Here’s a side project of mine that I’ve fortunately completed today!
For some time, mainly after watching some of the Marvel superhero movies and hearing about the “tesseract”, I looked it up and was disappointed that the movie tesseract has nothing to do with the geometrical concept, which is much more interesting. Disney portraits the tesseract as a simple, glowing blue cube with an animated fluid substance inside.
In geometry, the tesseract or hypercube is the "four-dimensional analog of a three-dimensional cube. It is to the cube, what the cube is to the square". (source)
This is very cool. I was looking for a way to take 3D slices through 4D shapes and found this on Github: https://github.com/orybkin/Tesseract . It uses a Jupyter Notebook to visualise a 3D slice through 9 hypercubes (not all are visible at once)
Unfortunately it runs in Cpython, not in Ironpython and uses Scipy for key elements so no way to easily integrate it with Rhino / GrassHopper
It takes any geometry, and stereographically projects it from flat 3-space to the 3-sphere, rotates it in 4d, then projects it back.
In 2 dimensions there are many possible conformal mappings (angles are preserved, and circles stay circular), but in 3d these Möbius transformations are the only conformal transformations possible.
Since we’re only looking at a 3d slice or projection anyway, I think it can sometimes be more illuminating to explore 4d rotations through their effect on familiar 3d objects rather than just higher dimensional polyhedra.
You can also get all sorts of interesting curves and surfaces by sweeping or arraying points or curves through these 4d rotations
As always, astonishing stuff, Daniel!
I barely managed to get the projections for the cube right, but your example with the horse is just great, and has meme potential.
