Econodome

[biagiodicarlo]

Dear Taff,

please can you add a levelled triangulation 

to the base  band of the 4v Econodome ?

Thank you,

Biagio

[TaffGoch]

​​

Biagio,

The

​​

Econodome is a

​​

Kruschke tessellation, which, by design, has level truncations on "small circles." The 4v subdivision provides three level truncations that can be

​employed as

 a ground footprint.

​ (In the 4v subdivision, the equatorial truncation is a "great circle.")

This 3D SketchUp model, for th

​is​

 Kruschke 4v sphere, is attached.

​​

-Taff

[biagiodicarlo]

A very interesting Kruschke design.

Thank you very much Taff!!

Biagio

Il giorno 29/ott/2015, alle ore 23:46, TaffGoch <taff...@gmail.com> ha scritto:

​​

Biagio,

The

​​

Econodome is a

​​

Kruschke tessellation, which, by design, has level truncations on "small circles." The 4v subdivision provides three level truncations that can be

​employed as

 a ground footprint.

​ (In the 4v subdivision, the equatorial truncation is a "great circle.")

<Kruschke; 4v.png>

This 3D SketchUp model, for th

​is​

 Kruschke 4v sphere, is attached.

​​

-Taff

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<Kruschke; 4v.skp>

[TaffGoch]

Charlie,

I know I've posted this before, but it's easier to repost than it is to find it in past discussions:

SketchUp model attached.

-Taff

[norm...@gmail.com]

thanks!  I've seen that picture before but didn't see the skp file

[biagiodicarlo]

Grande Taff!!

- Biagio

Il giorno 31/ott/2015, alle ore 00:45, TaffGoch <taff...@gmail.com> ha scritto:

Charlie,

I know I've posted this before, but it's easier to repost than it is to find it in past discussions:

<Kruschke spheres.png>

SketchUp model attached.

-Taff

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<Kruschke spheres.skp>

[norm...@gmail.com]

thanks again taffgoch.  I exported the skp into a file blender can open and I've started working on a 6V 'weave' model:

My apologies for taking this offtopic as your econodome here deserves recognition too.

You really do such amazing work.  thank you again 

[TaffGoch]

Charlie,

Upon quick review, I can only surmise that the "bamboo" domes are different because they employ triangles as "nodes" in the overall hexagonal structure, while Kruschke domes do not. Kruschke domes are more "restrictive" in original parameter establishment, in that the lesser circles MUST cross-over at "shared" node points. 

The lesser circles in the bamboo domes don't have to cross-over at a shared point (node.) They cross-over to form triangles, which can vary in size. This provides for variability in the spacing between parallel lesser-circles. So, essentially, the bamboo domes are less restrictive in the placement of the parallel lesser-circle spacing (planes.)

Whether from Gerry's calculations, or my constructions, the Kruschke dome chord factors are virtually identical (to several decimal places.) Our bamboo domes (or anyone else's) can be, however, substantially different, with varying sizes of triangular "windows" that serve in the stead of "point" nodes.

I hope my description makes some sense in explaining the possible variations in triangular-window bamboo domes.

-Taff

[TaffGoch]

To interested readers,

Charlie's quest is to produce a dome of overlapping planks that "weave" in-and-out, as depicted in this photo:

(You may have to view the photo full-size to see the "weaving" nature of the dome.)

I know that there are smaller, lesser-frequency, domes of this type, such as Bucky's "tensegrity dome" on Bear Island:

Also, re-created, not too many years ago (in France):

This is why Charlie refers to the "tensegrity" quality of this type of woven-plank dome.

Also see Charlie's experiment (four photos, to date):

https://www.flickr.com/photos/118440047@N08/

One of his photos:

-Taff

[TaffGoch]

Another photo of the tensegrity dome, in Lyon, France (2012):

The same art exhibit also included this bamboo version:

-Taff

[TaffGoch]

Okay, I guess we do have to consider this discussion properly "hijacked."

Here's a clean representation of the geometric basis for the "4v" tensegrity dome:

Each of the "disks" have 10 equal-length "segments" defining its perimeter. (I started this construction with 20-segment disks, to find the proper spacing between parallel disks, ensuring that the crossing points/vertices coincided with vertices shared by intersecting disks.)

Subsequently, the "disks" can be edited to insert parallel edges, on the surface of the disk, to define "just kissing" edges, where other intersecting disk edges "penetrate" the subject disk, leading to this:

Now, all straight segments are equal, and cross-over regions are "just kissing." It looks like the Lyon artists didn't subject their design to sufficient analytical assessment.

Charlie, I assume that this is the geometry you are looking to achieve. It doesn't quite conform to Kruschke specification, as you noted earlier, although it's close. Pursuing Kruschke conformity is a "wild goose chase."

