### Post by wgardner on Jun 16, 2019 10:54:19 GMT -5

Jotting down a few thoughts on this stuff:

The Fibonacci series is computed recursively by just adding the previous 2 numbers to get the next number. 1,1,2,3,5,8,13,21,34...

The Golden Ratio (last chapter title) is the ratio of neighboring fibonacci numbers as the series gets large: 13/8, 21/13, 34/21, ... converges to the Golden Ratio phi. The Golden Ratio phi = (1+sqrt(5))/2. It's not a rational number, so its digits form a pseudo-random series of digits (perhaps a perfect key for a Vigenere-type code?) and this is true not just in base 10 but in any base system (well, except for base phi which evidently is a real thing).

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Fractals are also computed recursively, much like the Fibonacci series.

For example, let's take the top fractal triangle on page 73. Let's call the 3 directions Up=1, LeftDown=2, RightDown=3.

Let's start with the big upside-down white triangle in the middle (I'll call that triangle 0, size 1, the biggest size).

The next 3 biggest triangles (size 2) are in 3 directions from this biggest triangle. I'll label them triangle 1 (size 2, Up), 2 (size 2, LeftD) and 3 (size 2, RightD).

We create the size 3 triangles by again putting them on the 3 sides of each size 2 triangle. So we have 3 triangles around triangle 1 (which I'll label 11, 12, and 13), 3 triangles around triangle 2 (which I'll label 21, 22, and 23), and 3 triangles around triangle 3 (which I'll label 31, 32, and 33). etc. etc.

To find and label a specific triangle, you can just start in the middle and then specify the series of directions you take (1, 2, or 3), much as you might tell someone to walk in a city by going north for a block, then west, etc...

As an example, I see 5 shaded triangles in the top of page 73. Using the labeling scheme I describe, I'd label these triangles:

12222

2312

23121

3122

33211

Clearly the shadings must have some meanings here. I'm wondering if some encoding of the triangles like this is used as part of a code/key later in the decoding process...

The Fibonacci series is computed recursively by just adding the previous 2 numbers to get the next number. 1,1,2,3,5,8,13,21,34...

The Golden Ratio (last chapter title) is the ratio of neighboring fibonacci numbers as the series gets large: 13/8, 21/13, 34/21, ... converges to the Golden Ratio phi. The Golden Ratio phi = (1+sqrt(5))/2. It's not a rational number, so its digits form a pseudo-random series of digits (perhaps a perfect key for a Vigenere-type code?) and this is true not just in base 10 but in any base system (well, except for base phi which evidently is a real thing).

-------------

Fractals are also computed recursively, much like the Fibonacci series.

For example, let's take the top fractal triangle on page 73. Let's call the 3 directions Up=1, LeftDown=2, RightDown=3.

Let's start with the big upside-down white triangle in the middle (I'll call that triangle 0, size 1, the biggest size).

The next 3 biggest triangles (size 2) are in 3 directions from this biggest triangle. I'll label them triangle 1 (size 2, Up), 2 (size 2, LeftD) and 3 (size 2, RightD).

We create the size 3 triangles by again putting them on the 3 sides of each size 2 triangle. So we have 3 triangles around triangle 1 (which I'll label 11, 12, and 13), 3 triangles around triangle 2 (which I'll label 21, 22, and 23), and 3 triangles around triangle 3 (which I'll label 31, 32, and 33). etc. etc.

To find and label a specific triangle, you can just start in the middle and then specify the series of directions you take (1, 2, or 3), much as you might tell someone to walk in a city by going north for a block, then west, etc...

As an example, I see 5 shaded triangles in the top of page 73. Using the labeling scheme I describe, I'd label these triangles:

12222

2312

23121

3122

33211

Clearly the shadings must have some meanings here. I'm wondering if some encoding of the triangles like this is used as part of a code/key later in the decoding process...