$21=9+12$: PG$(2,4)$=AG$(2,3)$+DAG$(2,3)$.

Abstract

In this paper, we provide a construction of $\mathrm{P}\mathrm{G}(2,4)$ by a collage of $\mathrm{A}\mathrm{G}(2,3)$ and its dual $\mathrm{D}\mathrm{A}\mathrm{G}(2,3)$. Moreover, we prove that the construction is unique.