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Exploring Fractals with ABACABA

Years ago, my three-year-old son asked me, “Why do numbers start at zero but go to infinity?” Much to my bewilderment, his young mind was somehow able to grasp this incongruity. For my reply, I used a number line to show him that for every positive number there is actually a negative number on the “other side” of 0, also to infinity. His response was the first time I heard his particular giggle. A giggle I would hear many times over the years and the sign he’s learned a new concept that tickles his brain. Using another number line for inspiration (a ruler), this post explores some interesting opportunities to explore fractals.


According to Wikipedia, a fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. One well-known fractal pattern called ABACABA, has long been used by artists, musicians and writers for creative inspiration. My own nerdlings have enjoyed exploring this mathematical pattern while gaining a deeper understanding of the potential inter-relationships between mathematical concepts and the arts.

The ABACABA pattern on a ruler demonstrates this pattern in its simplest form. In the space of one inch, the long mark dividing the inch in half represents the letter C (see picture below). The two shorter marks dividing the space on either side of the C in half represents the B’s. Finally, the shortest marks represent all of lines divided in half again, represented by A’s. Spelling out the pattern ABACABA.

Imagine this pattern continuing indefinitely, splitting lines in half until they are too small to see.

ABACABA inspired Art – Sierpinski Gasket



A Sierpinski gasket (above), created by the mathematician Waclaw Sierpinski (1882-1969), uses the ABACABA fractal pattern for the amazing picture above. Ask your nerdling to try zooming in on different parts of the gasket, she will see the same basic shape appearing no matter now far in she zooms, as the pattern continually repeats itself.

Binary Trees

Creating binary trees is a great place to begin playing with the ABACABA pattern. Follow the steps below to create a ABACACA binary tree:
Step 1: Draw a long line (this represents the longest line) in the example this is the complete inch.
binary tree

Step 2: Make two branches coming off of the trunk. These lines represent the C (and should be shorter than the tree trunk).

Step 3: Make two lines coming out of each branch to represent the B’s (also shorter than the previous lines).

Step 4: Draw two shorter lines attached to each of the B branches (these are the A branches)

Step 5: Feel free to continue to make smaller and smaller lines until the lines are too small to draw!

Paper Structures


Source: provides instructions and other information for fractal cutouts needed to make the pop-up above. Another website, also has loads resources, including this pdf (, with instructions to another pop-up structure on the last page (page 13).

Fractal Software

Making fractal art with markers or crayons can be a complicated process, but perhaps worth the effort to make an awesome picture. With modern technology, nerdlings can easily explore the world of fractal pictures with software apps designed for smart phones and computers.

Fractal picture I created with Frax

Fractal picture I created with Frax

I made the fractal picture here with an inexpensive iphone app ($.99) called Frax. It is a straightforward program that folks of any age can use to make inspiring art that can be saved, or even printed and hung on a wall. For a list of additional fractal software apps and a host of other activities visit

ABACABA in Music

Since the Baroque period, musicians have been composing with an ABACABA structure under the name of Rondo. This form of music has a principal recurring theme (often called a refrain) that alternates with different, and usually contrasting sections called episodes, digressions or couplets. Use the links below to play some classical and/or modern ABACABA structured music for your nerdling(s).

“Highway Star” by Deep Purple

“Every Breath you Take” by Sting and the Police

Abacab by Gensis

“March from the Nutcracker” by Tchailkovsky

Beethoven’s Sixth Symphony, last movement

“Eine kleine Nachtmusik,” K 525 by Mozart (last movement)



  1. Scott

    Huh, and now, having read the Wikipedia page, I now get why Genesis called the song and album “ABACAB!”

    • RexCelestis

      It took this article to get me to figure that out, too!

  2. Bonnie Balboa


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