Curtis' kitten

J. Conway and R. Curtis [Cu1] found a relatively simple and elegant way to construct hexads in a particular Steiner system $S(5,6,12)$ using the arithmetical geometry of the projective line over the finite field with $11$ elements. This section describes this. Let

$$\mathbf{P}^1(\mathbf{F}_{11}) =\{\infty,0,1,2,...,9,10\}$$

denote the projective line over the finite field $\mathbf{F}_{11}$ with $11$ elements. Let

$$Q=\{0,1,3,4,5,9\}$$

denote the quadratic residues and $0$ and let

$$L=<\alpha,\beta>\cong PSL(2,\mathbf{F}_{11}),$$

where $\alpha(y)=y+1$ and $\beta(y)=-1/y$. Let

$$S=\{\lambda(Q)\ \vert\ \lambda\in L\}.$$

Lemma 1   $S$ is a Steiner system of type $(5,6,12)$.

The elements of $S$ are known as hexads (in the ``modulo $11$ labeling'').

					$\infty$
					6
				2		10
			5		7		3
		6		9		4		6
	2		10		8		2		10
0										1

Curtis' Kitten.

The ``views'' from each of the three ``points at infinity'' is given in the following tables.

6	10	3
2	7	4
5	9	8

5	7	3
6	9	4
2	10	8

5	7	3
9	4	6
8	2	10

picture at $\infty$

picture at $0$

picture at $1$

Each of these $3\times 3$ arrays may be regarded as the plane $\mathbf{F}_3^2$. The lines of this plane are described by one of the following patterns.

$\bullet$	$\bullet$	$\bullet$
$\times$	$\times$	$\times$
$\circ$	$\circ$	$\circ$

$\bullet$	$\times$	$\circ$
$\bullet$	$\times$	$\circ$
$\bullet$	$\times$	$\circ$

$\bullet$	$\times$	$\circ$
$\circ$	$\bullet$	$\times$
$\times$	$\circ$	$\bullet$

$\times$	$\circ$	$\bullet$
$\circ$	$\bullet$	$\times$
$\bullet$	$\times$	$\circ$

slope 0

slope infinity

slope -1

slope 1

The union of any two perpendicular lines is called a cross. There are 18 crosses. The crosses of this plane are described by one of the following patterns of filled circles.

$\bullet$	$\bullet$	$\bullet$
$\bullet$	$\circ$	$\circ$
$\bullet$	$\circ$	$\circ$

$\bullet$	$\circ$	$\circ$
$\bullet$	$\bullet$	$\bullet$
$\bullet$	$\circ$	$\circ$

$\bullet$	$\circ$	$\circ$
$\bullet$	$\circ$	$\circ$
$\bullet$	$\bullet$	$\bullet$

$\bullet$	$\bullet$	$\bullet$
$\circ$	$\bullet$	$\circ$
$\circ$	$\bullet$	$\circ$

$\bullet$	$\bullet$	$\bullet$
$\circ$	$\circ$	$\bullet$
$\circ$	$\circ$	$\bullet$

$\circ$	$\bullet$	$\circ$
$\bullet$	$\bullet$	$\bullet$
$\circ$	$\bullet$	$\circ$

$\circ$	$\circ$	$\bullet$
$\bullet$	$\bullet$	$\bullet$
$\circ$	$\circ$	$\bullet$

$\circ$	$\circ$	$\bullet$
$\circ$	$\circ$	$\bullet$
$\bullet$	$\bullet$	$\bullet$

$\circ$	$\bullet$	$\circ$
$\circ$	$\bullet$	$\circ$
$\bullet$	$\bullet$	$\bullet$

$\bullet$	$\circ$	$\bullet$
$\circ$	$\bullet$	$\circ$
$\bullet$	$\circ$	$\bullet$

$\circ$	$\bullet$	$\bullet$
$\circ$	$\bullet$	$\bullet$
$\bullet$	$\circ$	$\circ$

$\circ$	$\circ$	$\bullet$
$\bullet$	$\bullet$	$\circ$
$\bullet$	$\bullet$	$\circ$

$\bullet$	$\bullet$	$\circ$
$\bullet$	$\bullet$	$\circ$
$\circ$	$\circ$	$\bullet$

$\bullet$	$\circ$	$\circ$
$\circ$	$\bullet$	$\bullet$
$\circ$	$\bullet$	$\bullet$

$\circ$	$\bullet$	$\bullet$
$\bullet$	$\circ$	$\circ$
$\circ$	$\bullet$	$\bullet$

$\circ$	$\bullet$	$\circ$
$\bullet$	$\circ$	$\bullet$
$\bullet$	$\circ$	$\bullet$

$\bullet$	$\bullet$	$\circ$
$\circ$	$\circ$	$\bullet$
$\bullet$	$\bullet$	$\circ$

$\bullet$	$\circ$	$\bullet$
$\bullet$	$\circ$	$\bullet$
$\circ$	$\bullet$	$\circ$

The complement of a cross in $\mathbf{F}_3^2$ is called a square. Of course there are also 18 squares. The squares of this plane are described by one of the above patterns of hollow circles. The hexads are

1. $\{0,1,\infty\}\cup \{{\rm any\ line}\}$,
2. the union of any two (distinct) parallel lines in the same picture,
3. one ``point at infinity'' union a cross in the corresponding picture,
4. two ``points at infinity'' union a square in the picture corresponding to the omitted point at infinity.

Lemma 2 (Curtis [Cu1])   There are 132 such hexads (12 of type 1, 12 of type 2, 54 of type 3, and 54 of type 4). They form a Steiner system of type $(5,6,12)$.