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I was talking recently with my daughters about non-planar graphs, like $K_{3,3}$, $K_5$, and the 7 bridges of Königsberg. They got pretty interested in it, and seemed to catch on to the core ideas. Then we tried to dive into the Petersen graph and the concept of linkless embedding in 3 dimensions. This wasn't so successful.

The idea of planarity can be illustrated pretty well on paper, but in 3 dimensions, one needs some sort of modeling toy to show what we're talking about in terms of vertices and edges that link together in cycles.

There are face-oriented toys out there for exploring geometry in 3D, like these which allow you to build certain polyhedra by attaching rigid triangular and square faces to each other edge-to-edge.

There are also rigid edge-oriented toys, like this one (unavailable), which offer more freedom in arrangement of vertices and edges.

But we want to explore graph theory in a way that's not limited to edges that are all straight and all the same length. In fact, sometimes experimenting with different "routes" for an edge is key to learning principles of graph theory. What are some good options for this?

This flexible rod play set would work, I suppose, though the joints supplied don't look very much like vertices. Something cheap would be nice. Pipe cleaners and clay seem to be a popular approach, as in the picture below, but I wonder what kind of clay is stiff enough to maintain a 3D configuration of more than a few vertices. Maybe pipe cleaners should be stuck into little foam balls? Any better ideas? They don't have to be brightly-colored pieces that are marketed as toys.

graph hypercomplex

(I moved this question from math SE on the advice of someone there.)

LarsH
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    Link to Mathematics Stack Exchange question: https://math.stackexchange.com/q/3442498/18398 – JRN Nov 20 '19 at 02:38
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    @RustyCore, "Sometimes, graphs are allowed to contain loops, which are edges that join a vertex to itself." – JRN Nov 20 '19 at 03:59
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    For a while I very seriously considered doing some work drilling holes into golf balls and then placing eyehole screws into the holes, connecting with some sort of flexible rope, for this very purpose. It was just too much work to do it myself - I was anticipating having to do quite a bit of trial and error as to the right materials. Sounds like a marketing opportunity if you build something like this from scratch! – kcrisman Nov 21 '19 at 02:18
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    You can try colored insulated thin wires and "wire caps" (I mean those elongated things with conical nuts inside so that you just twist the wires together and screw a cap onto them; if you google these two words, you'll see what I mean immediately). That holds great and using a few cap diameters, you can do reasonably high vertex degrees. If you want your vertices to look like balls instead of cones (though I see nothing wrong with imagining a vertex as a conical shape), you can place a piece of clay on each cap after the construction is finished. Clay then holds nothing but itself. – fedja Oct 31 '22 at 10:55

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