The $n$-simplex is defined to be $$ \Delta_n = \{x\in\mathbb{R}^{n+1} : x_i \geq 0 \text{ for all } i, \quad \sum_i x_i = 1 \} $$ This contains the basis vectors $e_0,\dotsc,e_n$, which are shown here as red spheres.

• $\Delta_0$ is just the single point $e_0$
• $\Delta_1$ is the line segment joining $e_0$ to $e_1$
• $\Delta_2$ is the triangle with vertices $e_0$, $e_1$ and $e_2$
• $\Delta_3$ is the tetrahedron with vertices $e_0$, $e_1$, $e_2$ and $e_3$