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I am currently reading "Basic Notions of Algebra" by Igor Shafarevich. In the first chapter example of a coordinatization of 4-point geometry is given.

Set of axioms:

  1. Through any two distinct points there is one and only one line.
  2. Given any line and a point not on it, there exists one and only one other line through the point and not intersecting the line (that is, parallel to it).
  3. There exist three points not on any line.

Finite affine plane of order 2, containing 4 points and 6 lines.

In this geometry we have 4 points A, B, C, D and 6 lines AB, CD; AD, BC; AC, BD. The families of parallel lines are separated by semicolons.

Let $\Bbb{0,1}$ be symbols with operations $+$ and $\times$ such that $$ \begin{array}{cc} \begin{array}{c|cc} \text{+} & 0 & 1\\ \hline 0 & 0 & 1\\ 1 & 1 & 0\\ \end{array} & \begin{array}{c|cc} \times & 0 & 1\\ \hline 0 & 0 & 0\\ 1 & 0 & 1\\ \end{array} \end{array} $$

The pair of quantities 0 and 1 with operations defined on them as above serve us in coordinatising the "geometry". For this, we give points coordinates (X, Y) as follows: A = (0, 0), B = (0, 1), C = (1, 0), D = (1, 1).

It is easy to check that the lines of the geometry are then defined by the linear equations: $$ \begin{array}{ccc} & AB: 1X = 0; & CD: 1X = 1; & AD: 1X + 1Y = 0;\\ & BC: 1X + 1Y = 1; & AC: 1Y = 0; & BD: 1Y = 1;\\ \end{array} $$

The question is: how does one should interpret this equations?

Any suggestions will be appreciated.

Aleksander Gurin
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    What do you mean by "interpret" that isn't already contained in the description of the question? One can think of these as the (affine) lines in $\Bbb F_2^2$, and hence identify this geometry with any affine chart for the Fano Plane, $\Bbb P (\Bbb F_2^3)$ – Travis Willse Dec 24 '15 at 03:51

2 Answers2

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So, now the base field, i.e. the number line instead of all the real numbers, consists only of $0$ and $1$, with $1+1=0$.

It is stated that the plane over this $2$ element field has six lines. Later those equations are just the equations of these 6 lines.

For instance, we have $A=(0,0)$ and $B=(0,1)$, and the line $AB$ is the one with equation $$X=0$$ and indeed, these two points satisfy this equation.
Or, take $AD$ with $D=(1,1)$, both points satisfy $X+Y=0$.
And so on.

Berci
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  • Thank you for your answer. I somewhat confused because I supposed that we specify line as a pair of points. As I understand from your answer we specify line as one symbol (0, 1) or two symbols (00, 01, 10, 11), thus we have 6 different lines and equations that corresponds to these lines. – Aleksander Gurin Dec 24 '15 at 09:33
  • No. The lines are just the pairs of the points: $AB,\ AC,\ AD,\ BC,\ BD$ and $CD$, exactly 6 of them. When the coordinates of the points $A,B,C,D$ are fixed, the given equations describe these lines in terms of coordinates. – Berci Dec 30 '15 at 17:29
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The equations describe the lines, that is, if a point satisfies the equation then it is on the line. For example the line $X = 0$ is satisfied by all points $(X,Y)$ with $X = 0$ (with the coordinates $X$ and $Y$ coming from $\mathbb{F}_{2}$, the finite field of order 2). This gives an algebraic way to describe the lines.

You can think of the lines as having a slope of either $0$, $1$, or $\infty$ ($Y$ coefficient divided by $X$ coefficient), these slopes determine your parallel classes. This should let you go from a pair of points to the equation by determining a slope and then using a point to get the remaining value to describe the line as $AX+BY=C$.

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