# Euclidean realizations of a configuration of $27$ points and $45$ lines

Let $$GQ(2,4)$$ denote the abstract configuration (=incidence structure) consisting of $$27$$ points and $$45$$ lines, with $$3$$ points on leach line and $$5$$ lines through each point, which can be described in any one of the following ways:

1. it is the classical generalized quadrangle with parameters $$(2,4)$$, i.e., it consists of the points and lines lying on a smooth quadric $$Q \subseteq \mathbb{P}^5(\mathbb{F}_2)$$ defined by a quadratic form of "minus type" (=Witt index $$2$$; or for definiteness, $$x_0^2 + x_0 x_1 + x_1^2 + x_2 x_3 + x_4 x_5 = 0$$),

2. it is the incidence structure obtained by considering the $$27$$ lines lying on a smooth cubic surface in $$\mathbb{P}^3$$ (over an algebraically closed field) and intersecting with a plane in general position (the $$45$$ lines come from intersecting the tritangent planes of the surface).

[See also this related question (note 1) for a combinatorial description of the same incidence structure.]

By an "(extended) Euclidean realization" of $$GQ(2,4)$$ I mean a set of $$27$$ points and $$45$$ lines in the Euclidean plane (resp. the projective completion of it) such that the incidences are precisely those prescribed by $$GQ(2,4)$$.

Description nº2 above shows that $$GQ(2,4)$$ indeed has a Euclidean realization, and in fact one in which all $$27$$ points lie on the same smooth cubic (namely the intersection of the cubic surface):

Question A: Is it the case for every (extended) Euclidean realization of $$GQ(2,4)$$ that all $$27$$ points lie on some cubic? (Stronger variant: does it come from the construction described above?)

Question B: Is does there exist an (extended) Euclidean realization of $$GQ(2,4)$$ having a Euclidean symmetry group of order $$\geq 3$$?

(The two questions are related in that I have a sort-of-argument why a positive answer to question A implies a negative one to question B.)

More generally, any comment as to Euclidean realizations of $$GQ(2,4)$$, or pointers to where they have been considered in the literature, are welcome.

• I don't see a 2d realization as it looks like at most 12 of the 45 lines are not parallel to any given line. Perhaps there is a 3d realization involving skew lines? Gerhard "Finite Geometries Fit Me Better" Paseman, 2017.02.18. – Gerhard Paseman Feb 18 '17 at 22:00