Owen Biesel

For a future blog post, I’ve been thinking about how sometimes, when we have two alternatives, one is really a special or “limiting” case of the other, the way a square is just a special case of rectangle. I’m still working on that post, but meanwhile I’ve been distracted by thinking about other shapes that are special cases of each other. Rectangles are special cases of parallelograms, which are special cases of trapezoids… Here, let me draw a diagram:

Sometimes this kind of “special case” information is organized as a Venn diagram (or more accurately, an “Euler diagram,” name rhymes with “boiler”) with the more specific categories shown completely contained within the more general categories:

This presentation makes it clearer that not only is every square both a rectangle and a rhombus, but every rectangular rhombus is also a square.

It’s a bit odd, though, to use ellipses to represent these classes of quadrilateral. Why not use… the quadrilaterals themselves! Then you don’t even need to label each category — the shape of the category is its label!

For example, the brown rectangle and the teal rhombus intersect as the pink square, because the set of squares is the intersection of the set of rectangles and the set of rhombi. Similarly, the teal rhombus is the intersection of the red parallelogram and the blue kite, because the rhombi are exactly the kite parallelograms. I think the result is quite nice!