"Clockwise" order of n-dimensional simplices

Question

Given a triangle $(A_1,A_2,A_3)$ in $\mathbb{R}^2$, it is easy to determine whether the order $A_1,A_2,A_3$ is "clockwise" or "counter-clockwise", by imagining a real clock sitting on top of the triangle.

Moreover, if we triangulate a triangle to smaller triangles and travel the vertices of each triangle in a clockwise order, then each internal edge is travelled exactly twice, in two opposite directions, while each boundary edge is travelled exactly once, in a clockwise direction; see below:

Is there a generalization of this fact to $n$-dimensional simplexes? I.e, if $(A_1,\ldots,A_{n+1})$ is a simplex in $\mathbb{R}^n$:

• Is there a way to determine whether the order $A_1,\ldots,A_{n+1}$ is "clockwise" or "counter-clockwise"?
• If an n-dimensional simplex is triangulated to smaller $n$-dimensional simplexes, and the vertices of each sub-simplex are enumerated in a clockwise order, is it still true that each internal sub-simplex of dimension $n-1$ is enumerated twice in two opposite directions?

Answer 1

Here's a brief account of how this works.

Let me use $\Delta = [v_0,v_1,...,v_n] \subset \mathbb{R}^m$ as the notation for an oriented $n$-simplex whose vertices are $v_0,v_1,...,v_n$, and whose orientation is the equivalence class of the ordering $(v_0,v_1,...,v_n)$.

The $n$-simplex $\Delta=[v_0,...,v_n]$ has $n+1$ faces each of dimension $n-1$. These faces can be listed and assigned orientations by cyclically permuting $\Delta$ and deleting the last vertex of the list: $$[v_0,...,v_{n-1}], \, [v_1,...,v_n], \, [v_2,...,v_n,v_0], ... , [v_n,v_0,...,v_{n-1}] $$ So in your example the three oriented faces of dimension $1$ are $$[A_1,A_2], [A_2,A_3], [A_3,A_1] $$ I'll refer to the orientation on each of the faces of $\Delta$ as the boundary orientation relative to $\Delta$.

Any $n$-simplex $\Delta$ can be subdivided, in many different ways, into a union of simplices such that the intersection of any two is either empty or a smaller dimensional simplex in the subdivision. In your example the $2$-simplex $[A_1,A_2,A_3]$ is subdivided into four $2$-simplices, nine $1$-simplices, and six $0$-simplices. When $\Delta$ is subdivided in this manner, you can prove that each subsimplex $\Delta'$ of $\Delta$ has a unique orientation that is "induced" from the orientation on $\Delta$. One way to state what this means is to proceed inductively, something like this:

1. For any sub $n$-simplex $\Delta' \subset \Delta$, if there exist $n-1$ dimensional faces $\Gamma' \subset \Delta'$ and $\Gamma < \Delta$ such that $\Gamma'$ is a sub $n-1$-simplex of $\Gamma$, then the orientation on $\Gamma'$ is the orientation induced from the orientation on $\Gamma$ (notice the inductive structure of this statement: $\Gamma$ and $\Gamma'$ have one dimension lower than $\Delta$ and $\Delta'$).
2. For any two sub $n$-simplexes $\Delta',\Delta'' \subset \Delta$, if $\Gamma = \Delta' \cap \Delta''$ is an $n-1$ simplex then the boundary orientation on $\Gamma$ relative to $\Delta'$ is the opposite of the boundary orientation on $\Gamma$ relative to $\Delta''$.

In your example, you can see item 2 in action by looking at each interior $1$-simplex $\tau$: the arrows on opposite sides of $\tau$ point in the opposite direction.

If you want to read about these things in detail, I would recommend one of the older algebraic topology books. Some of these concepts are required in any algebraic topology book including the current textbook of Hatcher, but older books generally contain more details regarding things like subdivision of simplices.

Also, these issues of oriented simplices are very closely connected with the theory of oriented manifolds which is covered in most differential topology books.