Bored one day in a staff meeting, I took to playing around with numbers – a nice way to pass the time. I wondered if chaotic numbers might exist; that is, numbers whose digits at first might look quite random, but hidden within this apparent disorder would be the signature of order that lies at the heart of chaos. I had big notions that maybe the digits making up famous irrational numbers like $\pi$ or $\sqrt{2}$ might be such numbers, but I decided to start with more simple numbers, those of the recurring decimals. I took the fraction 1/7 as my starting point.
The first return maps
Taking the decimal expansion of the fraction 1/7 = 0.142857…, I used consecutive pairs of digits as coordinates and plotted them on a simple x–y grid. This gives (1, 4), (4, 2), (2, 8), (8, 5), (5, 7), then back to (7, 1) before the pattern repeats. Note that I use each digit twice. This is the essence of a return map: one reading relates to the previous and, in the case of chaos, data that might initially appear to be very random will show unexpected order on the graph. A return map of the digits of a decimal fraction I shall call a fractogram. See overleaf for the unconnected fractogram for the fraction 1/7 – unconnected as the points are not joined together.
Unconnected fractograms for 1/7=0.142857… (left) and 1/17=0.0588235294117647… (right)
Note that the six points make two approximately straight, parallel lines. The shape of the six points also has rotational symmetry of order two. Next to it is the unconnected fractogram for 1/17. It also has sets of points making roughly straight parallel lines and has rotational symmetry of order two. Not all fractions give this result of course. For example, 1/3 being 0.3333… gives only one point (3, 3); while 1/11 being 0.0909… gives two points (0, 9), (9, 0); and 1/37, which is 0.027027…, gives three points and so on. Some decimal fractions truncate, such as 1/8, which is 0.125, so after plotting the points (1, 2) and (2, 5) it comes to an end. It is left as an exercise for the reader to try others and discover the wide variety of patterns!
Raising the resolution
The almost straightness of the lines formed by the coordinates on the 1/7 and 1/17 fractograms intrigued me. Why were they almost straight? Why were they not completely straight? I decided to play another game. What would happen if instead of plotting consecutive pairs of digits as coordinates, I plotted consecutive pairs of double digits? Taking the fraction 1/7 again, I obtained (14, 42), (42, 28), (28, 85), (85, 57), (57, 71) and (71, 14); which, when plotted on a fractogram, gave the pattern shown on the next page.
Unconnected fractogram for 1/7 with double digit coordinates
Taking consecutive pairs of the decimal expansion of 1/7 and plotting them as pairs of double-digit coordinates has made the six points lie closer to two straight lines. Well, what if triplets of digits were used, giving the coordinates (142, 428), (428, 285), (285, 857), (857, 751), (751, 514) and (514, 142)? The six points lie even closer to the straight lines they appeared to follow in the previous approximate cases. Indeed, it can be shown that if this process is continued the points would eventually lie exactly on one of two lines, each of gradient −1/2. Whether that was to be expected or not, it is curious to see that the gradient of the lines is a very simple fraction. Why should that be? For the 1/17 fractogram, the gradient of the lines approaches 3/2.
Joining the dots
The next step in the experiment was to see if there were any further patterns hidden in these almost orderly sets of points. I wondered if the order in which they appeared on the graph was relevant, so I did the equivalent of the children’s picture-drawing pass time of ‘joining the dots’ for the fraction 1/7. I call this a connected fractogram. Doing this for our original fractogram, where the coordinates were given by a single digit, results in a curious shape. It has the same rotational symmetry of order two but the lines do not follow the original approximately straight ones:
Connected fractograms for 1/7=0.142857… (left) and 1/17=0.0588235294117647… (right)
The connected fractogram of 1/17 is also shown above and it too has a symmetrical set of lines that have nothing to do with the rough straight lines that appeared at the beginning.
Each fraction on a connected fractogram has its own pattern. Some are symmetrical shapes like 1/7, the lines of which do not usually follow those approximately straight lines from the first experiment at all. Some are simple polygons: 1/37 is a triangle, while 1/101 is a square. Some merely show a random set of connected lines making a closed shape. To assist this process, which done by hand is slow, my son, Fabien Duncan, kindly wrote a program that does the job in a fraction of the time. See below for just two examples (1/97 and 2/79) with the grid lines removed. Doesn’t one of them look a little like Bart Simpson?!
The connected fractograms 1/97 (left) and 2/79 (right)
Superposed fractograms
If one takes the fraction 1/55, which is 0.018181…, it only has 3 different digits, hence only three points on a fractogram. By increasing the numerator by one to 2/55, one gets 0.03636…: a different set of three points. By themselves these fractograms are not so interesting but if one were to take all of them—1/55, 2/55, 3/55, up to 54/55—and plot them all on the same graph then one obtains a more fascinating pattern, which I call the superposed fractogram n/55. n/55 has loose ends and is not cyclic but it almost has rotational symmetry. Next to n/55 is the superposed fractogram for n/101: it is made up of squares and has rotational symmetry of order four plus line symmetry of equal complexity!
The superposed fractograms n/55 (left) and n/101 (right)
Where do we go from here?
This is only the beginning and there are so many other areas to explore and beautiful patterns still to discover! For example, we could try to tessellate the unconnected fractograms, placing them next to one another like bathroom tiles to see what large scale pattern they make. Or we could investigate what other convex polygons one could make with connected fractograms. How many of these are regular? What shapes are possible? Pentangles? Steps? Spirals? Are any shapes impossible? We could make an excursion into other bases aside from the base 10 that we are so used to. What would our fractograms look like in a different base, like base 7 or base 13? Or let’s extend our fractograms into three dimensions, representing our friend 1/7 by the six points (1, 4, 2), (4, 2, 8), (2, 8, 5), (8, 5, 7), (5, 7, 1) and (7, 1, 4). Perhaps we’d like to make a movie by creating a series of fractograms for a given denominator d (1/d, 2/d, …, (d-1)/d) and then playing them one after another to see how the shape evolves. Or… the list goes on; we are limited only by our own imagination.
And in case you’re wondering: no, there is no hidden pattern to the digits of $\pi$ when plotted as a fractogram! But I like to think that what I have found is not a bad consolation prize. So, have I discovered something new in mathematics or am I just being irrational?
More from Chalkdust
The first negative dimension
A tribute to alumni Roly Drower by Hugh DuncanFlo-maps fractograms: the game
Try out these flo-maps for yourself: fractions speak louder than words
Flo-maps fractograms: the prequel
Hugh Duncan returns with the long-awaited prequel in which he further explores the geometric patterns hidden behind the fractions.
Reproduce or die
Hugh Duncan explores an exciting variation on Conway's Game of Life
Overturned polygons: shapes with less than two sides
Hugh Duncan explores polygons with a shortage of edges
Frawks
Exploring non-random walks using fractions