Knex Geometry
When I was a kid we didn’t have LEGO in the house; we had Knex.
One of the benefits of Knex over Lego was that it’s hard plastic and could make a five-foot tall superstructure like this. I was just the right age when they were releasing new kits, and mass-producing kits with solar panels (which I later used as a demo in OLPC), remote-controlled motors, and paper magazines.
Never finding a market foothold, today they make some classic and classroom kits, and plastic toys and figurines with a box of Knex wrapped around it:
One of the basic concepts of Knex is that connectors have a set radius, and rods are color-coded with each size about 1.5x the previous one.
Now you might say, hey Pythagoras, it’s a triangle, a² + b² = c² , so we know that each rod is 1.414x the previous size.
But, as I discovered as a kid, that was not the case. In order to work with the radius of the connectors,(blue + 2*r)² + (blue + 2*r)² = (yellow + 2*r)²
Apparently there’s an Instructables with measurements for this already, but with some intuition you can see that you would need to calculate a specific ratio for each rod, and that the diagonal for the whole square above can be +1 to a red rod, or two blue rods and a connector as shown.
One summer I had the idea to build some of my kits with every piece scaled up to the next size. In my head I anticipated a problem that the rods would scale everything up ~1.5x while the connectors were the same size.
But everything fit together.
In 2020 when I was quarantining with family, I used some downtime to build a huge one of these scaled-up structures. Again against intuition it all worked (except for a few parts which clip onto the structure at an angle). If it was the perfect ratio of lengths between each piece, then it didn’t make sense to me why or how scaling would be built so cleanly into the Knex system.
This was cycling around in my head while I had graph theory people at my day job, so I thought about the connectors as 0-dimensional points on a grid and it was only the rods which needed to scale up. That didn’t totally sit right, but it got it off of my mind
Today (summer 2022) while I was cleaning dishes, I was thinking about transporting or reselling or donating the Knex (is it awkward to donate a giant bag of 90s construction toys?). I was visualizing them and it led me to think about the problem again:
I should never have wondered if the level-up structure could be built, because I had already built a version of it ! It’s true that the connectors do not scale up, but the difference in sizes works, because each next-level piece is already sized to fit a diagonal. Instead of visualizing a whole structure or grid scaling up, I could think of the diagonal structure rotating and becoming the main structure.
This explains why the ratio of lengths between pieces worked so neatly for my scaling project, why pieces attached at other angles didn’t follow suit, and how it connected back to the original triangles.
Anyway it is very late and now I don’t have to wonder about this anymore.
