Uniform tiling of a circle

Been googling on this for a bit to find that most packing problems are concerned with getting as many circles as possible into given shapes.

I'd like to do the reverse and pack triangles (polygons) into a circle. I need:

- the edges to be the same lengths as much as possible
- a symmetry, either axial or radial (latter preferred)

I need to be able to set circle diameter and edge lengths with sliders.

I'm using Daniel's MeshMachine and tend to get results like these:

The bottom one has some symmetry I suppose, but what I want is more like this, the only positive result I have managed to achieve:

I struggle to make Kangaroo help me produce several possible solutions to choose from.

How would you do it?

Replies to This Discussion

Permalink Reply by enrique soriano on August 27, 2015 at 8:02am

Hi Duncan

small mesh but correct topology> subdivide>pull naked vertices to circle

Then you can play whether you want 7-valence vertices or not, and where do you want them.

Permalink Reply by enrique soriano on August 27, 2015 at 8:04am

the file

Attachments:

tilingCircle.gh, 18 KB

Permalink Reply by Duncan W on August 27, 2015 at 8:34am

Many thanks.

Results tend to be extremly hexagonal looking. To remedy, I've been trying to do this not on a disc but on a fifth or third pizza slice, and then array polarly. Not too successful.

Do I HAVE to use anchor points? Why can't I just tell Kangaroo to use only the circle to pull to, and increase faces dynamically depending on the edge length slider I give it?

Permalink Reply by taz on August 27, 2015 at 9:04am

It's not clear what you're trying to accomplish. It sounds like you are looking for a periodic tiling that can be perfectly inscribe in a circle? Mathematically I don't think this is possible with a single triangular tiling.

See Euclidean tilings of convex regular polygons.

You can create a repeating pattern and test for inclusion within a region...

Permalink Reply by Duncan W on August 27, 2015 at 9:22am

Here's some more of my own attempts that I hope will make clear what I'm after.

All of these have a 'hole' in the middle which I don't want, and the edge lengths are too different.

You can also see the anchor points I had to give. Using just a boundary curve doesn't work and the mesh is sucked up immediately. Why is that?

Ideally, I would not want to use anchor points. I want Kangaroo to choose their position and number for me. How?

Permalink Reply by taz on August 27, 2015 at 11:48am

From a regularized grid you can see the deformation required to conform to a circle. Below is a K2 setup but the results are similar to what enrique has already produced...