Page last updated: 26 Sep 2002

This entry is an explanation of the mathematics used to generate the graphics in Fibonacci Art (1-8) and Fibonacci Art, continued (9 and 10). The explanation focuses on #10 so far, and will be expanded to explain the others as time allows. (If you know what they are, just think different bases.)

Fibonacci Series and Generalizations

The Fibonacci Series is a well-known sequence of numbers, first described by Leonardo da Pisa Fibonacci¹ in 1202 to describe the growth of a population of rabbits. It is easy to generate, and does not involve any complex mathematics, but it seems to be endlessly correlated with branches of math ranging from number theory to fractals to approximations of π. Here is a link to more math involving the Fibonacci numbers than you can shake a stick at.

The first two terms of the Fibonacci Series are 0 and 1. Each subsequent term is the sum of the two previous terms, so the third term is 0 + 1, or 1 again. The 'nth' term is represented by the notation 'F(n)', so what we've said so far can be written symbolically as follows. (Note that the numbering begins with 0, not with 1):

F(0) = 0
F(1) = 1
F(2) = F(0) + F(1) = 0 + 1 = 1

Continuing the process, you can generate as many terms as you like, always adding the previous two to get the next one:

F(3) = F(1) + F(2) = 1 + 1 = 2
F(4) = F(2) + F(3) = 1 + 2 = 3
F(5) = F(3) + F(4) = 2 + 3 = 5
F(6) = F(4) + F(5) = 3 + 5 = 8

...and so on. Here is a list of the first 20 terms of the Fibonacci Series, F(0) through F(19):

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, . . .

Seeds

Everything depends, of course, on the first two numbers, or 'seeds'. If we started with something other than 0 and 1, perhaps 1 and 5, then the series would be different: 1, 5, 6, 11, 17, 28, 45, . . . Traditionally, the 'Fibonacci Series' is the name for the sequence of numbers generated by beginning with 0 and 1 as seeds. The sequences generated by any random pair of numbers could be called 'Generalized Fibonacci Series'

Final Digits

If one is adding numbers in column form, it is apparent that one only needs to know the final digits of the numbers being added in order to know the final digit of the sum. If one number ends with a '7', and another ends with a '9', then their sum will end with a '6', whatever other digits the numbers may have. (We are talking about whole numbers here - no decimals.)

Applying this to the Fibonacci Series, one can generate a list of the final digits of Fibonacci Numbers, without having to worry about carrying anything beyond the units column. Such a list would begin like this:

0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, . . .

...and so on and so on. One might ask whether this sequence will eventually repeat itself. It should be clear that, if somewhere down the line we have a '0' followed by a '1', then everything after that will be just as above. Generally, if any pairing of digits appears for a second time, then the sequence will repeat itself.

Now, since we're talking about pairings of digits between 0 and 9, there are only so many possibilities. Specifically, there are 100, ranging from (0,0) to (9,9). [Note that (2,3) and (3,2) are considered different pairings of digits, because the order matters in the Fibonacci Series.]

So, with only 100 pairings available, we will have to see repetition by the time we get to F(100). In fact, it happens sooner than that. F(60) ends in a '0', and F(61) ends in a '1', so the sequence of final digits begins to repeat itself after only 60 terms. This immediately leads to 2 questions: Which 60? and What about the other 40?

How to Visualize Digit Pairings

Since each pairing consists of 2 digits, we will use a grid with 2 dimensions, width and height, or 'x' and 'y', if you prefer. The grid only needs 10 units in each direction, because there are only 10 digits. This 10x10 grid will have 100 squares, each representing a different pairing. Let's look at one:

(The rows should be numbered up the side too, starting with row '0' at the bottom, but I couldn't figure out how to make the computer do that. Sorry. Also, depending on your browser, you might not be able to see the lines separating all the squares. I'm working on it.)

Notice that one square is filled in black. That square represents the pairing (3,4). Just start in the lower left hand corner, and count 0, 1, 2, 3 along the bottom, then count 0, 1, 2, 3, 4, working your way up.

Now we have a way of looking at the 60 pairings of digits that occur in the Fibonacci Series' final digits. Here they are, in red:

Notice the nice pinwheel pattern. If we take the entire grid and tile it repeatedly across the screen, then we can see that the pattern continues beyond the edges:

Other Seeds - Other Sequences

So, sixty out of one hundred pairings are accounted for by the red squares in the grid. What about the other forty? Well, those must be pairings that don't appear in the final digits of consecutive numbers in the Fibonacci Series.

We saw above, however, that we can begin a generalized Fibonacci Series with any pair of numbers. Why not pick a pairing that isn't coloured red above, start a Generalized Fibonacci Series with it, and see what happens with the final digits? Maybe we'll get the other forty that way.

To choose one at random: (4,2). Beginning with 4 and 2 as seeds, and ignoring all but the final digit, we get:

4, 2, 6, 8, 4, 2, 6, 8, 4, . . .

Whoa! That one repeated after only 4 steps! That gives us 4 more pairings, but there are 36 still unaccounted for. Continuing to plug in different pairings, we can generate 4 other sequences of different lengths:

20 pairings: 0, 2, 2, 4, 6, 0, 6, 6, 2, 8, 0, 8, 8, 6, 4, 0, 4, 4, 8, 2, (0, 2, . . . )

12 pairings: 2, 1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9, (2, 1, . . . )

3 pairings: 0, 5, 5, (0, 5, . . . )

1 pairing: 0, (0, 0, . . . )

So, we've got a total of six sequences, each with different numbers of pairings. The original has 60, and here we've just seen sequences with 4, 20, 12, 3 and 1 pairings in them. Those numbers add up to 100.

Just as all the pairings in the first sequence were coloured in red, each of the other sequences can be assigned a colour, and the whole square filled up that way.

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¹ Italian Mathematician, ca 1170-ca 1240