A emph{sign pattern (matrix)} is a matrix whose entries are from the set ${+, -, 0}$. The emph{minimum rank} (respectively, emph{rational minimum rank}) of a sign pattern matrix $cal A$ is the minimum of the ranks of the real (respectively, rational) matrices whose entries have signs equal to the corresponding entries of $cal A$. A sign pattern $cal A$ is said to be emph{condensed} if $cal A$ has no zero row or column and no two rows or columns are identical or negatives of each other. In this paper, a new direct connection between condensed $m times n $ sign patterns with minimum rank $r$ and $m$ point--$n$ hyperplane configurations in ${mathbb R}^{r-1}$ is established. In particular, condensed sign patterns with minimum rank 3 are closed related to point--line configurations on the plane. It is proved that for any sign pattern $cal A$ with minimum rank $rgeq 3$, if the number of zero entries on each column of $cal A$ is at most $r-1$, then the rational minimum rank of $cal A$ is also $r$. Furthermore, we construct the smallest known sign pattern whose minimum rank is 3 but whose rational minimum rank is greater than 3.
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