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I am interested in the affine version of the well-known Hessenberg's Theorem (saying that Pappian projective planes are Desarguesian). First I introduce all necessary definitions. Definition L. A line …

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\ …

The answer to this question is negative: Jut take $\alpha$ such that $2^\alpha=3$. Taking into account that $2^n\ne 3^m$ for any natural numbers $n,m$, we can prove that the number $\alpha=\log_23$ i …

Under one additional condition, the answer to this problem is affirmative. The proof involves the following implication of the Affine Desargues Axiom: The Affine Moufang Axiom: for every parallel li …

Now I have understood how to draw the classical unital, $U_q$ at least for $q=3$. It is known that this unital is isomorphic to the subset of the projective plane $PG(2,9)$, defined by the equation $X …

The answer to this question is "No". A non-affine biaffine Cartesian plane can be constructed as follows. First, we fix a suitable terminology. Every function $F:X\to Y$ is identified with its graph …

I would like to understand the geometry of the classical unitals. They are block designs containing $q^3+1$ points and whose blocks have cardinality $q+1$, where $q$ is a prime power. For $q=2$ (if I …

This question concerns the (synthetic) geometry of linear spaces. Definition 1. A linear space is a pair $(P,\mathcal L)$ consisting of a set $P$ whose elements are called points and a family $\mathca …

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\ …

This is yet another idea to to resolve this resistant problem. Let us parametrize everything. Identify the plane containing the moving square with the complex plane $\mathbb C$ and let $\mathbb T=\{z\ …

For a (topological) field $F$ by $FP^2$ we denote the projective plane, i.e., the quotient space of $F^3\setminus\{0\}^3$ by the equivalence relation $\vec x\sim\vec y$ iff $\vec x=\lambda\vec y$ for …

It seems that the proof from the book posted by @ihromant follows the lines of the original Hessenberg's proof from his paper in Mathematische Annalen of 1905: