__Square Tessellations__* *

Twenty-five
years ago I embarked upon a university course called Vision and Structure, a
cross-curricular mathematics-art study where tessellations played a large part.
My collection of university notes has dwindled since leaving, until now I possess
only a few remaining pages, all of which come from that one unit. Perhaps they
represent the gold left after years of panning: whenever I chance across a
tiling now, I can feel a distant part of me waking up. The other day, I chanced
across the tessellation shown in Figure 1:

Fig. 1

This can be
seen as a tessellation of the tile in Figure 2.

Fig. 2

This tile is
built from a 1-square, a 2-square and a 3-square (throughout this article an ‘n-square’
means a square of side n units.) Thus the tessellation in Figure 1 contains
equal numbers of 1-squares, 2-squares and 3-squares. Now I knew that a tile built
from a 1-square and a 2-square can tessellate (see Figure 3) - there are many
patios up and down the land that testify to that, mine included.

Fig. 3

The question
arises, can you always find a tessellation constructed from a tile containing
equal numbers of 1-squares, 2-squares, ... up to n-squares, for any natural
number n?

Let's start by
looking at n = 4. We can certainly arrange our four tiles into what we might
call an L-shape tile, shown in Figure 4:

Fig. 4

This certainly
tessellates (see Figure 5):

Fig. 5

Indeed, it is
easy to convince yourself that every L-shape tile will tessellate in this way. So
how far can we go with this? 1, 2, 3, 4 and 5-squares form the L-shape tile
shown in Figure 6.

Fig. 6

However, try
as we might, we cannot form an L-shape tile from a 1-square, a 2-square… up to
a 6-square (have a go!) But we can get what we might call an S-shape tile from
these squares (see Figure 7):

Fig. 7

Does this
tessellate? It does, forming the attractive tessellation in Figure 8.

Fig. 8

We can add a 7-square
to get another S-shape (Figure 9), which also tessellates (Figure 10).

Fig. 9

Fig. 10

We might be
forgiven for thinking that all S-shapes tessellate, but this is untrue, as experimenting
with the tile in Figure 11 shows:

Fig. 11

At this point
we seem to come to the end of the road. However we play around with the tiles,
we can't seem to put a 1-square through to an 8-square together in a way that
will create an L-shape, or indeed a helpful S-shape. Perhaps a different tack
is needed.

Suppose that S_{n}
is the statement: “There exists a rectangle made up of equal numbers of
1-squares through to n-squares.” If S_{n} is true for all n, then as a
rectangle clearly tessellates, we will have a tile that meets our requirements.

S_{1}
is clearly true, as is S_{2} (Figures 12 and 13).

Fig. 12 Fig. 13

If we now add
two 3-squares to Figure 13, we can make two rectangles that will not fit
together to make a third.

Fig. 14

However, if we
take 2×3 = 6 copies
of Figure 14, we can then build a single rectangle from the result.

Fig. 15

So S_{3}
is true. Can we carry out this procedure in the general case? An argument using
induction shows that we can.

Suppose S_{n}
is true and there is a rectangular tile (called A) containing equal numbers of
1-squares through to n-squares (suppose also there are k of each, and that A is
p by q). We can make k (n+1)-squares
into a k(n+1) by (n+1) rectangle (call this B).

Fig. 16

Now take
(n+1)p copies of the diagram (we may not need as many as this.) We can combine
the copies of A into a (n+1)p by pq rectangle, while the copies of B will form
an (n+1)p by k(n+1)^{2}
rectangle. These may be simply combined into a single rectangle,
containing equal numbers of 1 through to (n+1)-squares (there will be kp(n+1)
of each.)

So if S_{n}
is true, then S_{n+1 }is true, and by induction, a tile that
tessellates made from equal numbers of 1-squares through to n-squares is
possible for all n. Maybe a challenge if you happen to need a new patio?

4 April 2006, Jonny Griffiths,