Is there a name for the general class of triangles for dimensions other than two?

Question

Triangles differ from all other two-dimensional polygons in that their angles are rigidly fixed when the side lengths are known. It occurs to me that a triangular pyramid has the same property in three dimensions that a triangle does in two-dimensions. Is there a name for this phenomenon in general, or for the class of shapes that share this property?

Answer 1

A triangle in $n$ dimensions is known as an n-simplex.

Answer 2

The $n$-dimensional simplex has $n+1$ vertices and also $n+1$ facets, all of which are $n-1$-dimensional simplices in turn.

In fact, the count of elements of an $n$-simplex is being given by the $n+1$-st row of the Pascal triangle, i.e. the number of $k$-dimensional elements of an $n$-simplex always is just ${n+1\choose k+1}$. Eg. the (2D) triangle has both 3 vertices and 3 sides. This property is easily verified via inductive construction by means of the identities ${n+1\choose k}={n\choose k-1}+{n\choose k}$ as well as ${n+1\choose 0}={n+1\choose n+1}=1$.

The 2D simplex is known to be a triangle or trigon. A 3D simplex sometimes is spoken of as a (general) tetrahedron, esp. (for sure) when it is considered to be fully regular. Quite similar a 4D simplex will be called a (general) pentachoron, a 5D one then a (general) hexateron, a 6D one then a (general) heptapeton, a 7D one a (general) octaexon, etc.

--- rk