> What’s interesting to me is that lots of other dots that lie close to this line are also relatively dark. Why does this happen?

I think I have one possible answer. Points with Fibonacci coordinates can be obtained by starting with coordinate (1, 1) and then climbing upwards via the x(n+1) = x(n) + x(n – 1) rule.

But we can start with any (a, b) coordinate where a and b are co-prime. This will give us:

x(0) = a

x(1) = b

x(2) = a + b

x(3) = a + 2b

x(4) = 2a + 3b

etc.

For example, for a = 1 and b = 3 we get: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, etc, which are still pretty bad for Euclidean algorithm. And the ratio between the neighbours also tends to the golden ratio (e.g. 123/76 = 1.618), in fact almost by the very definition of golden ratio. Indeed, at infinity we have:

ratio = x(n+1) / x(n) = x(n) / x(n – 1)

Rewriting this via x(n + 1) = x(n) + x(n – 1) we get:

(x(n) + x(n – 1)) / x(n) = x(n) / x(n – 1)

or (p + q) / p = p / q — which is how the golden ratio is defined.

So, regardless of the starting coordinate (a, b) our generalised Fibonacci climb will eventually bring us close to the dark side of the Euclid’s Orchard 🙂

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http://www.extension.umn.edu/environment/trees-woodlands/forest-management-practices-fact-sheet-managing-water-series/making-and-using-measurement-tools-basal-area/

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Claim: if and only if is a primitive root of unity.

Proof:

Consider the point , where .

Suppose . Then , so is a root of unity. As and is a root of unity, is not a primitive root of unity.

Suppose . Suppose that is not a primitive root of unity. Then there is some such that is an root of unity, i.e., . Thus . As , we must have . This is a contradiction, since . Therefore must be a primitive root of unity.

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