The n-Category Café

You should have explicitly mentioned that your last graphic is dynamic and people should drag their mouse cursor over it. That this might happen does not seem a possibility unless one has gone off to play with the earlier examples. From inspection this is achieved with an <object> that includes a separate .html file.

Thanks, Simon — that’s nice!

Is this 600-cell orthogonally projected from 4d space down to the plane? No fancy perspective tricks?

If so, I’m surprised and enlightened: I hadn’t imagined that the 600-cell could have shadows that were so far from round!

Just in case anyone lurking here likes to create beautiful animated gifs, here’s one I’d really like to see. I’d like to see a looped animated gif of the 3d polyhedra formed as as ‘slices’ of a 600-cell as a hyperplane passes through it.

In more detail:

We can take the 600-cell to have vertices

$$\displaystyle{ (\pm \textstyle{\frac{1}{2}}, \pm \textstyle{\frac{1}{2}},\pm \textstyle{\frac{1}{2}},\pm \textstyle{\frac{1}{2}}) }$$

$$\displaystyle{ (\pm 1, 0, 0, 0) }$$

$$\displaystyle{ \textstyle{\frac{1}{2}} (\pm \Phi, \pm 1 , \pm 1/\Phi, 0 )}$$

and those obtained from these by even permutations of the coordinates, where

$$\displaystyle{ \Phi = \frac{\sqrt{5} + 1}{2}}.$$

A nice choice of hyperplanes to slice this with are those where the first coordinate takes a chosen value $t \in [-1,1]$. As $t$ goes from $-1$ up to $1$, the polyhedron we get will morph from a point to a dodecahedron to an icosahedron to a dodecahedron to — at its largest — an icosidodecahedron:

and then back down to a dodecahedron, an icosahedron, a dodecahedron and finally a point again. So, this will make for a nice looped gif.

For more details see:

• The 600-cell, Azimuth, December 16, 2017.

From this you can easily read off the values of $t$ at which the slices I mentioned occur.

Since this posts asks if “someone can suggest a more efficient workflow”, the 3D viewer used here now has the option to “Save as HTML” for embedding graphics in web pages. It is also the default 3D viewer for the current version (9.0) of SageMath.

Great stuff, thanks for sharing this.

Three.js is a brilliant graphics library. Javascript is super fast in modern web browsers. A satisfying thing to do would be to take your models you made in Blender, save them in a suitable 3d file format, and then load that file into three.js. You will have super-reactive embeddable 3d swallowtail diagrams in your webpage!

The best tutorial I found for playing with three.js and doing the above is this one.