What is the maximum number of vertices a simplex in $\mathbb{R}^n$ can have?

Question

I'm currently studying Bloch's A First Course in Geometric Topology and Differential Geometry.

I'm having a hard time understanding the definition of a simplex and affine independance. A simplex in Bloch is defined as

Let $a_0, ... ,a_k \in \mathbb{R}^n$ be affinely independent points, where $k$ is a non-negative integer. The simplex spanned by the points $a_0, ... ,a_k$ is the convex hull of these points, and is denoted $\langle a_0, ... , a_k\rangle$; the points $a_0, ... ,a_k$ are called the vertices of the simplex.

That's all well and good, but I'm a little confused on what affine independance means. I think it means something like they're not all coplanar, but I'm still a little confused.

I read on Wikipedia that

A set $X$ of points of an affine space is said to be affinely independent or, simply, independent, if the affine span of any strict subset of $X$ is a strict subset of the affine span of $X$.

If we had, say $\mathbb{R}^3$, shouldn't there only be a maximum of $4$ points? If we had five points, we could construct two spans, or "copies" of $\mathbb{R}^3$ that would span the same space, which would contradict the second definition given here, correct (that is, a strict subset of $\mathbb{R}^3$ could generate something that was not a strict subset of the affine span of $X$.) Or, is there something that I am not seeing?

In short, shouldn't a simplex in $\mathbb{R}^n$ have at most $n+1$ points?

Answer 1

You are quite right that a simplex in $\Bbb{R}^n$ can have at most $n+1$ vertices.

Given the notion of the convex hull, $\mathsf{hull}(A)$, of a subset $X$ of $\Bbb{R}^n$, an alternative definition of affine independence is to say that $A \subseteq \Bbb{R}^n$ is affinely independent if there is no proper subset $B \subset A$ such that $\mathsf{hull}(B) = \mathsf{hull}(A)$. (If you translate $A$ so that one of its points is the origin, then affine independence reduces to linear independence: if you choose $a_0 \in A$ and let $A' = \{ x - a_0 \mid x \in A \setminus \{a\}\}$, then $A$ is affinely independent iff $A'$ is linearly independent.)