Steve Whealton








Coxeter's Frieze Patterns of Integers

Professor H. S. M. Coxeter of the University of Toronto is one of the great mathematicians of this century. He has presided over the rebirth of geometry that has occurred in recent decades in large because of computers and of their need for new geometric ideas.

Coxeter has written many books and articles. They range from the abstruse and difficult to the historical and popular. He has also maintained a lively interest in mathematical games.

More than twenty years ago, he developed a group of interesting new ways of placing whole numbers into a specific type of regular array. As he explored his invention more and more deeply, others joined with him and devised extensions and variants of his ideas.

For the Second Edition of his Convex Polytopes (Cambridge University Press), Professor Coxeter gathered together a full lesson's-worth of information and explanation about these frieze patterns. Anyone wishing to know more would do well to borrow or purchase that wonderful book.

 

For now, let us merely sketch out the bare essentials.

Our first step will be to define the empty shell, or mesh, that our whole numbers are going to fit into. It is rather like an ordinary rectangular array, except that it has been rotated 45 degrees.

Think of the Pascal Triangle. In fact, our fpoi has much in common with the famous triangle. Much of the general idea of how number patterns are begun, carried on, and finished, is similar.

As with the triangle, a sea of identical numbers is presumed to exist beforehand. With the triangle, it is assumed that a horizontal line is drawn and that each cell above that line contains a zero. A 1 is placed in a cell on the row below that line. Throw in the formula for calculating new numbers for new cells, and all else follows.

With Coxeter's frieze patterns of integers, the situation is similar. Now, it is not an endless sea of zeroes that is presumed to exist above some arbitrary horizontal line. Rather, the entire horizontal line itself is filled in with 1s.

Pascal's triangle needed only a single 1 to start it off and to determine its full course. Coxeter's frieze patterns need more. Unlike Pascal's unique pattern, Coxeter's friezes come in great variety.

What we are going to do is to create a left-right-running line of ones and then consider it as the top border of our frieze-pattern-to-be. Our next step will be to select one of the 1s in the top border row; one of them over towards the left. Then below that particular 1, we will fill in a zigzag course downwardˇfill it in with more 1s.

We can zig where we like and we can zag where we like. We can continue downward for as long as we like. But what we must do is keep all those zigging and zagging 1s connected. We cannot go way off to the left or to the right. We have to choose adjacent cells.

Back to our chosen 1, we now look on the row just beneath it on the page. Because of the stagger between adjacent rows, our diamond-shaped 1-containing cell abuts precisely two cells in a downward direction. One of these cells could be said to be its southeast neighbor, and one to its southwest neighbor. We will call a move to the southeast neighbor cell a "zig left" and a move to the southwest neighbor cell a "zag right."

Let us make our first choice and zig. Having done so, we now place a 1 in the cell to the southwest of that cell on the top row that we chose in the first place.

It is now time to choose again. Let us opt for yet another zig. Two zigs. Let our next choice be a zag and our next a zig. Then two more zags.

At about this point, we should choose to stop. We could go on and on and on, but the longer the left border we create, the larger the integers that will ensue, and the more of them. Let us keep their size down, at least here at the very beginning.

To recap, we have drawn in a left border from the top down. It reads like this: Left, Left, Right, Left, Right, Right, stop.

We are now six rows below the horizontal row of ones that we drew in way back in the beginning. Our pattern looks something like this:

1

 

1

 

1

 

1

 

 

1

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

The next step is to draw in another horizontal row of ones there at the bottom, right at our stopping point:

1

 

1

 

1

 

1

 

 

1

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

1

 

1

 

1

 

1

 

 

 

 

 

 

 

 

 

To make the next steps in the creation of our first frieze pattern, we will rely on a formula. It is a bit like the formula used in Pascal's Triangle, but a little bit more complicated. Instead of simply summing the two numbers to the northwest and northeast of any given cell (as with the Pascal Triangle), one must consider a triangular pattern of four cells. We will call them a, b, c, and d:

 

a

 

b

 

c

 

d

 

Let us think of the above pattern as a set of four variables, like an unknown "x" in an algebra problem.

Our a, b, c, and d can stand for any four numbers standing in a diamond relationship in our fpoi pattern.

The only thing needed now before we can proceed to create a frieze pattern out of the top and left borders of ones that we have drawn in is a formula. Here it is:

(a * d) + 1 = (b * c)

Within our frieze pattern, any diamond of numbers will satisfy the above formula. Whenever 3 out of 4 numbers are known, a fourth can always be calculated.

Where do we have diamonds with 3 of the necessary four cells filled with numbers? There are two such spots, and in each case a, b, and d are 1 and c is missing. So we calculate c.