(SketchUp file attached.)

-Taff

[TaffGoch]

Charlie,

If you're interested, here's how the 20-segment disks intersect, before "whittling down" to 10-segment disks:

Here's a "fair-to-middling" depiction of planks:

Some "kissing" is affected, but should be readily accommodated in construction.

(Revised SketchUp file attached.)

-Taff

[biagiodicarlo]

Thank you very much.

- Biagio

Il giorno 18/nov/2015, alle ore 03:28, TaffGoch <taff...@gmail.com> ha scritto:

Charlie,

If you're interested, here's how the 20-segment disks intersect, before "whittling down" to 10-segment disks:

<Tensegrity; disk, 20.png>

Here's a "fair-to-middling" depiction of planks:

<Tensegrity; planks.png>

Some "kissing" is affected, but should be readily accommodated in construction.

(Revised SketchUp file attached.)

-Taff

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<Tensegrity dome basis.skp>

[Adrian Rossiter]

Hi Taff

On Tue, 17 Nov 2015, TaffGoch wrote:
> Each of the "disks" have 10 equal-length "segments" defining its

> perimeter. *(I started this construction with 20-segment disks, to find

> the proper spacing between parallel disks, ensuring that the crossing

> points/vertices coincided with vertices shared by intersecting disks.)*

The spacing provided by uniform decagonal antiprisms is pretty good

poly_kscope ant10 -s I -y 4 -c vef | antiview -v 0.04 -m spread
off_util std_ant10 -K f0,1 -x f | poly_kscope -s I -y 4 | antiview -v 0.05 -E ivory -B black

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

[TaffGoch]

Charlie,

The planks now have holes "drilled" in them, defining the length. (Revised SketchUp file attached.)

The plank length-to-width ratio is critical. The thickness, not-so-much (within reason.) I think that you can see that a plank that is too wide is going to bind. Too narrow, and no "just kissing" character. So, the width of a plank will establish the length. In other words, you should scale the model according to the width of the planks that you will be using.

-Taff

[TaffGoch]

Adrian,

On close examination, I find that my model has some binding, or "overlap" in some of the cross-overs. Charlie commented that, with Kruschke subdivision, two plank length definitions are required. That's likely true, to fine-tune my model, as well, but the difference is much less than a Kruschke subdivision. A dome/sphere built from my model will accommodate the minor binding, due to flexibility in the plank, and may, indeed, make for a "tighter" construction.

I suspect that close examination of your antiprism model will reveal the same slight (innocuous) discrepancy.

-Taff

[Adrian Rossiter]

Hi Taff

On Wed, 18 Nov 2015, TaffGoch wrote:
> On close examination, I find that my model has some binding, or "overlap"
> in some of the cross-overs. Charlie commented that, with Kruschke

...

> I suspect that close examination of your antiprism model will reveal the

> same slight *(innocuous)* discrepancy.

Definitely, it just gets fairly close with an easy construction.

Regarding an optimal solution, the icosahedral compound of twelve
decagons has two degrees of freedom, angular and translational,
but with four different strut separations to be made equal. There
is opportunity for pursuing good solutions, but I would suspect
that there isn't an exact solution.

The poly_kscope command allows the free variables of a compound to
be specified, and so it is possible to explore the forms with a
command like this, which makes the uniform antiprism form

poly_kscope pol10 -s I -y 2,0:9:0.4311985019297292 | antiview -v 0.05 -x f -E ivory -B black

The '9' is a the angle (degrees) that the base polygon is turned,
and the 0.4311985019297292 is the translation distance. (It also
colors the two vertex types.)

For another example, in the following command I have set the translation
to 0.4, and adjusted the angle to 12 so that vertices align at the
small triangles, but the other vertices are now quite a bit off
[image attached]

poly_kscope pol10 -s I -y 2,0:12:0.4 | off_color -v S| antiview -v 0.05 -x f -E ivory -B black

For two edge lengths, the base polygon can be created from a point
by poly_kscope as follows, with the 20 indicating that one edge has
a 2*20=40 degree central angle (the other has (360-5*40)/5 = 32
degrees) [image attached]

off_util -A v1,0,0 null | off_trans -R 0,0,20 | poly_kscope -s Cv5 | conv_hull | poly_kscope -s I -y 2,0:7:0.25 | off_color -v S | antiview -v 0.03 -x f -E ivory -B black

[TaffGoch]

Attractive construction, Charlie, despite having to use "cheat" notches!

Joking aside, I've used notches, as well:

http://taffgoch.deviantart.com/art/Rotegrity-Woodwork-201870495

Actually, using additional notches in the design of a dome like yours could, possibly, be employed to increase structural strength and stability. Might be worth considering....

