The Fano Plane

# The Fano Plane

So far we have only looked at the real projective plane. We will now look at an example of a projective plane over a finite field.

Consider the set $\mathbb{Z}_2 = \{ 0, 1 \}$ with the operations of $+$ and $\cdot$ defined for all $x, y \in \mathbb{Z}_2$ by:

(1)
\begin{align} \quad x + y = (x + y) \mod 2 \quad \mathrm{and} \quad x \cdot y = xy \mod 2 \end{align}

With these operations, $\mathbb{Z}_2$ forms a field. Define the projective plane with the notation $\mathbb{P}^2 (\mathbb{Z}_2)$ where the elements in this projective plane are the points $\mathbf{x} = [x_1, x_2, x_3]$ where $x_1, x_2, x_3 \in \mathbb{Z}_2$, not all $0$. What exactly does $\mathbb{P}^2(\mathbb{Z}_2)$ look like?

Incidentally, $\mathbb{P}^2(\mathbb{Z}_2)$ will contain the points $[1, 0, 1]$, $[1, 1, 0]$, $[0, 1, 1]$, $[1, 1, 1]$, $[0, 0, 1]$, $[1, 0, 0]$, and $[0, 1, 0]$. We see that there are hence $3$ points at infinity. Furthermore, $\mathbb{P}^2(\mathbb{Z}_2)$ will contains the lines $<1, 0, 1>$, $<1, 1, 0>$, $<0, 1, 1>$, $<1, 1, 1>$, $<0, 0, 1>$, $<1, 0, 0>$, and $<0, 1, 0>$

Using the operation $+$ defined on the field $\mathbb{Z}_2$ we see that the following points are contained on each of these lines:

(2)
\begin{align} \quad <1, 0, 1> = x_1 + x_3 = (x_1 + x_3) \mod 2 = 0 \quad \mathrm{contains \:} [1, 0, 1], [0, 1, 0], [1, 1, 1] \end{align}
(3)
\begin{align} \quad <1, 1, 0> = x_1 + x_2 = (x_1 + x_2) \mod 2 = 0 \quad \mathrm{contains \:} [1, 1, 0], [0, 0, 1], [1, 1, 1] \end{align}
(4)
\begin{align} \quad <0, 1, 1> = x_2 + x_3 = (x_2 + x_3) \mod 2 = 0 \quad \mathrm{contains \:} [0, 1, 1], [1, 0, 0], [1, 1, 1] \end{align}
(5)
\begin{align} \quad <1, 1, 1> = x_1 + x_2 + x_3 = (x_1 + x_2 + x_3) \mod 2 = 0 \quad \mathrm{contains \:} [1, 1, 0], [0, 1, 1], [1, 0, 1] \end{align}
(6)
\begin{align} \quad <0, 0, 1> = x_3 = x_3 \mod 2 = 0 \quad \mathrm{contains \:} [1, 1, 0], [1, 0, 0], [0, 1, 0] \end{align}
(7)
\begin{align} \quad <1, 0, 0> = x_1 = x_1 \mod 2 = 0 \quad \mathrm{contains \:} [0, 1, 1], [0, 1, 0], [0, 0, 1] \end{align}
(8)
\begin{align} \quad <0, 1, 0> = x_2 = x_2 \mod 2 = 0 \quad \mathrm{contains \:} [1, 0, 1], [1, 0, 0], [0, 0, 1] \end{align}

So each of the $7$ lines in this projective plane contain $3$ points and $\mathbb{P}^2(\mathbb{Z}_2)$ can be represented with the following diagram:

The projective plane above is given a special name which we define above.

 Definition: The Fano Plane is the projective plane over the field of integers modulo $2$ denoted $\mathbb{P}^2 (\mathbb{Z}_2)$ and is the smallest projective plane with $7$ points and $7$ lines.