Welcome to Abacabax, a world built on the fantastic Abacaba fractal pattern. Here you’ll find information on the world of Abacabax, including the fantasy novel, music, art, projects you can create yourself, connections and more.

Abacabax – a new fantasy novel by Mike Naylor


Max Teller is on the run for his life on Abacabax, a world of incredible patterns, danger and beauty that stands on the brink of destruction. Four families, each with unique abilities, are locked in a struggle for power and revenge while dark creatures have united with deadly purpose. Max is lost and alone, not knowing whom to trust and unable to awaken the powers that everyone assumes he has. Armed only with a broken abacus and his unconventional imagination, he must make unlikely allies and battle the growing awareness that he may be the one destined to destroy the world.

Abacabax book

What they’re saying

“A fascinating story, well written and interesting. I wish it was twice as long. Recommended to anyone with an imagination and mandatory to those with an imagination and an affinity for numbers, systems and maths in general.” – goodreads

“This was a very interesting, exciting and capturing book to read. Something totally different from books in this genre that I have read before. From start to end you are taken in to a world of beauty, magic and mathematics. It is filled with strange, strong characters with different abilities, using their powers for the good and the bad. It is playful, scary, fun and with a lot of surprises. A stunning, breathtaking adventure. A real page-turner – I couldn’t put it down!” – goodreads

I truly enjoyed reading this book. It is not only exciting… it inspired me to learn more about mathematics. The reader does not need to know much about math or even like it in order to become captivated by the story – and bewitched by the power of math or the power of recognizing patterns behind the surface … in everything from nature, the landscape, and the tools used by its inhabitants, to the hieroglyphics and the music. Abacabax seduces the reader to see the beauty and similarity behind different forms … Having read Abacabax, I bought myself an abacus for playing with numbers in a new way.” – amazon.co.uk

Available now

Paperback, 370 pages. Includes chapter illustrations, maps and addenda detailing the reckoning methods of the four families. Learn to reckon like a Dijin, Bone-Thrower, Calculist and Abacist!

Available now on Amazon in both paperbook and eBook format.

Click here to go to Amazon UK

Abacabax is available on all Amazon stores – check with the store nearest your country!

More about the story

Abacabax is the exciting new novel in a fantasy world like no other. The denizens of this world have powers they use by “reckoning,” a form of calculating. The original family, the Dijins, use simple finger reckoning. Bone-Throwers, the mysterious and seldom-seen second family, calculate using human bones. The third family, the Calculists, use stones and stone tablets and have mighty military power. The fourth family, the Abacists, use abacuses and are the most powerful, though very few in number following a recent catastrophe.

It is nearly the time of the Great Divide when a fifth family will be born. Despite the marked hierarchy of powers and abilities of the families, each family has a member who has developed new methods of reckoning and stands as a candidate to become the New Origin, the leader of the newest and most powerful family on the planet. Some will stop at nothing to be sure their family is the next to ascend …


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.

Download pdf in A4 paper size

Download pdf in US letter paper size

Music. Listen to the music of Abacabax. This simple melody is based on the Abacaba-fractal pattern.

Listen to a fully orchestrated version here:

Abacaba patterns are embedded not only as notes, but in the rhythms, stereo effects and more. (Download the fully orchestrated version as an mp3.)

Download sheet music for version 1 so you can play it yourself on piano.

Learn to reckon. Download the addenda from Abacabax. Full instructions on how to calculate using your fingers, bones, a stone tablet and the abacus.

About the Abacaba-pattern

The Name

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.

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:


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!

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!

image source: Wikipedia commons

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.

Music machine

Here’s a machine that plays the Abacaba pattern as it counts in binary from 0-127.



Sign up for news from Abacabax. We’ll send you (very occasionally) news, secrets, updates, insider information and things we think you’ll enjoy hearing about. We won’t spam you and we’ll never share your contact information with anyone outside of Abacabax.

Send a message

Questions, comments or requests? We’d love to hear from you! Contact the author here.