What could be more intuitive than the notion of space itself? Space is how we know here from there and all that entails. Which is everything. Imagine some existence without here and there and everything, all of us, shared a single-dimensional point, somehow. You, me, President Barack Obama, Taylor Swift—a single point in space.
It’s just as hard to imagine the opposite, a space of more than three dimensions. A fourth dimension for time isn’t so bad, where we might still imagine our three-dimensional world, but only with a sort of inner motion or current: moving without moving. Or maybe imagine time like a computer screen, a static reality that is always refreshing as time progresses. The frequency with which it refreshes, that’s time.
But five dimensions? Fuck.
There’s an easier way to look at it, courtesy of my old Calculus III professor. We think of space as space, heres and theres, but these three dimensions are really a commingling of three variables, each one describing something different: length, width, height, x, y, z. There could be other spaces too, maybe one of red, green, and blue, or one of temperature, pressure, and humidity, etc.
So: three things connected by some equation of three variables. RGB, red-green-blue, color is properly considered as a space, the RGB color space. There are other color spaces, where we might imagine a color as a point given by three axes, with each one representing one constituent color. Usually, we imagine the color, not the space.
It doesn’t matter what the variables, or axes, of a space are so long as they’re related by some function, an equation relating all those variables. It’s easier to see the possibility of other dimensions then: temperature, pressure, humidity, and then, say, water vapor pressure, mixing ratio, CO2 ppm, x, y, z, a, b, c. This doesn’t make the graphical representation of extra dimensions easier to visualize, but the idea is maybe easier to get a hold of when it’s not literal heres and theres.
(To really push the bizarre intuition of spaces, consider the additional feature that, as we add axes/variables to our function, each new axis has to be at right angles to all of the other axes. So, look at a three-dimensional graph and try to figure that out.)
In the quantum world, things get really, truly wrong. Here we have spaces of infinite dimensions, by necessity. Spaces of possibilities.
First, imagine that a point in some space is described by a list of variables, and this list is stored inside of a vector, which is a just a stack of values, one for every dimension. To describe me in some 10-dimensional space, I could be described by the coordinates (x1,x2,x3 … x10) all lined up neatly in a vector, like a silo for numbers.
In non-infinite space, we can do things like find angles between these vectors and the vectors that might result if we add them together. We can do this for any finite number of dimensions, like a billion. There is an angle between two vectors with a billion dimensions.
Infinite dimensions doesn’t mean quite what it feels like it should mean, which is some infinitely huge thing, like a train that doesn’t have a beginning or end, just length that expands forever and ever. Infinity can be contained within something finite too.
When I play a note on my viola, that note can be described by a wave with some frequency and amplitude. So just think of a wave spanning space from point A to point B. The thing about waves is that they’re always the sum of other waves. My note is a wave, but it can be described as two (or more) other waves, but also two other other waves where one wave is a little bigger and the other a little smaller. And so on.
This decomposition becomes arbitrary at a certain point, which is a bit like saying it becomes infinite, or a space of infinite possibilities.
Read More: Here