Reviewing my deviantART productions, I realized that this one is the same geometry as the Bear Island design (and yours):

http://taffgoch.deviantart.com/art/Reciprocal-Frame-206427744

(That one, I produced years ago, and had forgotten about it -- go figure.)

-Taff

[Adrian Rossiter]

Hi Charlie

On Mon, 23 Nov 2015, norm...@gmail.com wrote:
> Can someone explain this further or in different terms? I'm not quite
> getting it.

>
> Regarding an optimal solution, the icosahedral compound of twelve
>> decagons has two degrees of freedom, angular and translational,
>> but with four different strut separations to be made equal.

Here is an animation to show some of the angular and translational
freedom in the compound

http://www.antiprism.com/misc/anim_comp_12_decs.gif

A decagon has a 10-fold rotational symmetry axis. If you rotate
the decagon around the axis and/or translate it along the axis
then the transformed decagon has exactly the same 10-fold symmetry
axis as it did in its original position.

If you align this 10-fold axis with a 5-fold axis of icosahedral
rotational symmety then the decagon is mapped onto itself five times
by icosahedral symmetries. Other icosahedral symmetries carry it
to other locations, and create a compound of 12 decagons.

The rotation and translation of the decagon on its axis don't lead
to more copies in the compound, but change the shape of the compound.


These steps can be followed in Antiprism as follows. The decagon
model is called pol10. Its symmetry axis is the z-axis, and it has
a vertex on the x-axis. It can be rotated 8 degrees and translated
0.45 units on its axis like this (antiview is set to look at the
origin with -C 0,0,0)

off_trans pol10 -R 0,0,8 -T 0,0,0.45 | antiview -C 0,0,0

The Antiprism icosahedral symmetry group has a 5-fold axis on
(1, phi, 0), and the z-axis can be rotated to this axis to carry
the decagon into position

off_trans pol10 -R 0,0,8 -T 0,0,0.45 -R 0,0,1,1,phi,0 | antiview -C 0,0,0

Finally, add the symmetric repeat

off_trans pol10 -R 0,0,8 -T 0,0,0.45 -R 0,0,1,1,phi,0 | poly_kscope -s I | antiview

You can vary the 8 and the 0.45 to make all the different possibilities.


Now, looking at an example compound of decagons you can see that it
has two different kinds of vertex. These can be coloured differently
(and the decagon faces removed), like this

off_trans pol10 -R 0,0,12 -T 0,0,0.431 -R 0,0,1,1,phi,0 | poly_kscope -s I | off_color -v S | antiview -x f -v 0.05

Each vertex involves two struts above another, and the separations
should be the same for a snug fit. The two types of vertex mean four
separations need to be equalised (but one of these is independent
so that there are only three degrees of freedom.)

I have attached an image of this model with arrows pointing to the
two kinds of vertex, and short blue lines indicating the four kinds
of vertex separation that need to be equalised.

However... the solution to get touching cylindrical coplanar
struts won't apply to get exactly touching rectangular section
overlapped struts. I don't know if the solution is close enough
to be useful in a physical model using the rectangular section
struts, it may not be.

[Paul Kranz]

Normalson:

DId I ever tell you that the Greek word ECONODOME translates to "a building?"

Paul sends...

On Fri, Nov 13, 2015 at 9:58 PM, <norm...@gmail.com> wrote:

Taffgoch, sorry to be a pest but I was wondering if you could solve my confusion here on the 4V Kruschke vs the lesser circle skps?

I've made models of both and both 'seem' to work

On Saturday, November 7, 2015 at 9:42:36 PM UTC+8, norm...@gmail.com wrote:

Hi again,

Taffgoch, I had a couple questions.  What is the difference between the bamboo dome skp that you made for this thread: https://groups.google.com/forum/#!searchin/geodesichelp/lesser$20circle/geodesichelp/960EgZfW7hw/E_YJmOcPVkQJ

and the Kruschke 4V in the skp file provided here once it is truncated?

I'm getting two sets of numbers for each:

the bottom pic is the bamboo dome, the top one is the 4V Kruschke.  As you can see in the picture, I'm looking for the data for the thin triangles that are formed similar to my 6V drawing above.

For the 4V there are two different triangles.  When you have a second could you open up sketchup work your wizardry, and respond back with the most precise numbers you get?

In case the pictures aren't clear, the bamboo skp gives me a triangle with 0.6247, 0.3346, and 0.298 for sides, and a triangle with 0.6111, 0.3346, and 0.2839 sides.

The 4V Kruschke gives me a triangle with sides 0.6253, 0.3091 and 0.3249, as well as a triangle with sides 0.5624, 0.2596, and 0.3091

thanks for the help I really appreciate it.

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Very high regards,

 

Paul sends...

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