TITLE: Maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1). Proof?
QUESTION [13 upvotes]: How to prove that the maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1)?

REPLY [8 votes]: Assume one of the points is the origin, the others are the vectors a1,...,an+1. If the pairwise distance is 1, then $a^2_i=1$ and $(a_i-a_j)^2=1$ (i < j)(scalar multiplication). This gives $(a_i,a_j)=\frac{1}{2}$. Now show that $a_1,\dots,a_{n+1}$ are linearly independent: if $\lambda_1,\dots,\lambda_{n+1}$ are scalars and $\sum \lambda_ia_i=0$, then scalar multiplication by $a_i$ gives $\lambda_i+\frac{1}{2}\sum_{j\neq i}\lambda_j=0$, then we have $\lambda_i=-\Lambda$ for each $i$ where $\Lambda=\lambda_1+\cdots+\lambda_{n+1}$, summing gives $\Lambda=0$ so each $\lambda_i=0$.