Fourth dimension

This post is not about math, it is about how I perceive the world.

I know, fourth dimension is easy, for example you can easily create vector v=(1, 2, 3, 4) … but somehow, this is just numbers and tricks, I can’t really feel the fourth dimension. I recently had some free time to think about random stuff, and, well, I thought about this:

Let’s start with: 1 = x² + y². As we all know, this equation draws a circle.

The next step consists of substituting the one for another variable: z = x² + y². Now we get one more dimension. Although it might not be completely clear on the first sight, this is paraboloid. If we assume z is a variable that we can set to anything we want, different values of z create different circles (z is the height at which we cut the paraboloid to get that particular circle).

We had a circle already, so let’s bring the circle to the third dimension and make a sphere: 1 = x² + y² + z²

Now, what do we get when we make the transformation we made in the first step – substituting the one for yet another variable?

w = x² + y² + z²

I don’t know about you, but I can finally feel the fourth dimension within my reach. ;-)

While I don’t know what exactly this “w = x² + y² + z²” looks like (neither know what it is called, for that matter), I can see that there is precisly the same relation – the same way circle is one “slice” of paraboloid, sphere is one “slice” of w = x² + y² + z² (for some concrete set value of w).