It turns out that my brother and I, in our work, both deal with a mathematical object called a simplex. A simplex is just the simplest thing with straight sides in any particular number of dimensions.

For example, in one dimension a simplex is a line. In two dimensions it is a triangle, and in three dimensions it is a tetrahedron (a three sided pyramid).

In each case, the number of vertices (or corner points) is one more than the number of dimensions. For example, a line has two vertices (one on each end), a triangle has three, and a tetrahedron has four.

In my work in computer graphics, I have often needed to create a simplex shape. But once things go beyond two dimensions, it can get hard to figure out where to put the vertices.

Or at least I thought so, until I talked about simplex shapes with my brother. Because he comes from the world of statistics, he had a whole different way of looking at the shape of a simplex. More tomorrow.

Don’t you just have to extend / extrude the previous vertices in the new dimension ?

eg. a stretched point is a line, a stretched line (segment) is a plane (quad), a stretched quad is a cube, ..etc ?

That’s how I always have imagined it but well, I’m still lost after the hypercube.

That is a great way to construct an n-Cube (that is: a line, a square, a cube, a hypercube, etc). You need a different construction for the simplex.