[Turtle]

___Hello. I am Turtle, your guide on this expedition. The attached image below is one place we are going, as I suggested that Tormod, expedition sponsor, may learn how to make a musical score out of it.
___We will be spelunking in the cave of mathematics, where all manner of patterns & forms, & structures abound. Now you may know that if you shine an ultraviolet light in a cave on certain minerals, they flouresce, & without the UV light the same minerals are indestinguishable from the others. In this expedition our UV light is the Katabatak Function, K(n). Call it the "K function" or "Big K", or just "K".
___I will define it presently, but a word about its origin first. I took the word from a Greek root which means to bring down, & this because the K function brings down integers & relates them to a single symbol.
___Before I post again & we start into the cave, you have some time to examine the image & decide if you care to come along. Until otherwise noted, dress for Base Ten, where the domain of the K function is all integers & the range is length nine. Range element symbols may appear as numerals or as colored squares. Welcome!

[Tormod]

I'm in for the ride...except it's midnight here in Norway so I'll catch up later. Maybe this will be the first official Hypography composition?

[maddog]

Originally Posted by Tormod

I'm in for the ride...except it's midnight here in Norway so I'll catch up later. Maybe this will be the first official Hypography composition?

I curious to know what Fractals would sound like...

Maybe try a symphony as a combination of the two...

Maddog

[Turtle]

You all look like a good group, so off we go. In deference to Tormod, we shall hasten slowly. I thought initially to take you into the cave by the main entrance, but Maddog mentioned fractals & I happen to know a discrete side entrance that he may like. This is a recreational expedition afterall. So, there it is, that small square passage; we have to crawl a ways so follow me. ... You wanted fractals & here we are inside a Sierpinski Sponge. Interesting texture in here don't you think? We just have to climb down a few iterations here... and here we are in the Base Ten Chamber of Integers.


___Now to the Katabatak function, K(n). Two principles, addition & repetition, make up the function. It is that operation of repeatedly adding an integer's digits until arriving at a single digit. As in K(1,908) = 1+9+0+8=18, but 18 is two digits so 1+8=9. Another example: K(3,454,671,304)= 3+4+5+4+6+7+1+3+0+4=37, but then 3+7=10, but then 1+0=1. In a bit more rigorous terms, the Katabatak Function is a conditionally recursive algorthm. Steady now; I know the function is the prime operater for that red-headed step-child of math, numerology. I know too that here in the Base Ten Chamber, the K function is congruent modulo 9.
Enough for now; I'm tired. Sleep on it & tomorrow we'll start fresh on a fractal path.

[Turtle]

___Here we are again then. I want to clarify that I said two principles in the K function, but mentioned three. Just so, three priciples:addition, repetition, & comparison, ie conditional testing. Because we soon will start compounding these principles, it is not incorrect to say the course is fractal inasmuch as it is self similar.
___Now strictly speaking, a proof is required for my claim K(n) is congruent n mod 9. Instead, I prepared a recreational demonstration. Down below, you see I arranged some little boxes, 3 rows of 28. The bottom row is open empty boxes, the middle row is closed empty boxes with blank tops, & on top, closed boxes of rocks with the first twenty eight integers expressed in Base ten. A pile of rocks on the ground in front which I gathered up from the cave floor. Finally, in the top row, the number of rocks in each box is the same as the number on its lid, & the ellipsis says of course add as many boxes as you please.
___This wouldn't be any kind of expedition (as if following a turtle into a cave isn't daring enough), if you didn't get your hands a little dirty, & I know everyone wants to try their shiny new K functions. So, take each written integer,n, from the top row lids & use the K funtion on it & write the result on the lid of the box directly below your integer n. Do this for the whole row.
___Now out of each box in the top row, take out all the rocks of one box, & seperate them into piles of nine. Any rocks left over, drop into the empty box in the third row directly beneath the box you took them from. Diascard your piles of nine on the pile on the ground.
___When you compare how many stones in a bottom row box(which of course is the remainder, or residue, from casting out nines), with the number written on the middle row lid above it, you see they agree, i.e. congruent. Now, just one last point before I give you a rest. The range of the K function does not return zero as does the modulo function, but rather 9. With the katabatak function, you get rocks in your box. Adieu until next time, when we start pointing our K functions at squares.

[Turtle]

___Hello again. See here on the ground I have filled in the box lids, added a label to the third row, & as promised, listed some squares & some of their transforms under K. Now I scratched a table in the cave floor with a stick rather than use boxes again for the squares; see that it is only a change in abstraction. Before I ask you to fill in the rest of the squares table for K(n), (I see some of you already have), I want to point out some things about my original first three rows of boxes.
___No matter to what degree or extreme of abstraction, we are really talking about quantity, i.e. how much of something. It's just a matter of efficiency to use the symbols; we simply can't carry all those rocks! As long as you see how my abstraction, the K function, truly represents some qualities of count, we can do away with the rocks. Moreover, because the K function truly represents the same quality of count as the modulo function, we can for the time being set it aside as well.
___I mentioned efficiency & in this venue, the K function is far more efficent than the modulus. Take this number:333,435,686,792,008,566,344,780,128. Now quickly, with pencil & paper, tell me what is the value of that number modulo 9? .......... Times up: it's 4. You see, I needed just 20 sec to sum those digits. I didn't use paper or pencil. Right then; off you go on your own then again until next time.

[Tormod]

Originally Posted by Turtle

Before I ask you to fill in the rest of the squares table for K(n), (I see some of you already have)

Gulp, how did you see that?

[Tormod]

Okay. I also got 4 (or rather, 121, which I then assume is 1+2+1 = 4).

Now please take a step back and point out to me why this is important. Remember I am a tourist who just stopped by for the ride with little mathematical insight.

[Turtle]

___Yes, if you mean that a 4 goes in the box under 121 because 1+2+1=4, you have it correct. I'll take the step back & we can camp on it a bit. This is important from several perspectives, & if I may have you again look at the top row of boxes labeled n. There is the begiining of the list of integers on the lids, written in base ten. Whenever you look at an integer(in base ten) you start, almost unconciousely, to take the measure of it by looking at the last digit. Right away you know even or odd (divisibility by 2), & whether it divides by 5 or 10. (Did you know also that if the last two digits of any number divides by 4, the whole number divides by 4?)
___The Katabatak function gives you another quick measure because it gives you more information on divisability. If k(n) = 9, you know the number divides by 9 (and 3 of course). If that same number happens to be even, then you know it also divides by 6. Similarly, if K(n) = 6, the number divides by 3, & again if the number is even, it also divides by six. Lastly if K(n) =3, the number divides by 3, & 6 if it's even.
___Finally, since K(n) is congruent modulo 9 (in base ten), you may transform any katabatak result to its modular form. We are just a few posts away now from making some patterns with this & trying to make some music!

[Tormod]

Aaargh. My brain is boiling. I need to reread this a couple of times...