Here is a much less tricky version: A car’s dimensions can include its length, height, and width, but also the amount of gas in the tank (4th), the position of the throttle pedal (5th) and brake pedal (6th), its weight (7th), the number of passengers (8th), etc etc.

I don’t think that works because the N-1 dimension is contained in the N dimension. But all those things described are disjoint. Now you might be able to say, “the mass of all objects in the universe” but is it a dimension? I dunno I’ll leave that to the topologists!

Edit: I think you have to have the similar type of units to increase the dimensions. Like the article talks about N dimensional space (e.g. point, line, plane, etc). To consider what mass of an object would be you’d have 1 dimensional mass; 2D mass; …; ND mass whatever that would be.

> I don’t think that works because the N-1 dimension is contained in the N dimension.

I'm not sure that's right. Each dimension is linearly independent [0] from the others, which means e.g. you can't add up a bunch of width and get height, or add up a bunch of length and get width. So in an important sense, they're not contained within each other.

You might be thinking of how a 2D plane contains the first dimension within it, but that's not the 2nd dimension... that's two-dimensional (a combination of two dimensions).

[0] https://en.wikipedia.org/wiki/Linear_independence

Hmm yeah that what I’m confusing. I suppose it’s misleading when folks say The Fourth Dimension since it depends on context! But in the case of the article the question becomes how do we visualize four or more dimensions? What the dimensions represent doesn’t necessarily matter unless there’s a specific problem being solved.

The car example works as a vector space, but not an inner-product space. There's no sensible way to rotate the length of the car to gasoline in the tank

That's right! The less pithy answer: objects are the things that stay invariant under a transformation. In that sense then car length and gas are somehow MORE than linearly independent - they are "geometrically independent".

(Note there is a relationship between the length of the car and the gas tank when the car is moving fast (close to c) in your reference frame - but AFAIK there aren't any useful geometric transformations that depend on that fact. :)

Linear independence doesn't guarantee that you can rotate, for that you need to know how to exchange one dimension for another.

Essentially you need to define a Pythagorean theorem for your space. You can do that for the car vector space, but there isn't a natural choice.

Isn't that the definition of rotation, the "exchanging one dimension for another"? In 2-D it keeps a point invariant, in 3-D a line, and in N-D it keeps the N-1 object invariant. (Side note: the more basic operation is reflection, since you can get rotation from two reflections)

>Isn't that the definition of rotation, the "exchanging one dimension for another"?

Maybe? I think it allows solutions not typically considered to be rotations, like the special relativity example.

You can't rotate around an arbitrary vector and have current brake pedal position determine the number of passengers.

There's a fundamental difference between a "set of independent variables" and a physical space with geometric transformations. The latter is far more complex to visualise.

I had a notion for a physical theory of everything, but I got hopelessly lost once I wandered into 6-dimensional knot theory.