Tensegrity

Sheared and Stretched

I’m kind of loving this project. In addition to the ability to 3D print anything that can describe mathematically with STL files, I can also do a lot of this work with the Zome construction toy. Zome is truly wonderful and what was startling to discover was that I can represent the basis vector that is used for the Cartesian to Body Centered Cubic Grid coordinate conversion directly with a Zome model. No need for doing any custom work. Straight up mass produced plastic construction toy. Love it.

The paper Body Centered Cubic Grid – Coordinate System and Discrete Analytical Plane Definition provides a magical, non orthogonal vector basis that we can use to convert between a grid composed of cubes and a grid composed of spheres. Here’s the key figure from the paper, illustrating the basis vector used in the paper to convert from Cartesion to/from BCC coordinates.

As mentioned, this basis is non orthogonal. It’s a bit hard to see from the above figure b, but basically we take the X and Y axis as is – your normal x, y plane we’re familiar with – scale that by a factor of 2. We then take the Z axis, which is orthogonal to both the X and the Y axis in Cartesian coordinates, and slant that fairly severely (sheer) and scale that axis by the square root of 3.

Which, I must admit, is an entirely non obvious thing from my point of view, which is why I love this Information Age we live in. So if you look at the figure above, it’s figure b in that figure that illustrates the basis vectors. And… again. Not really a great picture to illustrate just how weird this basis is and what horrible torture was performed to get this transform to and from Cartesian coordinates to all be integers….

So I built the illustration above using Zome.

Anyways, been working on the graphics stuff. Finally got some stuff working using Rotors from Geometric Algebra, rather than quaternions and I must say I’m very impressed with it (narrator: he is easily impressed). Very simple and I spent a decent amount of time viscerally coming to grips with the operations a Rotor does. Very cool.

I ordered a photo box thingy so I can take far better photos of my Zome and other models, but that shows up on Friday, so… better looking photos in our futures.

That’s enough for today’s post.