The Queen of Hearts Plays Noughts and Crosses Queen of Hearts Plays Noughts and Crosses ... geometry...
Transcript of The Queen of Hearts Plays Noughts and Crosses Queen of Hearts Plays Noughts and Crosses ... geometry...
The Queen of Hearts Plays Noughts and Crosses
David Butler
PR
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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.
AF
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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