Modeling the Weaire-Phelan Structure
Becoming popular after the Beijing 2008 Olympics National Aquatics Centre‘s facade (which is believed to be a Voronoi subdivision, as an epic mistake), the Weaire Phelan structure is a solution of equal volumes with minimal surface area. This nice website briefly explains the phenomenon. There is also an implementation on Grasshopper by Jon Mirtschin. Here, I will be modeling the Tetrakaidecahedron of the Weaire-Phelan structure.
First, we define the overall size by drawing a 20×10 rhombus.
Copy the rhombus by 10 upwards, and rotate 90 degrees.
Connect the vertices.
Also, connect the vertices sideways to create equal triangles. Then, create a 10-sided polyhedron, as shown below:
Continue by drawing the short diagonal of the rhombus and copy it to 3.7 units (for 10) in both directions. It is interesting to think about how this 3.7 would work, really.
These intersections will create the irregular hexagon of the polyhedron. We’ll transfer the edge lengths at the far corners of the rhombi.
Put two spheres centered at the far vertices of the rhombi, by pointing to the new (3.7) intersections to define radii. You will transfer these radii like this;
Select the edges shown above and intersect the spheres to get those points.
You will use these three points to define cutting planes. But now, you’ll have to repeat the above steps for the second rhombus as shown below.
Finally, you’ll get something like this;
Then, trim all four sharp corners.
First cut,
Second cut.
Third cut.
The fourth and final cut. Cap the solid and finish the Tetrakaidecahedron.
If you had the patience to go this far, just pack them all with regular dodecahedra, and rotate 3d them.
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