## June 11, 2016

### How the Simplex is a Vector Space

#### Posted by Tom Leinster

It’s an underappreciated fact that the interior of every simplex $\Delta^n$ is a real vector space in a natural way. For instance, here’s the 2-simplex with twelve of its 1-dimensional linear subspaces drawn in:

(That’s just a sketch. See below for an accurate diagram by Greg Egan.)

In this post, I’ll explain what this vector space structure is and why everyone who’s ever taken a course on thermodynamics knows about it, at least partially, even if they don’t know they do.