A Simple Recipe to Generate Phi

Here is an interesting little recipe I found in an old notebook. Unfortunately, I did not write down where I got it from, or if I just thought it up, or what.

Start with “0”. At each step, if a character is a “0”, replace it with “1”. If “1”, replace it with “10”.

So, the results are:

Step 1. 0

Replace the “0” with “1”, so we get:

Step 2. 1

If a character is a “1”, replace it with “10”, so we get:

Step 3. 10

If a character is a “0”, replace it with “1”. If “1”, replace it with “10”. So, the “1” becomes “10” and the “0” becomes “1”, to give us:

Step 4. 101

(Do you see what happened there? The “1” was replaced by “10” and then the “0” at the end was replaced with a “1”.) Repeat again: “0” -> “1” and “1” -> “10”. This gives us:

Step 4. 10110

Step 5. 10110101

Step 6. 1011010110110

And so on and so on.

Two interesting things: One is that each step (Step 6) is simply the text from the previous step (Step 5) with the text from the step before that (Step 4) added to the end. So:

Step 7. 101101011011010110101

Also, the ratio of 1’s to 0’s in each step approaches phi, the Golden Ratio.

In Step 4 there are three 1’s and two 0’s.  3/2 = 1.5

In Step 5: the ratio is 5/3 = 1.6667

In Step 6: the ratio is 8/5 = 1.6

In Step 7: the ratio is 13/8 = 1.625

These are all Fibonacci Numbers: 2, 3, 5, 8, 13, and the ratio of one Fibonacci Number to the previous one approaches the Golden Ratio, which is the ratio of Adults to Kids in an extended Lotka-Volterra model of predator-prey dynamics.

It seems exceedingly strange to me that the ratio of 1’s to 0’s approaches an irrational number even though the recipe only involves swapping each “0” with “1” and each “1” with “10”.
Now I really wish I had written down where I got this recipe from.

~~~~
This post from BoingBoing says that the aspect ratio of human eye’s field is about 16:10 width-to-height (or 1.6, which is maybe why people find the Golden Rectangle aesthetically pleasing). In some previous photo posts of mine, including this picture of an insect and this one of a car, I used a photo size of 550 x 170, which is 2 times Phi.

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