---- > [!definition] Definition. ([[simplex]]) > A (geometric) **$n$-simplex** is a $n$-dimensional polytope that is the [[convex hull]] of its $n+1$ vertices. Image from Wikipedia: > ![[CleanShot 2024-10-02 at [email protected]|500]] > The **standard $n$-simplex** is the subset $\Delta^{n}$ of $\mathbb{R}^{k+1}$ obtained as the [[convex hull]] of the standard [[basis]] vectors $e_{1},\dots,e_{k+1}$: $\Delta^{n}=\left\{ (t_{0},\dots,t_{n}) \in \mathbb{R}^{n+1} : t_{i} \geq 0, \sum_{i} t_{i}=1 \right\}.$ $\Delta^{n}$ lives in the [[plane|hyperplane]] $x_{1} + \dots + x_{k+1}=1$.[^1] Sometimes denoted $\Delta^{k}$. >![[Pasted image 20250405141659.png]] > \ > The [[convex hull]] of any nonempty subset of $\ell+1$ of the $k+1$ vertices is called an **$\ell$-face** of the simplex. Faces are themselves simplices: > - $0$-faces are just the vertices themselves, > - $1$-faces are **edges**, and > - $(n-1)$-faces are just called **faces**. > > The $i$th face of $\Delta^{n}$ is $\Delta^{n}_{i}=\{( t_{0},\dots,t_{n}) \in \Delta^{n}: t_{i}=0 \}.$ > $\Delta_{i}^{n} \subset \Delta^{n}$ is evidently [[homeomorphism|homeomorphic]] to the standard $(n-1)$-simplex $\Delta^{n-1}$, via the **$i$th face map** $\begin{align} > \delta_{i}: \Delta^{n-1} &\to \Delta^{n} \\ > (t_{0},\dots,t_{n-1}) & \mapsto (t_{0}, \dots, t_{i-1}, 0, t_{i}, \dots, t_{n-1}). > \end{align}$ > > ^definition > [!definition] On Orientation. > Ordering the vertices as $v_{0},\dots,v_{k}$ induces an **orientation** on a simplex: edges are directed from lower to higher index, triangles are oriented according to the right-hand rule about increasing indices. (need to be more precise). Once an ordering is available, the notation $[v_{\lambda_{1}},\dots,v_{\lambda_{\ell}}]$ can be used to denote the $(\ell-1)$-simplex obtained from the vertices $v_{\lambda_{i}}$ (here, $\lambda_{1} \leq \dots \leq \lambda_{\ell}$ represent multiindices). ^definition ---- #### [^1]: As characterized by having unit normal vector proportional to $[1 \ 1 \cdots \ 1]=\sum_{i}e_{i}$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```