Same size Hexagon on spheroid

If I’m not mistaken a normal sphere can be covered with same size hexagons if you allow for 12 5-sided cells. I would already not know how to do this in Rhino, but my task is even more difficult as I have to create it on a spheroid. I’ve tried many things all day like trying to do something with paneling tools or flowonsurface. Nothing I know can get the job done. Any advice?

Hello- I am not sure if that one is covered here but it is worth a look:

-Pascal

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Thank you for the quick reply. I’ll dive into it. Cheers.

May be this one:

Truncated icosahedron

-Pascal

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But how to make it in Rhino?

Using that plug-in you can find it in the list and click OK, then at the command line set the type of output you want - curves, (lines) surfaces or mesh. Set the center and radius…

Of course none of this helps if the overall shape is not basically a sphere… that would be a different and very possibly insoluble problem…

-Pascal

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It’s a beautiful plugin, but it seems to me it cannot do what I need on a spheroid as I would need to scale it non uniform. Thus my hexagons will not remain the same size. Right?

So why don’t you post a file with your spheroid?

True. This geometrical problem does not have a solution. (Large carbon cages called fullerenes are made of hexagons and pentagons. Oblate and prolate fullerenes do not exist for the same reason.)

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If you’re willing to work in grasshopper, you may be able to generate one of the dual geodesic icosahedra from RhinoPolyhedra, scale it in 1 or two dimensions to fit your spheroid, and then use the Ngons plugin to planarize the faces of the squashed dual geodesic icosahedron.

I’ve done this recently, playing with the latest release of Ngons. There are limits to how much you can warp that polyhedron before it becomes impossible to planarize it with the Ngons tools. But it did work well to generate a geodesic spheroid section.

maybe there’s a method to do this with Kangaroo that might also be able to balance out the areas of the faces as much as possible? Not my area of expertise, but it seems like a Kangarooish thing to do.

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Hi,
I’ve done the bucky dome by hand, you need to use orient 3 point and rotate 3d and using circles find the intersection points of the faces then rotate your faces and then use orient 3 point.
I’ve attached the model you speak of, maybe you can cage edit it to fit your spheroid or use Gh to transform it. It’s not easy or hard to make but time consuming. Long before Dale posted his polyhedra plugin I was modeling them by hand. At some point Miko posted a tutorial that showed how to use circles to find the intersection points that helped me get started but it’s been many years since I tried to model these so I don’t recall my exact steps.
RM
bucky3.3dm (87.3 KB)

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For strictly regular hexagons (plus 12 pentagon’s), meeting without gaps, you can’t even make them tangent to a sphere when you have more of them.
The Goldberg polyhedra (https://en.wikipedia.org/wiki/Goldberg_polyhedron)
get close, but have multiple types of slightly irregular hexagons.

So all strictly regular hexagons is out (except for the special case of the truncated icosahedron). Equal areas or equal edge lengths is probably possible through optimization.
If you gave more context in your question about what this is actually for though, you’d probably get more helpful replies.

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Thank you and others for great replies. What I’m basically trying to do is cover a shape with hexagon tiles. There is a tile joint that can give some space for freedom. Before I wanted to cover it with different sizes hexagon, but now I want to use only one size. I’m working on a building design. I’ve included the file with the correct dimensions. I would actually like the tile also to stay planar.shape and tile.3dm (3.9 MB)

Are you building a UFO?

Do you need the whole thing or maybe just half of it?

Here’s a go at this.
It relaxes a triangular mesh to get as close to regular as it can, then takes the incircular dual, then fits a perfectly regular (and flat) hexagon to each cell of this.
The base mesh before subdivision I used here is a hexagonal antiprism, which I believe gives a better fit for this shape than simply an icosahedron. It still has only 12 pentagons, but in a different arrangement. I think any other coarse mesh with more irregular faces is likely to result in worse gaps.
Realistically, when covering a curved surface with only a single size and shape of tile, there will always have to be significant gaps. I think it might be possible to improve the result here a little, by using a slightly different energy in the relaxation, but to get much closer fit, you’d need to vary the size/shape of the tiles.


hexagons_ellipsoid.gh (60.5 KB)

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Depending on what portion of the spheroid you can actually see, you might want to follow an other approach?

A projected hexagonal grid can look more regular from above or below. It does of course have pentagons on the circumference of the base, but there are no pentagons within the semi sphere or whatever that would be called.

It’s a different approach and the size of the hexagons is too big of course but I wanted to show this option.

20_10_24_hex_on_spheroid_mrtn.gh (305.5 KB)

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Yeah, flying to the moon. Half won’t do I’m afraid. Gets chilly up in space.

Too funny last night I was thinking after I posted, that if there is one person on this planet who knows how to do this or if it is even possible that would be Daniel Piker.
RM

Wow. I just saw the grasshopper definition. You’re either a wizard or spent a lot of time on that. Very much appreciated. I would send you flowers if I had your address. (seriously not fishing) Thanks a lot.

Thanks a lot, looks great. How many irregular or pentagons in this model?