New Nexorade/Rotegrity project

[TaffGoch]

I've been playing with new nexorade geometry that is not too complex, but just a bit more intricate than previous rotegrities that I've modeled. I started with a Class-III, frequency 3v{2,1) tessellation, produced the dual, then slightly rotated the edges (or "nexors" in nexorade parlance):

It took some fine-tuning to get the equal-thirds nexor subdivision, but the initial nexor rotation got me pretty close. For those new to nexorades, the above animation depicts the general theory of how nexorades are produced.  The "rods" in the depiction can be substituted with planks, "springs" or straps:

Note above, that the nexors can be rotated (about their midpoints) in either, a clockwise, or counter-clockwise direction. Conveniently, the nexor definitions (third-subdivisions) do not change. Either sphere can be constructed, using the same nexors. There are, by the way, four different-length nexors, totaling 210 count. I'm currently considering constructing a snug-fitting strap "cage" for a large beachball...

...maybe putting a bright LED light source in the center, for backyard "moon" ball lighting (suspended overhead, between two trees.)

-Taff 

[TaffGoch]

By the way, if you want a larger version of the second (straps) image to study, you can find it here:

http://taffgoch.deviantart.com/art/Nexorade-3v-2-1-615949183

-Taff

[TaffGoch]

These are the nexor variations that I mentioned:

Straps

Rods

Planks

Springs

Simplified depiction of the basic geometry of the nexor "thirds" arc-subdivisions:

-Taff

[TaffGoch]

On Friday, July 1, 2016 at 8:21 PM, Robert Clark wrote:

Beautiful nexorade.  Maybe you can cover it with heat-shrink boat plastic.  Do you know if it is possible to have just one or two lengths of nexors?

_________________________

Robert,

Three of the nexors are pretty-darn close to being the same length. The pentagon nexors are noticeably shorter. In physical construction, it may not affect the spherical shape too much, if only two definitions are used. Only physical experimentation will tell....

-Taff

[TaffGoch]

Robert,

Nexor comparison:

-Taff

[TaffGoch]

Robert,

"...that's not really much to look at..." -- unless you know what you're looking at! 

I found it to be a good physical representation of what could have been iteratively-calculated in a spreadsheet. Excellent depiction, actually. 

-Taff

[TaffGoch]

I just noticed that one of the examples from Clinton's "Equal Central Angle Conjecture" paper is a 3v{1,2} dual,...

 which should be good initial geometry for rotating nexors(edges,) to produce this particular nexorade.

Perhaps equal nexor length is, indeed, practical -- will give it a try.  (Although, equal-third subdivision may not behave. It might be possible to meet one criteria, but not both.)

-Taff

[TaffGoch]

Initial test of hypothesis doesn't bode well....

It is, nevertheless, a good starting-point for fine-tuning, but since I've already tuned the original, to achieve "thirds" with varying-length nexors, pursuing this (above) 3D exercise would end-up producing the same results. 

-Taff

[TaffGoch]

Regarding the original nexorade post:

These chord factors will produce a sphere of unit (1.000) radius. Straps are depicted, but values are valid for each of the previously-depicted nexorades. (For color-coding, refer to the "rod" or "spring" depictions.)

-Taff

[TaffGoch]

Color map of nexor chords...

...should you find such more convenient than prior depictions.

-Taff

[TaffGoch]

I've posted the SketchUp model to the 3D Warehouse:
https://3dwarehouse.sketchup.com/model.html?id=5b5a1809-5e01-496b-b03d-060c82a72494

-Taff
(File attached, as well.)

[TaffGoch]

Had a thought, last night...

...could the same straps be used to assemble a "little brother" to the original nexorade?

Seems so! Just connect strap endpoints to the "other" mid-points.

-Taff

[muhammad karim Toury]

Hi

I made up a rotegrity sphere by aluminium tube  (8 mm in diameter) and joints were by means of wires. Joint holes on every element are linear and they are not at the right angle (photos attached). Now I want to make up a bigger one and join them by bolts and I need the right angles in order to drill the elements. I need help to find the angles.  

[muhammad karim Toury]

[TaffGoch]

Hang in there, Muhammad, I'm working on it!

I've done what you are proposing (bolt connectors,) and can relate that the resulting appearance is gratifying. I'm planning another construction, but with hidden pins, so that no connectors are visible, at all. You're correct that the proper angles and hole positions are critical.

I've completed the model, and will work on the bolt centerlines, next.

What diameter rod are you considering for your large project? I can model to conform to that diameter, and produce hole spacing diagrams for each of the two different-length rods (yellow, green.)

-Taff

[muhammad karim Toury]

Thank you so much Sir for your attention and patience.

I want to make a rotegrity sphere by tube 80 mm in diameter to gain sphere 4m diameter.

Thank you 

Muhammad Karim

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[muhammad karim Toury]

I want to now the difference of these two arrangement in rotegrities that I attached.

thanks

[muhammad karim Toury]

Hi

I built a 5-nexor fan to explicit the drilling angles by the real model but the bolts are so serious about little faults. I couldn't do that right. I want to examine it by smaller bolts into bigger holes.

On Sat, Dec 10, 2016 at 7:19 AM, TaffGoch <taff...@gmail.com> wrote:

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[muhammad karim Toury]

I do it by smaller bolts and could complete the mission.

