Downloadthis article in pdf format

Abacaba-Dabacaba!
by Michael Naylor
copyright 2005

One of the fundamental patterns in our universe is one which couldbecalled a “binary branching pattern.” This structure shows up in anamazing variety of places – we’ll explore some of the exciting ideaswhich all share this pattern, a path that will take us throughgeometry, number systems, art, music, poetry, higher dimensions, andmore!

A binary branching pattern is simply one in which one object dividesinto two, then each of those divides into two, and so on, like thebranches on this tree:

It is not hard to imagine that the branching and doubling could becontinued infinitely, and if we could only magnify our view enoughtimes, we would see the same patterns continuing forever and ever...

This same pattern is followed (in reverse) in a play-off schedule whereteams or players are paired off with the winner of each roundprogressing to the next while the loser is eliminated:

If we move from the top to the bottom of the playoff tree, we’ll noticethat the name closest to the top of the chart appears in column 1, thenext highest name is in column 2, then column 1, then 3, then back to1, and so on. The pattern goes: 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1,2, 1
Can you spot the same pattern again on an English ruler? Each unit isdivided into halves by a long mark, those halves are each divided intoquarters by smaller marks, and so on.




If the shortest marks are length 1, the next length 2, and so on, thepattern of the marks is the same as in the play-off tree.
This pattern is so important, it deserves a special name. Let’s givethe pattern a name, then explore some of the many surprisingconnections.
The Name

Instead of using numbers to describe the length of branches or marks onthe ruler, let’s call the shortest lengths “a”, the next longest “b,”then “c,” and so on. The pattern then becomes:

. . . or “Abacaba-Dabacaba!” This word sounds very much like themagician’s phrase “abracadabra,” a very apt resemblance given theseemingly magical properties of this pattern.
To understand the pattern a little better and see how to continue it,let’s see how this pattern grows. Start with an “a.” This is the firststep, the “tree trunk” if you like.

1. a

To grow the pattern, add the next letter in the alphabet and thenrepeat everything that has gone before (which is just the letter “a” inthis case.) The next step, then, is “aba,” which is like a trunk (“b”)with two branches (“a”).

2. aba

Continue by adding the next letter, “c,” and repeating the “aba.”

3. abacaba

The fourth step adds the letter “d” and repeats the pattern:abacabadabacaba! The next few steps are shown:

4. abacabadabacaba
5. abacabadabacabaeabacabadabacaba
6. abacabadabacabaeabacabadabacaba-fabacabadabacabaeabacabadabacaba

It’s fun to see how much you can say aloud. How long would it take tosay the word all the way to “z”?
Fractals

Abacabadabacaba is a fractal pattern – and it shows up in many otherfractal patterns. Shown here are three famous fractals. The KochCurve is formed by replacing the center third of a linesegment with two edges of an equilateral triangle. The SierpinskyTriangle (or Gasket) is formed by removing the center of atriangle and repeating, and Cantor Dust is made by successivelyremoving the center third of a line segment. Can you findabacabadabacaba in each of these?

Number Systems

The binary number system is the simplest and most significant of allpositional number systems.

In binary, only two digits are used, usually zero and one. The value ofa written number is determined by the arrangement of zeros and ones,with the value of each place being a power of 2 rather than a power of10 as in our familiar system. The number 19, for example, would bewritten as follows:

19 (base 10) = 0 1 0 0 1 1 digits of binary #
32 16 8 4 2 1
2⁵ 2⁴2³ 2²2¹ 2⁰

19 = 2⁴ + 2¹ +2⁰


The count from 0 to 8 in binary is: 0, 1, 10, 11, 100, 101, 110, 111,1000

The abacabadabacaba pattern is repeated at infinite levels in thissimple counting pattern. The start of one such pattern is shown to theright. Can you find others?

Puzzles and Legends

A popular puzzle called “The Towers of Hanoi” asks players to move astack of disks one at a time from one of three pegs to another peg.Only the top disc of a tower may be moved, and one may never place alarger disc on top of a smaller disc. The object is to transfer theentire tower to a different peg in the fewest number of moves.
A couple different versions of this game are included at the end ofthis article. The key to solving this puzzle lies in the abacabapattern.

There is a legend of the Temple of Abacabax, where monks move goldendisks on diamond spindles. When they have moved all 26 (or 64,eye-witness accounts vary) the universe will come to an end. Should weworry?
The Temple of Abacabax is high atop of tower of stone blocks. Theblocks are arranged as such: Draw half a square, cut along the diagonalto form a right isosceles triangle, and then draw the largest squarepossible inside of it. Continue by placing the largest square possibleinside of all the right triangles created, and repeat. When you decideto stop, you will have made a staircase of blocks similar to those atthe temple of Abacabax. As you climb these stairs, the size of theblocks you step on makes the abacabadabacaba pattern. In the staircaseat the actual temple, this process is carried out 26 times, the centralblock corresponding to the letter ‘z.’