(a * d) comes to (1 * 1). That's 1. Add 1 and you get a total of 2 for the left side of the equation. (b * c) will be equal to 2. If b is 1 (and it is), then c has to be 2. We write in a "2" in each of the cells where 3 ones abut on the left. We now have:

1

 

1

 

1

 

1

 

 

1

 

 

 

 

 

 

1

 

2

 

 

 

 

 

 

1

 

 

 

 

 

 

1

 

2

 

 

 

 

 

 

1

 

 

 

 

 

 

1

 

1

 

1

 

1

 

 

 

 

 

 

 

 

 

This gives us but one new triangle that we can fill in. It's the one with the two 2's at top and bottom and a 1 on the left. We multiply the 2's and add one. That gives us 5. 1 times 5 is 5, so we write in a 5.

1

 

1

 

1

 

1

 

 

1

 

 

 

 

 

 

1

 

2

 

 

 

 

 

 

1

 

5

 

 

 

 

1

 

2

 

 

 

 

 

 

1

 

 

 

 

 

 

1

 

1

 

1

 

1

 

 

 

 

 

 

 

 

 

Continuing for a ways we find something strange.

1

 

1

 

1

 

1

 

1

 

 

1

 

3

 

3

 

2

 

1

 

1

 

2

 

8

 

5

 

1

   
 

1

 

5

 

13

 

2

 

1

 

1

 

2

 

8

 

5

 

1

   
 

1

 

3

 

3

 

2

 

1

 

1

 

1

 

1

 

1

 

1

 

1

We have arrived again at the same zigzag pattern of ones we drew in back at the beginning. A border has appeared on the right, and it is identical to the border on the left.

There is but one more surprise in store. To see it, let us begin again, but this time let us draw in a left border that is lopsided. Our first example was so prim and orderly that it hid one of the loveliest features of the fpoi. We will go after it now. Rather than go through all of the steps, we'll show first the zigzag left border, and then skip immediately to the full frieze pattern:

Here is the pattern of ones:

1

 

1

 

1

 

1

 

 

1

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

Finally, the full frieze pattern:

1

 

1

 

1

 

1

 

1

 

1

 

1

 

2

 

2

 

3

 

2

 

1

 

 

 

1

 

3

 

5

 

5

 

1

 
 

 

 

1

 

7

 

8

 

2

 

1

 

 

 

1

 

2

 

11

 

3

 

1

 
 

 

 

1

 

3

 

4

 

1

 

 

 

 

 

1

 

1

 

1

 

1

In this new example, the border on the left again repeats itself on the right, but this time it is upside down. The possibilities for symmetry are many.

 

G. C. Shepherd of the University of East Anglia has discovered that if the role of 1 is given instead to 0, and if the role of multiplication in the formula,

(a * d) + 1 = (b * c)

is changed to addition, so that the overall formula now reads

(a + d) + 1 = (b + c)

then a symmetric overall pattern again emerges. Again, the border of zigzagging zeroes reappears on the right, but this time, it's right side up. There is another kind of symmetry at work.

Here are our two zigzag patterns, but this time using zeroes instead of ones, and addition in the formula, instead of multiplication:

Here is the first table:

0

0

 

0

 

0

 

0

 

0

0

0

 

0

 

2

 

4

 

5

 

3

1

0

0

 

1

 

5

 

8

 

7

 

3

1

0

0

 

3

 

8

 

9

 

6

1

0

0

 

1

 

5

 

8

 

7

 

3

1

0

 

0

 

2

 

4

 

5

 

3

1

0

0

 

0

 

0

 

0

 

0

 

0

0

0

Here is the second table:

0

 

0

 

0

 

0

 

0

0

0

0

 

1

 

2

 

4

 

5

3

0

 

0

 

2

 

5

 

8

 

7

2

0

 

0

 

4

 

8

 

9

5

1

0

 

0

 

1

 

6

 

8

 

6

3

0

 

0

 

2

 

5

 

4

3

0

0

 

0

 

0

 

0

 

0

 

0

0

0

So what can one DO with FPOI?

For graphics files, one begins by entering just enough information to define a frieze pattern completely. My method has been to use the fact that diagonal rows obey the rule that each number divides evenly the sum of its two neighbors. I enter a the numbers for a diagonal row that obeys this criterion.

From there on, the entire frieze pattern can be calculated, diagonal row by diagonal row. For each cell in each diagonal row, the number residing there can be squeezed, disguised, and variously distilled (as desired or as needed) into a number that falls into the range necessary for it to represent a color. An abstract graphic image is born.

In music, the very shape and nature of the frieze pattern of integers is highly suggestive. A direct transformation of the numbers within the pattern to an ostinato pattern for music is one obvious manifestation. Another is to bounce around along diagonals, using the numbers that are encountered to control pitch.

 








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