Joining two segments
My how time flies…
The join of two topological spaces and is basically the space of all line segments between every pair of points. As a nice embedded, visceral example of this, a tetrahedron may be viewed as the join of two skew line segments.
Indeed this is the illustration on Wikipedia.
It also arises naturally when you think about grid diagrams of knots. Remember this from way back?
Thinking along those lines, you might consider discretizing it a bit… Instead of taking the join of two entire segments, just take the join of points along each segment.
Here I’ve used . And that gives us the complete bipartite graph naturally embedded as a subset of the tetrahedron. Moreover, any grid number knot is a cycle in this particular embedding of the graph.
While it’s a rather simple object, I’ve nonetheless found it to be a rather pleasing one. Lots of emergent rhythms.
Kinda gotta hold it and spin it around to really appreciate it, so print one out if you’d like. I’ll talk about some of the others models there whenever I eventually get around to it…