Someone once told me that in the same way a 3D object casts a 2D shadow, a 4D object casts a 3D shadow. I just...can't. I can't wrap my head around that no matter how hard I try.
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'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).
(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. :)
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.
Maybe? I think it allows solutions not typically considered to be rotations, like the special relativity example.
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.
A 4-cube would look like just few identical 3-cubes from the side, if oriented face-to-3-space. If not, it will be series of prisms or platonic solid alikes.
It doesn’t help much with shapes that have complex 4d-edges, or with higher dimensions, but may give an idea. Also, open two gifs from this link http://www.math.union.edu/~dpvc/math/4D/folding/welcome.html side by side and you’ll quickly get the concept of rotation of a solid around 4th axis.
Here the fourth dimension is discrete, but you can imagine that for each city you get a 3D object.
It's not perfect. We can rotate in the normal 3D space but you don't get proper 4D rotation with this technique. Nor does this give you a good intuition about that shadow.
But it's something.
Gives you basically nothing when it comes to the 5th dimension, though, because I'm fresh out of spacetime at that point.
https://www.youtube.com/watch?v=vZp0ETdD37E
Same dev who also created 4D Toys toolbox: https://4dtoys.com/
Most people of average IQ say that.
> 4D object casts a 3D shadow. I can't wrap my head around that
No one can. I don't think you understand what a dimension is.
> Most people of average IQ say that.
I seriously question that. Can you point us to any evidence that people of average IQ are most often visual thinkers? (Or most often say that they're visual thinkers, if that's where you were going with this.)