I was trying to understand what a simplex was intuitively by constructing an example.
Consider only points in $\mathbb{R}^2$. From wikipedia the definition seems to be:
Choose k+1 points $u_0, ..., u_k \in \mathbb{R}^n$ s.t. are affinely independent (i.e. $u_1 - u_0, ..., u_k - u_0 $ are linearly independent).
So I choose $u_0 = <1,2>$ and $u_1 = <1, -2>$ and the "simplex" they seem to span seems to me more of a parallelogram than a triangle. Which I found confusing. I will admit I don't know a rigorous topological definition of a triangle, however, intuitively it doesn't look like a triangle.
It looks like this to me:
A simplex formed by two vectors is a line segment. The problem with your picture is that you shaded the region where $\theta_0$ and $\theta_1$ are between $0$ and $1$, but without the condition that their sum is $1$. Your picture should be the segment between the endpoints of the vectors.