If anyone wanted evenly spaced points in $\mathbb{R}^2$ in the region defined by $x+y=1$ having $x,y<1$, the points are easily generated $(0,1),(.1,.9),(.2,.8),\dots,(.9,.1),(1,0)$.
In $\mathbb{R}^3$, if we had $x+y+z=1$ and $x,y,z<1$ it's again simple to create a “grid” over the section (which is a triangle with vertices at $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$) and generate as many evenly spaced points on it.
Now the problem starts. I don't know how to do that in $\mathbb{R}^6$. Say I have $x_1+x_2+x_3+x_4+x_5+x_6=1$, and $x_1,\dots,x_6<1$. How could I create a grid of evenly spaced points?
I would love to have an algorithm in an answer that I could run in Python or something like that where I could say something like I want 20 evenly spaced points, and get a list of coordinates like $(.1,.4,.3,0,.2,0)$ that are evenly spaced and are spaced through the whole space.
It would be awesome if rather than a fixed list, I could ask freely if I want $n$ evenly spaced points.
The idea is that after finding a bit the area I'm interested in, I will keep generating a finer grid around such area and keep approximating.