Spiro-Fibonacci Sequences

Spiro-Fibonacci Patterns

...can look like this: Differencing the Fibonacci Sequence

One of the most striking features of the Fibonacci sequence is that the gaps between the terms give the sequence itself.

Although it is possible to define the nth term (a(n)) of the sequence as an algebraic function of n, the better known and original definition of the sequence is inductive. The first two terms (the zeroth and first terms respectively) are chosen as 0 and 1, and from then on we write down further terms one by one, in each case making a(n) equal to the sum of the two previous terms, i.e. of a(n-1) and a(n-2).

So we get:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, etc.

And the gaps are:

1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc.

For n > 0, the (n+1)th gap is the same as the nth term. This follows quickly from the fact that when n>1, a(n) = a(n-1) + a(n-2). Substituting n+1 and n+2 for n, we get a(n+1) = a(n) + a(n-1) and a(n+2)) = a(n+1) + a(n). So the (n+1)th gap, defined as a(n+2) - a(n+1), equals a(n+1)+a(n) - (a(n) + a(n-1)) = a(n+1) - a(n-1). But since a(n+1) = a(n) + a(n-1), this equals a(n) + a(n-1) - a(n-1), which equals a(n).

We get a 1 before the sequence itself reappears in the gap sequence, but this seems of little significance since it is simply the gap between the two specified initial terms, 0 and 1.

Taking higher-order differences (gaps between gaps, and then gaps between gaps between gaps, and so on) gives:

1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc. (first differences)

-1, 1, 0 , 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc (second differences)

2, -1, 1, 0 , 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc (third differences)

-3, 2,- 1, 1, 0 , 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc (fourth differences)

5, -3, 2,- 1, 1, 0 , 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc (fifth differences)

-8, 5, -3, 2,- 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc (sixth differences)

The sequence repeats itself at the start of each higher-order sequence, going ‘backwards’ from 0 and with alternating sign.This is the same as what we would get if we continued the basic sequence backwards too, working out each term by reverse application of the ‘add the two previous terms’ rule. This means that the identity between the two-directional infinite basic sequence and the infinite-order difference seqence is complete.

Taking literal ‘differences’, i.e. absolute values, gives:

1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc. (second absolute differences)

And we the order of difference increases, the sequence simply repeats the digits ‘0, 1, 1’ at the beginning. For example:

0, 1, 1, 0, 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc. (sixth absolute differences)

Subtractive and Differential Fibonacci Sequences

With regard to the difference sequences derived from the Fibonacci sequence, it might be said that repeatedly taking absolute differences between terms that are all either 0 or 1, does not seem especially interesting...

Indeed it seems little more interesting than if, instead of defining a(n) as a(n-1)+a(n-2), we define another sequence where a(n) equals a(n-1)-a(n-2). Let us call this the Subtractive Fibonacci sequence.

Once the first two terms are defined as 0 and 1, the sequence is:

0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, ... with a period of 6,

And if the definition is changed so that a(n) equals the absolute difference, abs(a(n-1)-a(n-2)), this becomes what we call the Differential Fibonacci sequence:

0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, ... with a period of 3, and again simply repeating ‘0 1 1’.

So in short, differencing the Fibonacci sequence is boring, and so is altering the definition by replacing addition with subtraction or with taking absolute differences, yielding the Subtractive Fibonacci sequence and the Differential Fibonacci sequence respectively.

But think again.

Generalising the Definition

Defining the Fibonacci sequence, we said that a(n) was equal to the sum of the two previous terms.

But what does the phrase ‘two previous terms’ actually mean?

The answer is that if we are considering a(n), the two previous terms are a(n-1) and a(n-2).

Not difficult.

But this is an algebraic definition, it is not geometric. If we want a geometric definition, we have to put things differently. Keeping the definition inductive, we could define the expression ‘two previous terms’ as meaning the two terms, among those we have already written down, that lie immediately behind the nth term, on the line on which we are writing down the sequence.

So if 610 is our nth term, then the two previous terms are those we wrote down immediately before (behind) it, namely 233 and 377.

... 233, 377, 610...

Picking this definition apart at the seams, we find ourselves holding two separately definable chunks. First, the terms must already have been written down. Second, of the terms that have been written down, they are the two closest to the nth term on the line. They must be the closest because we wrote them down immediately before we wrote the nth term.

Since we are talking geometry, it is no surprise that we are taking the term ‘line’ literally, i.e. spatially. Ditto the term ‘closest’.

If we really want to pare the geometric definition yet further back, we might consider more closely the commentary about the superlative adjective ‘closest’. In this context, ‘closest’ obviously means ‘immediately before’. But since we are only considering terms that have already been written down, we cannot get rid of the idea of ‘before’. Nor can we get rid of the idea of ‘immediately’, because, because...

...because when you write terms in a straight line, the closest terms to the nth term are the terms you wrote down just before you wrote down the nth term. Or in other words, because the line is straight.

So we can just say:

define the first two terms as 0 and 1, and, writing down successive terms along a straight line, define each term as the sum of the two physically closest terms that have already been written down.

Which leads us to a simple generalisation: make it optional rather than mandatory for the line to be straight.

We might as well keep the ‘line’ part, since however we write down the terms we can always conceive of writing them down along some line or other. But we can drop the ‘straight’ part. And if we do, we get:

define the first two terms as 0 and 1, and, writing down successive terms along a line, define each term as the sum of the two physically closest terms that have already been written down.

On this definition, the Fibonacci sequence is what we get if the line is straight. We can call it the Straight Fibonacci sequence.

