The Queen of Hearts Plays Noughts and Crosses

David Butler

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About this plane:

The lines look like this:

There are 13 points

and 13 lines.

Every line has 4 points.

Every point is on 4 lines.

Every pair of points is joined by a line.

Every pair of lines meets in a point.

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About this plane:

The lines look like this:

There are 9 points

and 12 lines.

Every line has 3 points.

Every point is on 4 lines.

Every pair of points is joined by a line.

Some pairs of lines don’t meet –

there are 4 sets of parallel lines.

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About this plane:

The lines look like this:

There are 9 points

and 8 lines.

Every line has 3 points.

Points can lie on 2, 3 or 4 lines.

Some pairs of points are not joined by a line.

Some pairs of lines don’t meet.

A plane you already know:

This poster is about the geometry of planes. The familiar

plane that uses straight lines on paper is called the

Euclidean plane. You would be used to its rules: there is

a line joining any two points; there are infinitely many

points on a line; two lines can either meet in a point or be

parallel; and given any point you can find exactly one line

through the point parallel to any particular line. However,

a plane doesn’t have to have these rules. In fact, you

already know one that doesn’t.

A plane you didn’t know you already knew:

If you know the game of Noughts and Crosses, then you

already know a plane with rules quite different to the

Euclidean plane.

In Noughts and Crosses, you start with a 3×3 grid. One

player takes O and one takes X, and you take it in turns

putting a symbol in a square. The first to get a whole

row—horizontal, vertical, or diagonal—wins. In the

picture to the left, the player with O has won.

The winning rows here can be thought of as the lines of

a plane, and the squares of the grid as the points. As you

can see, there are exactly three points on any line.

However, not every point is on the same number of lines.

As any seasoned player of this game knows, the centre

square has the most lines through it. This makes the

centre square the most powerful square. An O or an X

there can control the whole board. Because of this, the

first player can usually win Noughts and Crosses, and

the best the second player can hope for is a draw. We

will try to make the situation fairer by adding some

missing lines to this plane.

Filling in the missing lines:

The major point of difference between Noughts and

Crosses and the Euclidean plane (apart from one being

finite and the other infinite) is that in Noughts and

Crosses, there are some pairs of points with no line

joining them. If we add these missing lines, we can make

sure that every point is on the same number of lines, and

perhaps the game will be a bit fairer.

The lines we will add are the wraparound diagonals. To

make a wraparound diagonal, you start at any square

and move diagonally. When you leave the grid on one

side, you come in on the other. The four wraparound

diagonals are shown in the picture to the right.

The new plane we have made is called an affine plane,

and its rules are the same as the Euclidean plane. In

particular, any two points are now joined by a line; and

you can find a line through any point that is parallel to

any particular line. (Here, “parallel” means that the two

lines don’t have a point in common.) Also, every point is

now on the same number of lines, so we have removed

the power of the centre square.

Unfortunately, we have made the game completely

unfair: the first player always wins on the fourth turn. Try

it for yourself. (In the picture here, the player with O

started the game and won using a wraparound diagonal.)

Filling in the missing points:

Since Affine Noughts and Crosses is unfair, we will try to

fix the game again. If we add more lines, it will just be

easier for the first player to win, so we will add some

more points. The lines that are parallel at the moment will

meet at these new points. That is, we’ll add a point

where the three vertical lines meet, and a point where

the three horizontal lines meet, and a point for each set

of diagonals. To complete the plane, we will use these

four new points to make one new line. All the lines of our

new plane are shown in the picture to the right.

Now, every point is on four lines and every line has four

points. Every pair of points is joined by a line and every

pair of lines meets in a point. A plane with these rules is

called a projective plane.

Projective Noughts and Crosses has no square that is

most powerful, since all points are on exactly four lines.

Also, the first player doesn’t have to win. This is because

it’s easier for the second player to block when all the

lines meet. In the game shown, the second player was

able to force a draw.

Where the queen of hearts comes in:

There is something else to do with games that can be

made into a plane: a pack of cards. The cards shown

along the bottom of the poster are grouped into hands of

four. If we make these hands the lines and the card

values the points, then the pack of cards represents the

same plane as Projective Noughts and Crosses.

Each hand has four cards, and each card appears in four

hands. If you choose any two cards, there will be exactly

one hand containing both. For example, there is exactly

one hand with both a Queen and a six (it’s the one on

the far right). If you choose any two hands, there is

exactly one card in both. For example, the far-right hand

and the far-left hand have exactly one card in common,

and it’s a two.

By putting in “point” instead of “card”, and “line” instead

of “hand” in the above paragraph, you will come up with

the same set of rules as Projective Noughts and

Crosses. So, our pack of cards represents the same

plane.

What geometry is:

The fact that we can represent the same plane in such

wildly different ways tells us something about what

geometry is. Geometry is not about points and lines at

all—

it’s about relationships. If any things are related to

each other in the right way, you can call them points and

lines and then you’ll have a plane. This is what makes

geometry so interesting and so much fun.

To find out more about the fun and interest of geometry,

I would recommend these two books:

● Marta Sved, Journey into Geometries, Mathematical

Association of America, Buffalo, 1991

● Ian Stewart, Flatterland: Like Flatland only more so,

Macmillan, London, 2001