Shower Rods and Higher Dimensions
It’s moving day. You’ve got your entire house packed up into boxes all ready to go, and are all nervous/excited to move into your new home. Just one last item: the shower rod. You’ve got one of those fancy-schmancy extendable ones, so you start to wonder: “if I were to place my shower rod at one corner of my box, and extend it until it reaches the opposite corner, how long would it be?”
As it turns out, the answer to this question is either really simple or a complete mind-bender depending on one thing: how many dimensions you are living in.
A two dimensional box is a rectangle, so if you paid even the slightest amount of attention in middle school, this should be easy.
What’s the diagonal distance? Draw the diagonal line, and use the Pythagorean Theorem. In this case,
What use the flatlanders have for shower rods? The world may never know.
3D is slightly harder, but totally manageable. This takes some spacial visualization, and because I have no faith in my ability to convey pictures through text, I will cut out the middleman:
We want the distance from A to B.
Start with what we know. The distance from A to C is sqrt(x2+y2) just like before:
So how does this help us? Let’s cut a plane through the box along the distance we just calculated and point B:
Both A and B lie on this plane, so draw a line between them:
If we focus on just the plane, we can see that our new line forms a triangle! Not just any triangle, but a right triangle,1 which means we can use the pythagorean theorem again:
Simplifying the squared square root, and taking the square root of both sides, we get
If you are stubbornly stuck in the real world, you may stop here.
This is way harder. As humans, we don’t have a physical intuition of 4D. You could try to make analogies with our 3D world, but they never really give you a feel for higher dimensions. Visualising this is impossible, so no pictures. We’re flying blind
Fortunately for us, math doesn’t care about pictures.
So how do we even begin? Let’s start by trying to recognise patterns about what we already know:
Consider a 2d box. When “changing” from 2d to 3d, what direction is this new side? Well, all sides of a box are perpendicular to each other (that’s kind of the definition of a box), so the new 3rd dimensional sides2 are perpendicular to every possible line we could draw on the 2d box.
So what happens if we apply this pattern to the transition from 3d box to a 4d box? Well, whatever the fourth dimensional sides looks like, they would have to be perpendicular to all lines drawn in our 3d box. (stop trying to visualize it, just go with me)
Take the line along the fourth dimensional edge that touches the 3d line (that we already know the distance of). We know it has to be perpendicular to the 3d line, because of the rule above. We also know that it touches the 3d line, because the meet at the save vertex.
And what do we get when we have two perpendicular lines that meet?3 A right triangle!
Using our friend Pythagoras:
This is the exact same process I used in the 3d example, only there we had helpful diagrams that lead us to the answer. If you wish, you can continue this argument indefinitely for higher and higher dimensions. This gives the nice general case3 for n dimensions:
Oh, and just to be cute, we can do this formula for one dimension:
1Angle C is right.
2Ok technically, there isn't such thing as a "third" dimension, (mintue physics video on that) but let’s just pretend that one corner of this box is on the origin and all “sides” are on axies
3Assuming one point is on (0,0), so perhaps not as general as it could be