At the end of this article you’ll also find a template to make apop-up version of this staircase.
Hyperspace

Navigating higher dimensions? Don’t get lost! Call the left-rightdirection “a,” the up-down direction “b,” and theforward-back direction “c.”

Moving a will get you from one point to another – you travel aline segment.

Moving aba will move you around the corners of a square.

Try moving abacaba on the cube - did you visit all of itsvertices?

Let’s add another direction, the here-there direction, andcall it d. To travel to all of the vertices of this fourdimensional hypercube, just remember the magic word: Abacabadabacaba!

Here’s a 4D and a 5D hypercube. Are you up to the challenge?

                       
    

Philosophy and Poetry

It’s fun to think about how every decision we make leads us in a newdirection, as if our lives are an infinite fractal tree. Here’s a poemthat reflects those decisions... it has the structure abacabadabacaba.

Naylor, M. "Tree Diagram," CollegeMath Journal, 32, 3 (May 2001).

The Complex Plane

Abacaba-Dabacaba is written all over the Mandelbrot set. The M-Set is afamous fractal with infinite variety and complexity, generated byrepeating very simple rules. Many free computer programs are availableto explore this fantastic structure.

Zooming in on the “nose” to the far left reveals one abacaba pattern asshown here – there are many more you can find.

Music

Abacabadabacaba sounds like it could represent a series of notes.Indeed, if abacabadabacaba is played on piano, the result is beautifuland haunting. Download a copy from my website: http://www.wwu.edu/~mnaylor/abacaba/music.html.There is also a fully orchestrated version available for free download.

And On and On and On ...

Abacabadabacaba shows up in lots of other places as well. If you findone, won’t you please let me know?
The following pages contain puzzles and activities referred to earlier.Enjoy!

References

Gardner, M. “Mathematical Puzzles and Diversions,” Simon andSchuster, Inc., 1959.
Naylor, M. “Tree Diagram,” College Math Journal, 32, 3 (May2001).

(The following pages print better from the pdf file.  Download this article inpdf format)

Roly-Annie

“Roly-Annie” is a solitaire card game named after the card queen ofMississippi riverboats, “Roly” Annie Keim. To play, use the cards acethrough seven. Make the odd numbered cards black and the even numberedcards red, and stack them in order so that when the stack is face up,the ace is on top.

You are allowed to make two additional stacks in this game, a rightstack and a left stack. You’re allowed to move the top card ofany stack (on the table or in your hand) to the top of any other stack(on the table or in your hand). However, you may move only one cardat a time, and you may never place a higher numbered card on top of alower numbered card; you may only place a lower number on a highernumber (the ace has a value of 1). You may play any card on anempty stack.

The object of the game is get all of the cards into either theleft or right stack. The start of a game is shown below.

1. What is the minimum number of moves required to win this game?
2. List at least the first 20 moves. (You may wish to devise your ownnotation.)
3. Describe any patterns you found while investigating this game.Describe a strategy that will allow you to win every time withoutmaking a mistake.
4. You and a friend decide to race. You split a deck of cards andnumber the cards in each half from 1 to 26. Playing by the same rulesas before, about how long would it take to decide a winner?

POP-UP FRACTAL

This eye-popping fractal is a big hit with kids (and grown-ups, too)!It’s easy to create something complicated and beautiful using ideas offractals. The following page contains the pattern for making this workof art.

Supplies:

Copies of the handout on the next page
Scissors
optional: 9” x 12” construction paper and glue sticks

1. Have students fold the page in half along the dotted line. Fold sothat the print is on the outside of the folded page.

2. Cut along the two heavy line segments that connect to the crease andfold the flap upwards. The corners of the flap will fit neatly in the“L” brackets printed on the page.

3. Open the page, reverse the fold in the center of the flap and closethe page so the flap is now completely (and neatly!) inside the foldedpaper.

4. Two move heavy lines are now seen to reach the creases in the centerof the paper. Cut along these also, then crease, using the “L” bracketsto help. After folding the flaps up, you can flip the paper over andfold the flaps in the opposite direction. This will make it easier toreverse the flaps and pop them inside.

5. When you’ve popped the two smaller flaps inside, you should now beready to make two more cuts along the heavy lines that now meet thefour creases. Continue in this manner.

Point out that the first cuts made one flap, the second cut madetwo flaps, and the third made 4. How many flaps are made by the finalcut?

When it is done, you will have a beautiful 3-D pop-up form. This can beglued to a folded sheet of construction paper to make a lovely piece ofart or an eye-popping greeting card.

Questions for discussion:

How is this a fractal? How is this self-similar?
How many of each size block are there?
What other properties can you investigate?
What do you notice when you look at your model from differentperspectives?

Michael Naylor – mnaylor@cc.wwu.edu