal
places

http://www.antiprism.com/misc/geo_2_2_unit.wrl

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

[TaffGoch]

Adrian,

 

Great wrl !

 

Woohoo -- with all the undocumented Antiprism options, you can really party! :)

 

-Taff

[Hector Alfredo Hernández Hdez.]

This sound like a good option.
See you.

--

[TaffGoch]

 

I've posted the Lobel-frame model at the 3D Warehouse:

 

http://sketchup.google.com/3dwarehouse/details?mid=d707c5785e8945e15267ed5eeb5c0a1c

 

Lobel frames present nice prospects for simplified domes, composed of only equilateral-triangle panels

 

-Taff

[TaffGoch]

 

The Lobel frame website, www.equilatere.net, provides this diagram of the 2-2-2-3 CSM configuration:

[Alfred van Dijk]

If anyone was wondering: I'm still reading along, but don't have much
useful to add. I'll be busy studying the information given here.