  It is shown that the $E_{6(6)}$ symmetric entropy formula describing black
holes and black strings in D=5 is intimately tied to the geometry of the
generalized quadrangle GQ$(2,4)$ with automorphism group the Weyl group
$W(E_6)$. The 27 charges correspond to the points and the 45 terms in the
entropy formula to the lines of GQ$(2,4)$. Different truncations with $15, 11$
and 9 charges are represented by three distinguished subconfigurations of
GQ$(2,4)$, well-known to finite geometers; these are the "doily" (i. e.
GQ$(2,2)$) with 15, the "perp-set" of a point with 11, and the "grid" (i. e.
GQ$(2,1)$) with 9 points, respectively. In order to obtain the correct signs
for the terms in the entropy formula, we use a non- commutative labelling for
the points of GQ$(2,4)$. For the 40 different possible truncations with 9
charges this labelling yields 120 Mermin squares -- objects well-known from
studies concerning Bell-Kochen-Specker-like theorems. These results are
connected to our previous ones obtained for the $E_{7(7)}$ symmetric entropy
formula in D=4 by observing that the structure of GQ$(2,4)$ is linked to a
particular kind of geometric hyperplane of the split Cayley hexagon of order
two, featuring 27 points located on 9 pairwise disjoint lines (a
distance-3-spread). We conjecture that the different possibilities of
describing the D=5 entropy formula using Jordan algebras, qubits and/or qutrits
correspond to employing different coordinates for an underlying non-commutative
geometric structure based on GQ$(2,4)$.