If the line is curved, we get other sequences in the same general class as the Fibonacci sequence, but different. We can call them Curved Fibonacci sequences.

Spiro-Fibonacci Numbers

Spiro-Fibonacci numbers are what we get if we write down the terms in a ‘square spiral’.

A square spiral is a discrete version of an Archimedean spiral, where square boxes are chosen in a spiral, one after the other, from a tessellation of square boxes.

Like this: This is the spiral that Stanislaw Ulam used when he wrote down the positive integers and shaded the primes, thereby making the Ulam Prime Spiral.

But so much for the order in which the terms are written. Now let’s write some terms, keeping to the rest of the generalised definition. I.e. starting with 0 and 1 and working out each subsequent term as the sum of the two closest already-written terms.

The term that comes after the initial 0 and 1 can only be 1. Before we write it, 0 and 1 are the only terms already written, so they must be the closest terms (shaded yellow). The next term is also 1, since the two closest terms are again 0 and 1, even if the 1 is not the same 1 as before: The next term too is 1: And in fact we get seven 1s before eventually we get the eighth term, which is 2: The ninth term is 3: And the tenth term the 4, a number that does not appear in the Straight Fibonacci sequence at all: By the time we have filled in 25 terms, the sequence looks like this: Here are the first 81 terms: I.e. 0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 21, 24, 27, 31, 36, 42, 48, 54, 61, 69, 78, 88, 98, 108, 119, 131, 144, 158, 172, 186, 201, 217, 235, 256, 280, 304, 328, 355, 386, 422, 464, 512, 560, 608, 662, 723, 792, 870, 958, 1056, 1154, 1252, 1360, 1479, 1610, 1754, 1912, 2084, 2256, 2428, 2614, 2815, 3032, 3267, 3523, 3803, 4107

Differencing the Spiro-Fibonacci Sequence

The differences between the terms in the Spiro-Fibonacci sequence are not the same as the terms of the sequence itself:

1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 10, 10, 11, 12, 13, 14, 14, 14, 15, 16, 18, 21, 24, 24, 24, 27, 31, 36, 42, 48, 48, 48, 54, 61, 69, 78, 88, 98, 98, 98, 108, 119, 131, 144, 158, 172, 172, 172, 186, 201, 217, 235, 256, 280, 304, ... (first differences)

-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 2, 3, 3, 0, 0, 3, 4, 5, 6, 6, 0, 0, 6, 7, 8, 9, 10, 10, 0, 0, 10, 11, 12, 13, 14, 14, 0, 0, 14, 15, 16, 18, 21, 24, 24, ... (second differences)

The sixth differences are:

-1, 0, 1, -4, 6, -4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, 2, 2, -2, -2, 3, -2, 3, -2, -2, 3, -1, -1, 3, -2, -2, 3, -1, -1, 3, -2, -2, 4, -3, 0, -1, 8, -6, -5, 7, -2, -1, -4, 17, -12, -11, 16, -5, 0, -1, -8, 29, -20, -19, 28, -9, 0, -1, -12, 41, -28, -27, 40, -12, -1, -1, -2

No reflection. The tenth differences are:

27, -56, 70, -56, 28, -8, 1, 0, 0, 0, 0, 0, 0, 1, -7, 20, -28, 15, 7, -7, -16, 35, -40, 35, -15, -14, 27, -13, -13, 27, -14, -14, 27, -13, -13, 27, -13, -20, 42, -37, 28, -47, 71, -42, -28, 63, -45, 42, -102, 154, -84, -70, 148, -106, 32, 50, -180, 266, -140, -126, 260, -186, 56, 74, -260, 378, -196, -181, 367, -256, 84

Taking absolute differences gives:

1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 2, 0, 1, 1, 2, 1, 4, 5, 0, 1, 4, 5, 0, 1, 8, 9, 0, 1, 8, 9, 0, 1, 12, 13, 0, 1, 12, 12, 1, 1, 2 (sixth differences)

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 0, 10 (tenth differences)

There are a lot of 0s and 1s but not as the number of terms increases. And no periodicity.

Subtractive and Differential Spiro-Fibonacci sequences

Nor, unlike for the Straight Fibonacci sequence, is there periodicity when we define the nth term not as the sum but rather as the difference of the two physically closest terms that have already been written down.

All the terms we take differences of will still be 0s and 1s, but we get much more interesting patterns.

Taking signed differences (the most recently written down of the two closest terms, minus the less most recently written down) gives the following first 81 terms: Taking absolute differences, though, gives a sequence that is non-divergent because each term is either 0 or 1:

0,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1

or, on the spiral itself: The patterns appear better if the 1s are shaded. Here are the first 2500 terms, with squares containing 1s shaded red: And the first 10,000 terms: Note that if we work in binary, and use 0 and 1 as the first two terms, we get the same pattern, regardless of whether the sequence is Additive, Subtractive, or Differential. It follows from this that the 0s and 1s in the pattern are also the polarities of the numbers in the Spiro-Fibonacci sequence itself.

If the first two terms are defined as 1 and 0, rather than as 0 and 1, the pattern is different. For the first 10,000 terms: And if the first two terms are defined as 1 and 1, the pattern of the first 10,000 terms is: If corresponding terms from each of the first two patterns (with first terms defined as 0 and 1, and 1 and 0 respectively) are added together, the result is the same as the third pattern (with first terms defined as 1 and 1).

There is considerable scope for further investigation! :-)

Other articles at this site

Exploring Primeness Project - main page

Fernandez's Order of Primeness, F(p)

Some prime sequences related to F(p)

The prime-composite array, B(m,n), and the Borve conjectures