The homepage of one of the most fascinating patterns

The Name

Welcome!

“Abacaba” is a fascinating pattern that is found in all kinds of surprising places. Here you can explore the pattern, discover suprising connections, and experience music, art, poetry, craft projects and more.

The Name

The name of the Abacaba describes the pattern itself. Let’s build it!

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 over 67 million 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!

This word was published in 2011 as a 1600-page, 4volume work (printed mostly in 4-point font!), setting an unofficial world record for the longest published word.

Mathematics

Geometric lengths

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:

Trees

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.

Fractals

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!

Cantor dust is created by staring with a line segment and removing the center third. Each of the remaining pieces has the center thirds removed, and so on. Here are the first few stages… the holes make an Abacaba pattern:

When the process is carried out to its limit at infinity, Cantor dust has an infinite number of points, no line segments at all, and a total length of zero.

The 2d version is called the Sierpinski Carpet. Start with a square, divide into a 3×3 grid and remove the center square. Repeat the process on the remaining 8 squares. The sum of the edge lengths of a Sierpinski Carpet is infinite and it has an area of zero … it becomes a sheet composed entirely of holes!

Here is a mirror made with the first 5 iterations – there are over 4000 holes in the frame!

There is also a 3d version called a Menger sponge. It has infinite surface area and zero volume. Weird.

Number systems

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.

Hyperspace

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 four related figures in four different dimensions. 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!

Music

You can play ABACABA on an instrument. The letters of the name describe the notes. The melody is in the key of A-minor and it sounds quite nice.

Here’s a marble machine that plays the Abacaba pattern. The flippers are counting in binary from 0 to 127 as it plays from an A to an A one octave higher before repeating.

Poetry

Decision Tree

The Abacaba pattern is central in this tree fractal…

… which can also be drawn like this:

You can think of this a model of all of your paths in life. Every time you make a decision, the universe splits in two copies that are nearly alike, but depending on what your decision is, you could end up in two entirely different places! If only we could have a map of all of future decisions and where they lead, then we could see where we want to go and know exactly how to get there.

That’s the idea of this poem, Decision Tree. Start at the trunk of the tree… and choose wisely!

The poem was designed with the trunk as the question, the first branches as the decision, the next as the narrator’s reaction, and finally how the decision affects the future. In addition, movements to left are a bit devilish while those to the right are a bit angelic.

Interestingly, most of the paths create poems that are a little depressing, but there are two where you get the feeling the narrator is feels pretty good about him/her-self… the path entirely to the left and the path entirely to the right! In life, as in mathematics, we are free to make our own rules. After we’ve chosen the rules, we need to be consistent or we risk an unsatisfying result!

Entirely Nothing

The Cantor Set is a fractal made from a line segment. Remove the middle third. In each of the remaining parts, remove the middle thirds again. Repeat on these smaller parts and continue infinitely. The resulting shape has infinitely many points, but a combined length of 0!

At each step in making the Cantor Set, we find an Abacaba pattern. It’s in the spaces between the lines. As the original line segment shrinks to nothing, the Abacaba pattern grows infinitely!

Here’s a poem with this structure.

Activities

Paper folding models. Print out, cut and fold pop-up models of the fascinating Abacabax stairs. Download a pdf with full instructions and two different templates, an easy version and a more complicated one that looks like this:

The Tower of Hanoi puzzle is connected to the Abacaba pattern.

A set of disks is stacked on one of three pins, in order of size with the smallest on the top and the largest on the bottom. The goal is to move the entire tower from its starting position to one of the two other pins. But (1) you may only move one disk at a time and (2) you may never place a larger disk on a smaller disk.

You can play this using playing cards instead of disks. 5 cards is a good number to start with. Make a stack with an ace on the top, then 2, 3, 4 and 5. Start with the stack in your hand and picture two additional positions you can move them to, to the right and to the left. Now move cards one at a time between the three positions (right, left and in your hand), always building stacks and never placing a higher number on a lower number.

Tips: Make your stack alternating colors, for example with black A, 3, 5 and red 2 and 4. If you find a solution, pay attention to both the order of the numbers in the sequence of moves, and in the color of the cards being moved. There is a lot to discover!

Contact

This site is created and maintained by Mike Naylor. Have a question or a request? Send me a message!