M2C2A: Episode 12, The ABACABA, the biggest word in the mathematical dictionary

To write this word is a total mess, but to write the first 63 characters, this you can do.

Write an "a" every 2 spaces or (2¹) spaces. This should look like this:
a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a

Then, from the first unoccupied space, write a "b", every 4 spaces (2²) this time:

aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba

Repeat this last step, but with the "c", and now every 8 spaces (2³) .

abacaba_abacaba_abacaba_abacaba_abacaba_abacaba_abacaba_abacaba

If you continue the pattern, this you should get:

abacabadabacaba_abacabadabacaba_abacabadabacaba_abacabadabacaba
abacabadabacabaeabacabadabacaba_abacabadabacabaeabacabadabacaba
abacabadabacabaeabacabadabacabafabacabadabacabaeabacabadabacaba

Of course, if you continue this until the "z", you will get a word (2²⁷)-1 characters long, which is 134,217,727 letters long! 

And if you include all the character in the Unicode (the majority are Chinese characters and some are just representations as " ", but still, no discussing about that), the epic number of 109,449 you get:

5.40785978562894540705755027563290385780481607 × 10³²⁹⁴⁷ letters!!! 

But what does this apply to?

The Sierpiński Gasket!


Every time you see a black square, write "a", if you see a little white triangle, write "b", a bigger white triangle, "c"... etc.



And if all the "a"s in the triangle are one color, and the "b"s other... you get a colorful Sierpiński! (but that's another thing...)

And to really celebrate Sierpiński and the abacaba, I made the "rainbow gasket"!


Cool, huh?
-The Roaring Thunder