Lineland - Math & Physics in 1 Dimension
You're a point mass and you live on the x axis. That's your entire world. It's "Lineland". What's your life like?
First, as regards moving, you only have two directions, forward and backwards. And if you meet another point mass you cannot pass. So you can only know two other masses. You have just two friends maximum!
You have no reason to count objects beyond two, so you might be slow in developing the concept of integers. Or perhaps you never develop the concept at all. You simply have no need for it.
What about Physics in Lineland? You're a point mass, so you have a mass, let's say m. Another point mass could have a different mass, say M. So at least gravity exists, right? It does, but it has a very strange form. Newton's formula for the gravitational force F between two masses m and M in 3D space is..
where G is a constant and r is the distance between the two masses.
The r^2 term is good in a 3D space, but in general it's r^(n-1) where n is the dimension of the space. Putting n=1 for Lineland we get..
r^(1-1)=r^0=1 so F=G*M*m
Which means F is independent of distance! Gravity has the same strength no matter how far apart the objects are. So physics in Lineland is very different.
This is Lineland on the x axis. What if Lineland is the circumference of a circle? That's even more interesting. Would you be aware that Lineland had a "curvature"? What does gravity do now that Lineland is a closed loop? What happens if Lineland is a closed loop that intersects itself at several points? What happens at these intersection points and how do they contribute to gravity? How do things change as the number of point masses in Lineland changes? It turns out that even 1 dimension can be very complex!
Just think, there's probably a 4 dimensional world somewhere with math teachers looking for a nasty problem to set on an exam. Finally they come up with one, "explain how math would have developed if our world was constrained to just 3 dimensions".
Content written and posted by Ken Abbott firstname.lastname@example.org