Multiplying Non-Numbers

In last last week's episode of PBS Infinite Series, we talked about different flavors of multiplication (like associativity and commutativity) to think about when multiplying things that aren't numbers. My examples of multiplying non-numbers were vectors and matrices, which come from the land of algebra. Today I'd like to highlight another example:


It's true! To illustrate, here's the multiplication table for a point, a line, a circle, and a square.

See the pattern? Given any two shapes $X$ and $Y$, their product $X\times Y$ is the shape whose vertical cross sections look like $Y$ and whose horizontal cross sections look like $X$! Also notice that the point acts like a multiplicative identity: a point times a shape is just the shape.

I'm really stretching my art skills today, but the idea is hopefully clear: A 'circle $\times$ line segment' is a vertical cylinder: its horizontal cross sections are circles and its vertical cross sections are lines.

The idea behind the multiplication is that there's one circle for every point on the line segment (so, "a line segment's worth of circles") and one line segment for every point on the circle (so, "a circle's worth of line segments"). When you put it all together, the result is a cylinder!
Similarly, a 'circle $\times$ square' is a roundish-but-square torus thing. (For some reason it reminds me of a cronut, though they're not quite the same.) Its horizontal cross sections are circles, and its vertical cross sections are squares. (So what does a 'circle $\times$ torus' look like? Or a 'torus $\times$ torus'??)
Notice that this multiplication is not commutative! A circular torus with square cross sections is not the same as a square torus with circular cross sections! In other words, 'circle $\times$ square $\neq$ square $\times$ circle.' Likewise, a box lying horizontally is not the same as a box standing upright.

But what about associativity? Let's compare '(point $\times$ line segment) $\times$ circle' with 'point $\times$ (line segment $\times$ circle).' According to the multiplication table:
So, yes! It's associative! It doesn't matter which two shapes we multiply first---at least in this example. And it turns out that this multiplication-of-shapes is in general associative. By the way, this multiplication has a name. The product $X\times Y$ of two shapes is called the Cartesian product of $X$ and $Y$. So if you watched last week's video, you've now heard of at least four examples of multiplication* of non-numbers:
Now here's what's cool: shapes will make an appearance in our next epsisode on Infinite Series! As we saw last week, loop concatenation is not associative: the two ways of multiplying three loops are not equal. And in the next episode, we'll discover that there are infinitely many ways of multiplying three loops. AND - it gets better - all of those ways are encoded in a particular shape!

What shape, exactly?

You'll have to wait and see!


*While watching the video, you might've wondered why I referred to loop concatenation as a "multiplication" (or product) instead of "addition" (or sum). Technically, loop concatenation is a binary operation, which is a way to combine two things (the inputs) to get a third thing (the output). Addition and multiplication of numbers are examples of binary operations. In practice, mathematicians often use the word “addition” to describe a binary operation that is commutative. And although I don’t mention it in the video, loop concatenation is not commutative! So we call it a “multiplication” instead.

Thanks, John, for suggesting the idea for today's post!