In the world of computer graphics, it gets more and more difficult to place the vertices of a simplex as the number of dimensions increases. In one dimension, it’s easy. You can just draw a line from x = -1 to x = +1.
For two dimensions it’s a little more difficult, but still pretty easy. To create a regular triangle you go around a circle, marking off a vertex every third of the way around.
Starting with three dimensions it starts to get harder. To create a regular tetrahedron you need to resort to some fancy tricks. One trick is to first make a cube, and then use only half of its vertices. The edges of the tetrahedron are just diagonals of the cube’s faces.
But once you get to four dimensions, those tricks don’t really work anymore. Which is very discouraging if you like doing things with simplex shapes.
But then my brother told me how they make simplex shapes in his field, and it is completely awesome. More tomorrow.