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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:

  1. Any distinct points $x,y\in X$ are contained in a unique line $L\in\mathcal L$;

  2. Any line $L\in \mathcal L$ contains at least three points;

  3. $X\notin \mathcal L$;

  4. For every line $L\in\mathcal L$ and point $x\in X\setminus L$ there exists a unique line $\Lambda\in\mathcal L$ such that $x\in \Lambda$ and $\Lambda\cap L=\emptyset$.

For two distinct points $x,y\in X$ of an affine space $(X,\mathcal L)$ by $\overline{xy}$ we denote the unique line containing the points $x,y$.

Two lines $A,B\in\mathcal L$ are called parallel (denoted by $A\parallel B$ ) if either $A=B$ or $A\cap B=\emptyset$.

An affine plane $(X,\mathcal L)$ is finite if the set $X$ is finite.

Definition 2. An affine plane $(X,\mathcal L)$ is defined to have the doubling property if for every distinct points $a,b\in X$ and points $c\in\overline{ab}$ and $u,v,x,y\in X\setminus\overline{ab}$, the parallelity relations $\overline{xy} \parallel \overline{ab} \parallel \overline{uv}$, $\overline{ax} \parallel \overline{by}$, $\overline{xb} \parallel \overline{yc}$, $\overline{au} \parallel \overline{bv}$, imply $\overline{ub} \parallel \overline{vc}$.

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In affine planes with the doubling property we can duplicate the ``distances'' (in a suitable sense, of course).

Question. Is there a finite affine space without the doubling property?

It can be shown that the doubling property follows from the well-known

Little Desargues' axiom: for every parallel lines $A,B,C$ and points $a,a'\in A\setminus(B\cup C)$, $b,b'\in B\setminus(A\cup C)$, $c,c'\in C\setminus(A\cup B)$, the parellelity relations $\overline{ab} \parallel \overline{a'b'}$ and $\overline{bc} \parallel \overline{b'c'}$ imply $\overline{ac} \parallel \overline{a'c'}$.

On the other hand, the Moulton plane does not have the doubling property. But the Moulton plane is infinite.

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    $\begingroup$ spelling is "doubling", not "doubbling" $\endgroup$
    – YCor
    Commented Sep 22, 2023 at 12:09
  • $\begingroup$ @YCor Indeed, "doubling" is correct, see dictionary.cambridge.org/dictionary/english/doubling $\endgroup$ Commented Sep 22, 2023 at 12:41
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    $\begingroup$ Please edit the question into the body – it shouldn't be just in the title. $\endgroup$ Commented Sep 23, 2023 at 1:06
  • $\begingroup$ @GerryMyerson As far as I can see (at least at my computer) all the doubbling-s have been corrected by Tony Huynh. $\endgroup$ Commented Sep 23, 2023 at 8:14
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    $\begingroup$ @GerryMyerson Ups! Indeed, somehow I forgot to write down the precise question, I had in mind. Thank you for your comment. $\endgroup$ Commented Sep 23, 2023 at 19:17

1 Answer 1

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There is even stronger claim from which follows your answer.

Claim. Any affine plane obtained from Veblen-Weddenburn projective plane by dropping one line don't have "doubling" property.

Proof of this claim is made by computer. Let's start from definitions.

Veblen-Weddenburn plane is 91-point plane made from points A0...A12,B0...B21,...,G0...G12. Here is classic 1907 published article describing its properties. Lines are defined the following way: there are 7 basis lines:

  • [A0,A1,A3,A9,B0,C0,D0,E0,F0,G0]
  • [A0,B1,B8,D3,D11,E2,E5,E6,G7,G9]
  • [A0,C1,C8,E7,E9,F3,F11,G2,G5,G6]
  • [A0,B7,B9,D1,D8,F2,F5,F6,G3,G11]
  • [A0,B2,B5,B6,C3,C11,E1,E8,F7,F9]
  • [A0,C7,C9,D2,D5,D6,E3,E11,F1,F8]
  • [A0,B3,B11,C2,C5,C6,D7,D9,G1,G8]

Other lines are obtained by adding i=1..12 by modulo 13 to the indexes of line points. Obtained plane by following definition is projective where Desargues and Pascal theorem doesn't hold. Projective properties of this space are also checked by my code (see in the end of the answer). Now let us drop from this place first line [A0,A1,A3,A9,B0,C0,D0,E0,F0,G0]. Parallellity in the rest mean that two lines intersect in dropped line. Now let's consider points:

a=A2,b=B1,c=C1,x=B7,y=C4,u=A5,v=C5,

ab=[A1,A2,A4,A10,B1,C1,D1,E1,F1,G1],

xy=[A1,B3,B6,B7,C4,C12,E2,E9,F8,F10],

uv=[A1,A5,A6,A8,B5,C5,D5,E5,F5,G5],

intersection ab, xy and uv=A1, which is on dropped line

ax=[A2,B4,B7,B8,C0,C5,E3,E10,F9,F11],

by=[A11,B1,B9,C0,C3,C4,D5,D7,G6,G12],

intersection ax and by=C0, which is on dropped line

bx=[A6,B1,B7,D4,D9,E8,E11,E12,G0,G2],

cy=[A12,B2,B10,C1,C4,C5,D6,D8,G0,G7],

intersection bx and cy=G0, which is on dropped line

au=[A2,A3,A5,A11,B2,C2,D2,E2,F2,G2],

bv=[A3,B1,B6,C5,C8,C9,D10,D12,G4,G11],

intersection au and bv=A3, which is on dropped line

ub=[A5,B1,B12,D0,D6,F7,F10,F11,G3,G8],

vc=[A12,B2,B10,C1,C4,C5,D6,D8,G0,G7],

intersection ub and vc=D6, which is not on dropped line.

Counterexamples for other lines dropped are printed by test.

Code for checking correctness and Doubling property is this two Java classes. First - code for modelling Veblen model, second - code for testing which prints counterexample for all dropped lines Plane properties testing.

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