Tubes are 970 mm length and c/c of holes is 270 mm, holes are 20 mm in diameter and 4 bolts are 20 mm and 2 of them are 8 mm. 

[TaffGoch]

Muhammad,

I paused working on the hole angles, to answer your other question:

Here's the comparison:

The rotegrity "straps" (of the same color) are the same length/width (but bent to different radii, of course, producing different-size spheres.)

In 3D-modeling, I refer to the two different size rotegrities as "greater" and "lesser," to keep my model files organized. Technically, they are, both, Class-II, 2v rotegrities.

-Taff

[muhammad karim Toury]

Thank you Taff,

I will call them Greater and Smaller,

is there a special relation between radii of two sphere (that are made from same straps but in different arrangement) ?

Attached my today job.

Muhammad Karim

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[TaffGoch]

The ratio of sphere radii is about 1:¾

It varies, somewhat, for different tessellations. For example, this greater/lesser rotegrity pair...

...has a slightly different ratio -- still approximating 1:¾

-Taff

[TaffGoch]

Muhammad,

80mm diameter tubing will produce a nexorade diameter of 4.44 meters.

If this is in keeping with your proposal, I will scale the model, using 80mmØ tube dimension as the basis. I will then be able to provide precise measurements between holes, along the length of the two tube definitions.

(I've completed the hole centerline angle derivations.)

-Taff

[muhammad karim Toury]

Thanks a lot Taff

Is there any geometrical formula for these rotegrities ?

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[muhammad karim Toury]

Good job !!!

This is exactly what I want.

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[TaffGoch]

Muhammad,

Here is a reference/guide image of the nexorade:

The wedge shapes depict the plane of the strut, extrapolated from the sphere center. The wedge is used to define the zero-angle (centerline) alignment of the strut.

The angular-deviation of each bolt-hole is measured from the plane of the wedge.

Pent strut:

Hex strut:

NOTE: While the hex strut (green) CAN be flipped end-over-end,...

... the pent strut (yellow) CAN NOT! I have color-coded the end of the yellow strut, to depict which end is connected to the green strut. If you examine the top-most nexorade reference image, you will note that one end of the yellow strut is green, while the other end is yellow. (The green strut ends are both yellow.)

The metric-length dimensions are based on a strut diameter of 80 millimeters.

-Taff

[TaffGoch]

There are many articles, about nexorade derivation and mathematics, in this Geodesic Help Group topic:

https://groups.google.com/forum/#!topic/geodesichelp/vrDkGnHlX-Y

Fun reads!

-Taff

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[TaffGoch]

Muhammad,

Be forewarned:  The 3D-model that I produced is VERY tight! You will likely need mechanical "assistance" to pull parts into alignment, so that bolt-holes line-up. A small model should be easy, but your full-size project will be challenging (to say the least.)

Don't worry, even Kenneth Snelson uses a "come-along" to assemble his large-scale projects.
https://www.google.com/search?as_epq=Kenneth+Snelson

-Taff

[muhammad karim Toury]

Thanks a lot, Taff,

Everything is clear,

121 mm end of yellow struts will rest on greens. Am I right ?

And because both ends of greens rest on yellows we can flip them end-over-end.

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[TaffGoch]

Muhammad,

"...​121 mm end of yellow struts will rest on greens. Am I right ?

And because both ends of greens rest on yellows we can flip them end-over-end."

Your understanding of yellow/green strut orientation is correct

-Taff

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[muhammad karim Toury]

I extracted rotegrity sphere from 7v geodesic dome by AutoCAD.

[muhammad karim Toury]

Taff,

I have some questions about rotegrity geometry,

You sent me the lengths and angles of elements for a rotegrity in "greater" arrangement, can I make "lesser" type by means of these elements? Briefly I want to know that is the geometry of greater and lesser different or same?

Second question

In the dimensions that you sent me why the two ends of yellow strips are different? (108 and 121 mm, because these lengths are not affective on the form geometry) while these lengths are equal in greens.

Muhammad Karim

[Ashok Mathur]

Dear Karim,

Pl explore this book

[PDF]RECIPROCAL FRAME ARCHITECTURE

https://casaeco.files.wordpress.com/2012/03/reciprocal-frame-architecture.pdf

The title of this book, Reciprocal Frame Architecture, asserts that this is a book about ...... structures as 'nexorades', the name coming from the Latin word 'nexor',.

See if you like it.

Regards

Ashok

Regards

Ashok

[muhammad karim Toury]

Dear Ashuk

I had read this book,

and I couldn't to find my answers,

There isn't any information about rotegrity.

regards

Muhammad karim

[TaffGoch]

Muhammad,

"You sent me the lengths and angles of elements for a rotegrity in 'greater' arrangement..."

I have not provided you any rotegrity measurements or angles. I don't understand your queries.
___________________________

"In the dimensions that you sent me why the two ends of yellow strips are different? (108 and 121 mm, because these lengths are not affective on the form geometry) while these lengths are equal in greens."

Only the central 3 dimensions, of each nexor, are critical. You can make the end dimension any length you wish.

(I was "eyeballing" the end lengths, to relate to overhang. Since the angles between nexors are different on each end of a yellow nexor, I extended the end length at one end to provide roughly the same overhang.)
___________________________

Sorry for the delay in replying -- I've been in hospital.

-Taff