Carbon Structures - Creating a workflow for the generation of equilateral primitives and topological distortion

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Permalink Reply by August Lehrecke on February 7, 2018 at 11:35am

**An important topological side note**

The draft angle of the specific cone that I modeled was determined by 4 pentagons placed at the nose of the cone. The rest are equilateral hexagons spiraling outward and growing each row. I haven't really figured out how to get there yet with the script but thought it would be important to mention for all you geometry geniuses. 

Permalink Reply by August Lehrecke on February 8, 2018 at 7:44am

Kinda figured out a solution by turning the solids from the magnetic tile script into separate hinged shells. Still doing it with the gripper tool, will probably just load some pentagonal tiles to complete. Anyone have any ideas on how to automate it?

Permalink Reply by August Lehrecke on February 8, 2018 at 10:11am

Got the Pentagons on but still having trouble joining at the seams. Is there a way to weld the mesh together once the magnet snaps activate while Kangaroo is running? Ideally would like to be able to smooth with weaver bird after.

Permalink Reply by August Lehrecke on February 8, 2018 at 10:49pm

The system works with branching the form with heptagons as well.

Permalink Reply by Mathias Gmachl on February 9, 2018 at 4:59pm

Really nice work, August!

I have been fascinated by this simple system for a few years. Here is my earliest illustration:
https://vimeo.com/40389601

The principle could be described as this (for triangular/hexagonal meshes)

• Valence 5: A central element (vertex or polygon) surrounded by 5 neighbours creates an elliptical surface.
• Valence 6: A central element (vertex or polygon) surrounded by 6 neighbours creates an euclidian/flat surface.
• Valence 7: A central element (vertex or polygon) surrounded by 7 neighbours creates an hyperbolic surface.

In your example the round tip pf the cone has valence 5 connections, the rolled up sides have 6 and the branching point (where the surface is growing) has 7. 

In my example the actual meshes were modelled flat, observing to correct valence patterns only. And then relying on the geometrical/physical constraints simulated in Kangaroo to 'pop into shape'. I hope I can share a demonstration of this using your geometry soon.

By combining these principles it should be possible to recreate any surface (with more or less precision). The Lobel Frames you shared look fascinating. They seem to establish an archetypal set of shapes that can be modelled using this principle. I have to look into this further.

Whether we look at the triangular primal or polygonal dual of these forms/networks is open to us, they are completely interlinked (geometrically and energetically). On the computer it is often easier to work with the triangular primal representation as so many applications can handle triangular meshes.

Design tools, yes there should always be more good design tools. Daniel Piker's Mesh Machine is quite an incredible example as it can dynamically change the connectivity of the mesh to directly control the local surface. Here is a great paper Daniel shared earlier to discuss this in more detail
https://www.cs.sfu.ca/~haoz/pubs/aghdaii_gi12_567.pdf

and another one
http://people.physics.anu.edu.au/~sth110/10_1_ArchThRev.pdf

so much to say..
I hope there are some helpful pointers

Permalink Reply by August Lehrecke on February 15, 2018 at 6:47pm

Hi Mathias,

Thank you for your response and for sharing those links. I'm a huge fan of your work and it's really great to hear your thoughts on this.

Yes, it is quite an incredible phenomenon. For more complex structures do you also start with the 2d pattern shown in your video, or do you work from the form down with something like mesh machine? As I model more complex structures out of reed it becomes more difficult to imagine how to piece them together from the 2d primitives in Kangaroo but it seems to be the only way to ensure the symmetry of the polygons. 

Thanks again for the pointers, this particular project has been quite rushed and I've had to stop fiddling with the script and start cutting all the parts. I do have an updated script and many more questions which I will share after things get a little less hectic.

Permalink Reply by Daniel Piker on February 16, 2018 at 7:48am

Hi August,

In general with Kangaroo it is easier to change geometry during simulation than it is to change topology.

This is because of the way it interacts with the rest of Grasshopper and its directed graph system - generally you set which points a goal acts on upstream of the solver component, and since the data only flows one way this doesn't get changed by the solver.

There are some exceptions to this - like snaps or collisions, which are goals that act on large numbers of points, and which pairs among them are interacting can change during the simulation. However, for other goals like hinges, that aren't written with this sort of dynamic connectivity included, it isn't possible with standard components (though it is via scripting - you could either have a custom version of the hinge goal that selected dynamically the points it acted on by proximity, or you could have a custom solver script, which added and removed goals between iterations). 

For this reason, the approach Mathias shows works well - as long as your topology is what you want, you can start with very unequal edge lengths, and let them even out once the solver runs.

It can also be very helpful here to work with mesh subdivision - because this does not change the placement of the irregular vertices (corresponding to the pentagons and heptagons in the dual), it just adds extra regular vertices between them, so you can start by modelling the simplest level with just a few triangles.

Also, although topologically you can switch back and forth between them, when working with hexagonal networks like this, there is actually quite a bit more geometric freedom when relaxing them than with meshes of triangles with equal edge lengths.

Fixing the edge lengths of a triangular mesh removes most of the freedom of the surface to change shape, since it also fixes all the face angles.

With a hexagonal mesh though, the shape can deform without changing edge lengths, through changes in the angle between edges, and also via twisting about the edges(which can happen even keeping the angles fixed). It also looks like the sticks can bend a bit in your model.

I had a play using some snapping components to show one way of exploring these types of structures - definition below. (Probably for a more controlled design process though you still want to start by modelling the base meshes in Rhino, but it can be fun to play with.)

In the first file below I've used a combination of snaps and collisions, which have the effect of keeping the bonds straight.

In the second example I showed another approach using rigid bodies. Because they have a thickness, they also have some resistance to twisting about the connection, unlike in the first example.

In can sometimes be useful to switch on and off either the collisions or the snaps here (with the toggles connected to the strength multipliers), because the collisions can make it hard to pull parts together to make new connections, and if something snaps together unintentionally you can still pull them apart.

...I've also been thinking that one day it would be interesting to try making a script that lets particles store and update an atomic valence, so they snap together with other ones to form molecules until they saturate this. I'll let you know if I get this working.

Attachments:

• carbon1.gh, 27 KB
• carbon2.gh, 20 KB