# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 90: The Standard (Unit) Simplex

The standard $n$-simplex, sometimes denoted $\Delta^n$, is the $n$-simplex in $\mathbb{R}^{n + 1}$ with its vertices at the $n + 1$ ‘unit’ points

$e_0 = (1, 0, 0, \dots, 0)$

$e_1 = (0, 1, 0, \dots, 0)$

$e_2 = (0, 0, 1, \dots, 0)$

$\dots$

$e_n = (0, 0, 0, \dots, 1)$.

For example, $\Delta ^0$ is the point $1$ in $\mathbb{R}^1$$\Delta ^1$ is the line segment from $(0, 1)$ to $(1, 0)$ in $\mathbb{R}^2$, and so on. A formal description is

$\Delta ^n = \{(x_0, x_1, \dots, x_n) \in \mathbb{R}^{n+1} | \sum _{i = 0} ^{n} x_i = 1, x_i \geq 0 \}$.

One possible interpretation of $\Delta ^n$ is as the set of possible probability distributions of a categorical distribution on $n+1$ variables.