# A projective plane in the Euclidean plane

Problem. Is there a subset $$X$$ in the Euclidean plane such that $$X$$ is not contained in a line and for any points $$a,b,c,d\in X$$ with $$a\ne b$$ and $$c\ne d$$, the intersection $$X\cap\overline{ab}$$ is infinite and the intersection $$X\cap\overline{ab}\cap\overline{cd}$$ is not empty.

Here $$\overline {uv}$$ denotes the (unique) line containing distinct points $$u,v$$ in the Euclidean plane.

• Do you know the answer to the following question: Let $F$ be the field with two elements, and let $FP^2$ be the Fano plane. Let $\bar F$ be the algebraic closure of $F$. Take any embedding $FP^2\to \mathbb R^2$. Does it extend to an embedding of $\bar FP^2$? Aug 5, 2023 at 20:03
• @MikhailKatz No, I do not know the answer to this problem. I even do not see why the Fano plane does embed into the Euclidean plane. Aug 5, 2023 at 20:47
• No finite projective plane can be embedded in $\mathbb R^2$. Aug 5, 2023 at 20:49
• As to why no finite projective plane can be embedded in the real projective plane: this is because the projective plane structure recovers the field structure (once fixed arbitrary points on a line called $0$, $\infty$, $1$), and evidently $\mathbb{R}$ has no finite subfield. Aug 6, 2023 at 10:19

Let $$\ P^2(\mathbb Q)\$$ and $$\ P^2(\mathbb R)\$$ be the projective planes over rationals and reals. Let $$\,\ L\subseteq P^2(\mathbb R)\,\$$ be a straight line in the real plane such that
$$L\cap P^2(\mathbb Q)\,\ =\,\ \emptyset,$$
and let projective map $$\,\ F: P^2(\mathbb R)\to P^2(\mathbb R)\,\$$ map $$\ L\$$ onto the infinite line. Then
$$X\,\ :=\,\ F(P^2(\mathbb Q))$$