Help making a Goldberg polyhedron with equal-area hexagons?
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Mitchell Bandes
Aug 2, 2024, 4:56:52 PM8/2/24
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I am trying to generate a Goldberg polyhedron of 30-40 frequency, but I want all of the hexagons to be of equal area. But whenever I've tried to make such a polyhedron in Blender or Antiprism, I always run into the same problem; the hexagons neighboring the pentagons are always significantly smaller than the hexagons the furthest distance away from the pentagons (often, they're about half the size of the far hexagons!)
For my purposes, it's much more important to preserve consistency in area than to preserve consistency in edge length. If the pentagon edges need to be longer than the edges furthest away from the pentagons, so be it.
I don't necessarily need a polyhedron derived from an icosahedron, but I have noticed that the fewer vertices the starting shape has, the greater the distortion in the final product. Of course if I could get an octahedral Goldberg polyhedron with minimal distortion, that would be the most ideal situation.
Any help would be appreciated.
Adrian Rossiter
Aug 2, 2024, 9:27:56 PM8/2/24
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Hi Mitchell
On Fri, 2 Aug 2024, Mitchell Bandes wrote:
> I am trying to generate a Goldberg polyhedron of 30-40 frequency, but I
> want all of the hexagons to be of equal area. But whenever I've tried to
> make such a polyhedron in Blender or Antiprism, I always run into the same
> problem; the hexagons neighboring the pentagons are always significantly
> smaller than the hexagons the furthest distance away from the pentagons
> (often, they're about half the size of the far hexagons!)
>
> For my purposes, it's much more important to preserve consistency in area
> than to preserve consistency in edge length. If the pentagon edges need to
> be longer than the edges furthest away from the pentagons, so be it.
This is a basic F30 icosahedral geodesic sphere dual
antiview geo_30_d
I made a list of the face areas and sorted them (using unix utilities)
off_query FA geo_30_d | sed 's/.*,//' | sort | less
This key areas are
pentagons : 0.001236192105...
smallest hexagons : 0.001304551501...
largest hexagons : 0.001444701601...
In this model the hexagons are fairly ugly near the pentagons [images
attached near 5-way and 3-way axes].
Perhaps you were using the dual of a canonical model. It produces better
shaped hexagons, but has a variation in area similar to what you are seeing
(pent: 0.00031238476, small hex: 0.0006066296, large hex: 0.0015526128).
Adrian.
--
Adrian Rossiter - adr...@antiprism.com
http://www.instagram.com/adrian_rossiter
http://antiprism.com/adrian
On Fri, 2 Aug 2024, Mitchell Bandes wrote:
> I am trying to generate a Goldberg polyhedron of 30-40 frequency, but I
> want all of the hexagons to be of equal area. But whenever I've tried to
> make such a polyhedron in Blender or Antiprism, I always run into the same
> problem; the hexagons neighboring the pentagons are always significantly
> smaller than the hexagons the furthest distance away from the pentagons
> (often, they're about half the size of the far hexagons!)
>
> For my purposes, it's much more important to preserve consistency in area
> than to preserve consistency in edge length. If the pentagon edges need to
> be longer than the edges furthest away from the pentagons, so be it.
antiview geo_30_d
I made a list of the face areas and sorted them (using unix utilities)
off_query FA geo_30_d | sed 's/.*,//' | sort | less
This key areas are
pentagons : 0.001236192105...
smallest hexagons : 0.001304551501...
largest hexagons : 0.001444701601...
In this model the hexagons are fairly ugly near the pentagons [images
attached near 5-way and 3-way axes].
Perhaps you were using the dual of a canonical model. It produces better
shaped hexagons, but has a variation in area similar to what you are seeing
(pent: 0.00031238476, small hex: 0.0006066296, large hex: 0.0015526128).
Adrian.
--
Adrian Rossiter - adr...@antiprism.com
http://www.instagram.com/adrian_rossiter
http://antiprism.com/adrian
Mitchell Bandes
Aug 3, 2024, 3:14:51 AM8/3/24
to antiprism
Wow! I didn't expect such a helpful answer so quickly! Thank you very much!
Mitchell Bandes
Aug 3, 2024, 4:34:53 AM8/3/24
to antiprism
Hi Adrian,
Sorry for my lack of posting etiquette; I'm accustomed to posting in social media, and didn't realize that Google Groups uses email etiquette instead. Thank you again for your help so far; the polyhedron you showed me is precisely what I was looking for. Now all I need to do is import it into Blender. Could you point me to the command which lets me export the polyhedron as a mesh file?
Thank you,
Mitchell
On Friday, August 2, 2024 at 11:27:56 PM UTC-5 adr...@antiprism.com wrote:
Adrian Rossiter
Aug 3, 2024, 4:53:50 AM8/3/24
to antiprism
Hi Mitchell
On Sat, 3 Aug 2024, Mitchell Bandes wrote:
> Sorry for my lack of posting etiquette; I'm accustomed to posting in social
I didn't even notice! We are a very relaxed group.
> precisely what I was looking for. Now all I need to do is import it into
> Blender. Could you point me to the command which lets me export the
> polyhedron as a mesh file?
Look at
https://www.antiprism.com/programs/import_export.html
You could use off2dae, as the output is intended to be suitble for import
into blender
https://www.antiprism.com/programs/off2dae.html
Exclude vertex and edge elements like this
off2dae -x ve geo_30_d > geo_30_d.dae
If you want original polygons (not split into triangles), then
off2dae -x ve -t no_tri geo_30_d > geo_30_d.dae
On Sat, 3 Aug 2024, Mitchell Bandes wrote:
> Sorry for my lack of posting etiquette; I'm accustomed to posting in social
> precisely what I was looking for. Now all I need to do is import it into
> Blender. Could you point me to the command which lets me export the
> polyhedron as a mesh file?
https://www.antiprism.com/programs/import_export.html
You could use off2dae, as the output is intended to be suitble for import
into blender
https://www.antiprism.com/programs/off2dae.html
Exclude vertex and edge elements like this
off2dae -x ve geo_30_d > geo_30_d.dae
If you want original polygons (not split into triangles), then
off2dae -x ve -t no_tri geo_30_d > geo_30_d.dae
Mitchell Bandes
Aug 3, 2024, 6:29:13 AM8/3/24
to antiprism
Thank you for all your help; that was everything I needed.
Arturo G. Espindola
Aug 4, 2024, 1:39:27 PM8/4/24
to anti...@googlegroups.com
Have you try acidome?
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