New class of Goldberg Polyhedra

[Mack]

I was curious if anyone had attempted to model any of the polyhedra described here. These new polyhedra are special, and I would think useful for geodesic dome builders, because all of the edges are of equal lengths. These polyhedra were developed by neuroscientists Stan Schein and James Gayed working at UCLA. They published a paper in the Proceedings of the National Academy of Sciences, which is behind a paywall. However, this data supplement is not behind the paywall, and I believe contains all the relevant information to construct them. Taff, if you were looking for a project, I'd love to have a model of the polyhedra labeled T=48 (4,4) ! 
 

[Adrian Rossiter]

Hi Mack

On Tue, 25 Mar 2014, Mack wrote:
> I was curious if anyone had attempted to model any of the polyhedra

> described here.<https://www.sciencenews.org/article/goldberg-variations-new-shapes-molecular-cages> These

There is a program in Antiprism called minimax, that solves this
sort of problem.

One of the options is to find a polyhedron which is connected like
the input, but has regular polygons. It corrects for three things
during the solution: the edge lengths, the polygon planarity, and
the polygon radius.

However, the "weighting" used by the program for the polygon
radius correction is the same as used for the edge length
correction, meaning that the "equal angle" correction can't
be turned off with the program options provided.

Ill add a separate weighting for this, but turning off first the
planarity correction (with the program option) and then the polygon
radius corection (in the code) produces two models like in the paper.
The planar model produced has edges which are unit to 14 significant
figures, and a similar figure for the distance any vertex lies off
the plane of its polygon.

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

[Gerry in Quebec]

A few years ago Taff made SketchUp models of two of these equilateral Goldberg polyhedra, both icosahedral: I {2,0} and I {3,0}. They're in the public domain, the Google (or Trimble?) Warehouse.

 

https://3dwarehouse.sketchup.com/model.html?id=73df5cebd07d1f182137e32e8ebf2b1

https://3dwarehouse.sketchup.com/model.html?id=fbd64d9d014391602137e32e8ebf2b1

The original discussion took place on Yahoo's DomeHome discussion group, now called DomeTimes, in the context of Joseph Clinton's Equal Central Angle Conjecture.

I've attached pix of Taff's two models.

- Gerry Toomey in Quebec

[Gerry in Quebec]

For the record.... The original discussion about the equilateral Goldberg polyhedra (mainly between Taff and me) took place on the Dome Living group (not DomeHome, as I reported earlier), starting July 9, 2010. The group name was later changed to Dome Times.

 https://groups.yahoo.com/neo/groups/DomeTimes/conversations/topics/10050

 

In that thread, the link to the model images Taff posted is broken. Here's the new link:

https://groups.yahoo.com/neo/groups/DomeTimes/photos/albums/1696788196

 

- Gerry

[TaffGoch]

Here's an image export of the {4,4} polyhedron, as modeled in SketchUp:

Obviously, not spherical, but that's not unexpected, based on the premise stated in the Schein paper.

Edges are equivalent, and faces planar. With these two criteria, a comparable spherical polyhedron can not be produced. I've read a discussion post (elsewhere,) paraphrasing, that "it looks like it was constructed wrong...."

-Taff

[TaffGoch]

Adrian,

Are you going to make the mentioned changes (to minimax) public?

I'd be interested in further-exploring this class of polyhedra.

-Taff

[Adrian Rossiter]

Hi Taff

On Fri, 28 Mar 2014, TaffGoch wrote:
> Are you going to make the mentioned changes *(to minimax)* public?

>
> I'd be interested in further-exploring this class of polyhedra.

The changes are already available in the code repository

https://github.com/antiprism/antiprism/commits/master

I will make Windows binaries of the development code for the next
"Snapshot" release, which could be next week.

For reference, the command with the new option for the planar model
I posted is

minmax -a u -s 20 -l 20 -k 0 -n 100000 geo_5_d | antiview

[Gerry in Quebec]

Here are the key values I get from Excel Solver for the "Clinton Equal Central Angle Conjecture" version of Goldberg I {4,4}, which is nonplanar:

 

Central angle: 5.830089 degrees

Edge arc length (geodesic factor): 0.101754      

Chord factor: 0.101710

 

Can anyone confirm these numbers?

 

I deleted my earlier post this morning because the values were wrong.... I'd forgotten to include two edges in the Solver scenario.

 

- Gerry

[Mack]

Taff-

Will you please make this model available on the 3dwarehouse?

Thanks so much

-Mack

[TaffGoch]

Mack,

This SketchUp model is only a close approximation. I'm hoping to obtain precise datasets for all the polyhedra presented in Schein's paper, to make accurate models.

Give it a few days, and I may have a whole set to model and post. If not, I'll post this one, with caveat....

-Taff

[Adrian Rossiter]

Hi Taff

On Fri, 28 Mar 2014, Adrian Rossiter wrote:
> On Fri, 28 Mar 2014, TaffGoch wrote:
>> Are you going to make the mentioned changes *(to minimax)* public?

...

> I will make Windows binaries of the development code for the next
> "Snapshot" release, which could be next week.

I have made a snapshot

https://groups.google.com/forum/#!topic/antiprism/9d4KzN-1oUE

To make the planar models:

* Choose the base model, e.g. geo_5_d, geo_5_1_d, geo_o4_d

* Process the model with minmax (choose large -n, or reprocess the
output in a new minmax command, changing the file name each time), e.g.

minmax -a u -s 20 -l 20 -k 0 -n 100000 geo_5_d > geo_5_planar.off

* Check the model has edges close to 1, and that the maximum
deviation from planarity is small

off_report -SEF geo_5_planar.off

The key values from the above program run are

edge_length_max = 1.0000000000000215 (104,333)
edge_length_min = 0.99999999999999178 (300,302)

maximum_nonplanarity = 7.2497563508022722e-14

* View the model, e.g.

antiview geo_5_planar.off

[Gerry in Quebec]

Attached is a jpg of yet another version of the I {4,4} Goldberg tesselation of the sphere, with equal central angles, respecting the Clinton Conjecture. There was a glitch in the one I posted April 12 (due to an incomplete Solver scenario in Excel), so I deleted it. Pretty sure I've got it right now.

 

- Gerry in Quebec

 

 

On Saturday, April 12, 2014 12:19:44 PM UTC-4, Gerry in Quebec wrote:

I redid the Clinton Conjecture version of Goldberg I {4,4}. An Antiview image of the model, including the corrected central angle, geodesic factor, and chord factor, is attached.

- Gerry in Quebec