Theorem. If is a symmetric, non-trivial design, then .
Again, the non trivial, symmetric designs with are exactly the, eventually degenerated, finite projective planes.
The symmetric, non trivial, designs with are called biplanes. In concrete terms, we arrive at the following definition.
Definition. An incidence structure is called a biplane if the following two axioms hold.
Currently, there are only finitely many finite biplanes known. It's an open problem whether there exist arbitrarily big finite biplanes.