Building the pattern is easy. Start with the letter A. Double it and place the next letter of the alphabet in the middle to get ABA.
Repeat: Double ABA and place the next letter of the alphabet in the middle to get ABACABA. Continue doubling and adding the next letter of the alphabet to build the pattern as far as you want to go.
The pattern gets big … fast! By the time we reach Z, the word has 226 – 1 = 67,108,863 letters! If you could say the word non-stop a rate such that “Abacaba” takes 1 second, it would take over 3 months to say the whole word! (In the novel, the name of the world is this pattern up to and including the first X.)
This pattern shows up in a lot of surprising places. Here’s a few.
If we give each letter a length, doubling the length for each in order, we get this pattern:
This is the same pattern on an English ruler:
Rotate this 90° and we can see the shape of a play-off tree:
Rotate 90° again and we final a fractal binary tree. We could think of this as a family tree … everyone comes from two parents, each of them with two parents, and so on. That means every person is right in the middle of their own Abacaba pattern!
Here’s the same shape with the branches at 120° angles. You’ll find variations of this shape all over nature: in plants and in the human body for example.
Here’s another famous fractal, the Sierpinski triangle, made from removing the center of a triangle and then removing the centers of the remaining triangles, and so on. It’s full of Abacaba patterns …
Here’s a few of them. There are infinitely many more!
The Mandelbrot set is a well-known fractal built on the complex plane.
Zoom in on the nose to the left and you’ll see the Abacaba pattern rippling off to infinity!
Binary numbers are used everywhere. If you pay attention to the number of zeros at the end of each number as you count in binary, you’ll find the Abacaba pattern.
If we instead start with binary 00000 and switch the digits one by one in the position of the Abacaba pattern (digit 1, then 2, then 1, then 3, and so on) we get a number sequence called the Gray code. It contains all of the integers with none repeated.
This number sequence is used in devices that sense rotation. Because only one bit is changed at a time, errors are very small. You can see that Gray code also makes a fractal binary tree!
Here’s what happens when we connect the integer sequence with a curve that passes through the numbers in the Gray code sequence.
Planning to travel in hyperspace and worried about getting lost? Following the Abacaba pattern will take you to all of the corners of hypercube. Here’s a 4D hypercube. Label the direction of each dimension with a different number (or letter), start at any corner, and move according to the Abacaba pattern. You’ll visit every corner. It works in any dimension!
Several puzzles have solutions that follow the Abacaba pattern, such as the well-known Towers of Hanoi puzzle.
Here’s a machine that plays the Abacaba pattern as it counts in binary from 0-127.