Did you like playing with shape sorters as a toddler, but find them too simple as an adult? Well, I’ve got good news for you.
It started at February’s MathsJam, when the folk from East Dorset set a challenge for MathsJams around the world. They provided an example of an object that would cast a circular shadow from above, a square shadow from the front and a triangular shadow from the side. Could we, they asked, design objects that would cast a different set of shadows? Circle, circle, square maybe? Or circle, square, hexagon?
In Canterbury, we puzzled over this for a while. Circle, triangle, triangle was solved quickly, but circle, circle, square had us stumped. There was lots of hand waving and sketchy drawings. We could all see the shape, and I’m pretty sure we were all seeing the same thing, but none of us could define it in a tangible way. Until somebody said ‘it’s like the intersection of two drainpipes’.
That didn’t help us to draw the object, but it did mean that when I got home I could model it with OpenSCAD, which is perfect for jobs like this. With only six lines of code (two of which were closing braces) I had an onscreen model of the intersection of two cylinders, and a few hours later I had 3D printed it and could verify that, yes, it did cast a square shadow from above and circular shadows from the front and side.
I’ve since found out that this is called a Steinmetz solid; if you want to know more about them, Evelyn Lamb has written an interesting article.
But why stop there?
If circle, circle, square was the intersection of two cylinders, then maybe circle, square, hexagon would be the intersection of a cylinder and a hexagonal prism – and so it was. Well, almost. I couldn’t get both a square and a regular hexagon with perpendicular shadows; I had to settle for either a non-square rectangle or an irregular hexagon, but I counted that as a success. Soon I had a happy little family of objects that each cast its own set of shadows.
It still didn’t feel like enough, though. It’s hard to get a perfect shadow without fingers getting in the way and I wanted an easy way to show off the views of the objects. Another few hours with the 3D printer and I had it – the many-to-many shape sorter. Each object fits snugly through at least two holes, and each hole will accept one or more objects. Now I can show my shapes off as much as I like. There’s something immensely satisfying about seeing an object that doesn’t look at all hexagonal fitting neatly through a hexagonal hole.
If you would like to print your own many-to-many shape sorter, the files are all available on Thingiverse. Or why not play with OpenSCAD and see what other combinations you can create? You don’t need a 3D printer to use the software (which is free) – the objects can be rendered onscreen and you can rotate them to see the different views.Your inner toddler will love you for it.