Imagining multiple dimensions is hard.

I don’t think most people can intuit their way into a multivariate (i.e. multidimensional) problem. Our minds aren’t built for that and we have that pesky limitation in our senses that restricts us to 3-dimensional spaces. This was the kind of things I was thinking about while I was packing and moving. And while I was thinking about this, I remembered a very interesting assignment from back in the day when I was an undergrad. It demonstrates in a simple and elegant way that even our understanding of a concept as simple as the Euclidean distance (which is what most people think about when you say “distance”) can fail us when we move into higher dimensions.

Consider the following geometric scenario. We are in 2 dimensions here, so we will call the space \mathbb{R}^{2} . It has 4 circles of radius 1 centred at the (1,1), (-1,1), (-1,-1) and (1,1) coordinates, two tangent lines of length 2 (the diameter of each of the bigger circles) that touch each circle on two points forming a square and a small little circle centred at (0,0) which is itself tangent to the other 4 circles from the inside. So the thing looks like this:


The question to ask here is what is the radius of the small circle inside the bigger circles of radius 1, which are themselves inside the tangent lines of length 2 each forming a square. A simple answer is to calculate the (Euclidean) distance from the origin at (0,0) to the centre of the circle at (1,1). Notice that I chose the bigger circle in the positive quadrant arbitrarily. The argument works for a circle in any of the other quadrants. Since we know the radius of this bigger circle and where its middle point is, we can create a right triangle of  of leg size 1 (because its centre is at (1,1), calculate the hypotenuse that connects the coordinate (0,0) to (1,1) and, from that distance, subtract 1 (the radius of the bigger circle). So it looks like this:


The blue segment is the radius of the small circle which is simply \sqrt{1^{2}+1^{2}}-1=\sqrt{2}-1 \approx 0.4142136 . So far, so good.

Now let’s take it up a notch and bump this whole geometric setting one dimension from \mathbb{R}^{2} to \mathbb{R}^{3} , so everything is 3-dimensional now. It looks like this:


Our circles are now spheres and our square is now a cube. Something non-trivial has also taken place. In a 2-dimensional plane there are 4 quadrants and, hence, 4 circles. In a 3-dimensional plane there are 8 quadrants, 4 for the positive direction of the “height” dimension (which we call the Z axis) and 4 for the negative dimension of the Z axis. So now there are 8 larger spheres, all of radius 1, centered in the coordinates (1,1,1), (-1,1,1), (1,-1,1), (1,1,-1), (-1,-1,1),(1,-1,-1), (-1,1,-1) and (-1,-1,-1), the cube now has edges of length 4 in total, so it is 2 going in the positive direction of the Z axis and 2 going in the negative direction of the Z axis). The smaller sphere is still centered at the origin, now (0,0,0).

The question becomes again, what is the radius of the smaller, green sphere? Well, let’s again consider the bigger sphere in the positive quadrant, the one centred at (1,1,1) and calculate the (Euclidean) distance from (0,0,0) to (1,1,1) using the Pythagorean theorem. Notice that the cube is made up of planes tangent to the bigger spheres, all of length and width 2, the diameter of the sphere. This will become important later on. In any case, the setting now looks like this:


And again, the calculation is \sqrt{1^{2}+1^{2}+1^{2}}-1=\sqrt{3}-1 \approx 0.7320508 .

This problem seems rather uninteresting at first glance. If you move up on dimension, the number of spheres (we should call them “hyperspheres” now since they exist in dimensions greater than 3) increases as a function of the number of quadrants in the space. More specifically, for a space of dimensions \mathbb{R}^{n} you get 2^{n} hyperspheres. We saw that before. In 2 dimensions there are 2^{2}=4 circles, in 3 dimensions there are 2^{3}=8 spheres and so on. But now let’s focus on what happens to the smaller hypersphere, centred at the origin irrespective of the number of dimensions in relation to its distance to the outside hypercube (hyperbox?) that has a constant edge of length 2 (because, by construction it is tangent to the hyperspheres of radius 1).

Well, the distance from the origin to the centre of the sphere in the positive quadrant (or any other quadrant) is \sqrt{1^{2}+1^{2}+1^{2}+1^{2}+...}=\sqrt{n} Notice that even if you’re in the negative quadrants, the Euclidean distance requires squared terms so you are always going to end up with a 1 added to another 1 on and on as the number of dimensions grow. So the distance from the origin to the centre of any hypersphere is \sqrt{n}. Now since the radii of the bigger hyperspheres are always 1, the radius of the smaller hypersphere should be \sqrt{n}-1 for any arbitrary n. But wait a second… the sections of the hypercube at each quadrant that contain the bigger hyperspheres are also defined by these radii or, more correctly, the diameter of the hyperspheres which are always 2. So the relationship between the radius of the small sphere centred at the origin with the wider hypecube that encompasses all our geometric setting ends up looking like:

\sqrt{n}-1 \geq 2

\sqrt{n} \geq 3

n \geq |9|

So when we move up to \mathbb{R}^{10} or the 10th dimension the smaller hypersphere at the origin “pops out” (for lack of a better term) of the box. Strictly speaking, its radius is greater than the edges of box in the quadrant that contains it. Yet it is still bounded by the 2^{n} hyperspheres! How can this be!?

I remember this got a lot of people talking the next day and sparked an interesting class discussion. I jokingly called it “God’s sphere” because the little hypersphere centred at the origin exhibited a property that we usually only ascribe it to the divine, i.e. the ability to be in two places at once, both inside and an outside the box. Our prof used this as an example that we can’t always trust our intuition in Mathematics because there are quite a few results that are true yet difficult to understand unless you stick to the definitions. In my case, it also brought in an interesting insight: human beings just can’t think multivariately very well. Like, I don’t think anyone has a good insight of what “inside” and “outside” mean in multiple dimensions. I mean, the small sphere looks like it’s inside if you look at it from the perspective of the bigger hyperspheres but it looks like it’s outside from the perspective of the hypercube. But I don’t know how the inside of a cube looks like in dimensions greater than 3. This is one of the reason of why I like mathematical objects like copulas. They are able to, in a very rigorous and principled way, translate multidimensional questions into series of 1-dimensional questions so that we, as people, can at least try to glimpse how the multivariate space behaves.