Designing with Polyhedra

In a previous article I showcased digital fabrication of (art)work based on researching symmetry on the sphere and in hyperbolic space. In this article I’m expanding on this by introducing different ways to visualize polyhedra.

Spherical symmetry and polyhedra

Until now I had only explored spherical symmetry through the transformation of fundamental regions. A fundamental region in this context is a subset of the sphere, such that when a symmetry is applied to it it fully covers the area of the sphere -no more, no less. But we’ve seen that spherical symmetry can be applied to any object in 3D space. What happens if we apply spherical symmetry to polyhedra?

Octahedron symmetries

As all (uniform) polyhedra have some sort of intrinsic symmetry, using them as the fundamental object to which some spherical symmetry is applied gives some redundancy. This redundancy is seen because we have overlapping instances of the object. For example, given a spherical symmetry and careful placement of the polyhedron, we can fully overlap all instances of it to give the impression that we are manipulating a single object.

The design tool I’m building provides the ability to showcase 3D objects in a small-multiples fashion. This is a powerful approach to compare symmetries happening in euclidean, spherical and hyperbolic space. The images below provide all spherical symmetries for platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron).

Applying all spherical symmetries to all platonic solids

Although I like the chaotic appearance of some of these (like the 3,3 object in the icosahedron image above) I also developed an algorithm that would balance objects such that the distance between the multiple instances are maximal. You can see before (left) and after (right) examples of this below:

Providing a better balance on the symmetrical structures by maximizing the distance between instances of the polyhedra. Left: “unarranged” objects. Right: same objects after applying the balancing procedure.

I used polyhedra in the past in a map projection project to find optimal mappings over platonic solids. An optimal mapping means that the map is laid over the solid in a way that when the solid is unfolded we minimize land cutting. I adventured into new territory by expanding on this work to include all Archimedean solids.

Myriahedral Projections and Archimedean Solids

Optimal mappings on Platonic Solids. The map is laid out on the solid and unfolded in a way that land cutting is minimized

In the paper Myriahedral Projections, Jarke van Wijk studies optimal map projections on platonic solids. In this work the earth is mapped over the solid in a way that minimizes land cutting when unfolded.

I extended the original work to provide optimal mappings for all 13 Archimedean Solids. Although few of these mappings leave the land intact, I found it intriguing to see the unfolding of these more complex objects along with their optimal mapping.

Unfolding of the small rhombicuboctahedron

Recursive Subdivision

On Myriahedral Projections Jarke van Wijk also creates meshes out of recursive subdivision procedures such that high-res polys can be created by subdividing the faces of tetrahedrons, cubes, octahedrons and icosahedrons.

Recursive subdivision on platonic solids

As long as these solids consist of faces that can be recursively subdivided we can apply this same procedure to Archimedean solids. This leaves us with the cuboctahedron, small rhombicuboctahedron and snub cube as Archimedean solids that can be subdivided. Below we place the proper solid and the subdivided mesh next to each other which enables us to map the faces on each picture. Unfolding of the solid is different as recursive subdivision optimizes for keeping north up and south down instead of no land cutting, but we can see that the same number and type of faces have been unfolded.

Small rhombicuboctahedron as solid and as subdivided mesh. The weighting strategy is different but we can see the same amount (and type) of faces being unfolded

There are many other solids in this project that I recommend you to (re-)visit as new mappings and improvements have been made to the project.

Dual-layered Polyhedra

There are operations we can perform on solids to obtain new solids. For example, many Archimedean solids were created by truncating Platonic solids:

Optimal mappings on truncated versions of Platonic solids

There are other operations we can apply to a solid, including elevations and stellations, which were used by Escher and more recently Rinus Roelofs for mathematical art. Inspired by the work of Rinus Roelofs I applied these operations to a few solids in order to create dual-layered polyhedra. These are interconnected solids that consist of multiple layers.

Dual-layered polyhedra based on stellated dodecahedrons, icosahedrons and octahedrons

The WebGL design tool can use these transformations by also giving full customization of the design of the faces, which means many of these designs are unique. Excited by these new sculptures I decided to materialize some of them with 3D printing on Shapeways:

3D printed dual-layered models based on stellated icosahedron and dodecahedrons

Finally I generated more intricate models based on other polyhedra. I’m looking forward to exploring these objects in more detail and see what interesting sculptures I can create.

This wraps up work on spherical symmetries that tangentially also influenced an extension to previous work on myriahedral projections by adding optimal mappings and recursive subdivision on archimedean solids. The work also explored operations on solids like stellation and the creation of dual-layered polyhedra that were 3D printed in Shapeways. Looking forward to the next iteration of this work!