Katabatak Math-An Exploration In Pure Number Theory

[pgrmdave]

This is fun!

[maddog]

This is starting to look like numeralogy...

Maddog

[Turtle]

Yes... fun; this turtle means to put the "fun" back in "functions". I am i no hurry, (I am a turtle after all), so we'll just stay camped a while here in the Cave of Mathematics. Now this isn't just any 'cave of mathematics', this is THE Cave of Mathematics & THE 'Chamber of Integers; so where are all the other mathematicians & functions you may ask? Well, they are all here, but phase-shifted from us. If you remember we didn't use the main cave entrance, but rather came in through a Serpinski Sponge & following a turtle's tale no less. The result is a phase shift.
I confess, I did find this through numerology. I started reading on numerology & soon found it to be almost entirely arbitrary; except the math operation part. It is strict, it is umambiguous, it is congruent to the modulus, it is sometimes called 'fadic addition', it is sometimes called 'the rule of the pyramid', it is a conditionally recursive algorthm; it is the katabatak function.
Again, we are in no hurry here, so please don't hesitate to ask questions. I have been poking around in here with the k function a better part of a decade & I have sufficient fun at this rate for many months to come. May you all receive the most positive of influence & benefit the universe has to offer.

[Turtle]

Just a few more words as we sit here in the cave 'round my little flickering candle & before we retire to the dark. I just realized Tormod refered to 121 in relation to summing the digits of the long number I gave; it just so happens 121 is the square of 11 in the list of squares I asked you to fill in & I couln'dt figure out why you only mentioned that one.
Now I don't mean to try & frighten anyone, but the above brings to focus an element I haven't mentioned. That is danger, or rather safety perhaps. Already Tormod has experienced some disorientation in here & so some words of caution. While we stay in this chamber, the way out is of relative ease. When we leave however & I take you to unmapped areas, this spool of thread is your lifeline. Stay alert for low ceilings, drop offs, & loose rock; stay on the path.
'Till the morrow then; dream well.

[Tormod]

I'll go through it again.

Maddog - this is a fun exercise and I don't care what this is called - if it can be turned into a piece of music in the end then that's a worthwhile timekiller for me (plus I might actually learn something new).

[Turtle]

Good day again to you all. Before we move on later this afternoon, I have an apt analogy for the Katabatak function. Think of your K function as a machine; a machine that labels numbers. Into the top of a machine goes an integer, you turn the crank & out the side comes a label. With a base ten K machine, you may get out the side just one of nine different labels.
On its own, a number on a label might be confused for an integer that goes in the top, so we further mark the number on the label with an underline. (Look to the middle row of box lids).
So, every integer we use the function on gets a coded label & here is where we may start to make some music. Besides coding the labels with underlined numbers, I code the labels in color as I mentioned; you may add any other coding you like to the nine different labels & I see nothing wrong if Tormod labels his with music notation. Here is where I don't have enough musical knowlege & I turn the task over to those better qualified. 'Till this afternoon the. Wiedersehn.

[Turtle]

Without further comment for the time being, our diagram updated.

[Turtle]

All righty then, this brings us up to what's happenin' folks. We have this tool, the K function, which through repeated adding of an integer' digits, forms a label relation. Further, we can make other logical relations with the labels themselves. As you see I labled a row for Tormod to make musical relations to the labels, & I colored some of the labels to show how I relate colors.
Now if you will look at the diagram a little closer, you see I also added some broad vertical divisions on the boxes & the more abstracted list of squares. See too, that the labels within each division form a pattern, that repeats in the others exactly. Call this the Katabatak pattern. For the integers, the pattern is {1 : 2 : 3 : 4: 5 : 6 : 7 : 8 : 9}.
For the squares the Katabatak pattern is {1 : 4 : 9 : 7 : 7: 9 : 4 : 1 : 9}
(Turtle has fainted momentarily at the memory of his first discovery of the Katabatak pattern: he will be fine in awhile.)

[Turtle]

Every one else OK? I thought I was having a stroke & checking out. Life is short, so let's hasten slowly. Let's now take some time for an Arts&Crafts session.
I draw your attention once again to the K pattern of the squares {1 4 9 7 7 9 4 1 9}. We are going to take this ordered pattern & relate them to lengths; specifically to the lengths of strips of balsa wood 1mm thick & 1cm wide. So if you follow through, you should now have nine such pieces, two of length 1, two of length 4, two of length 7, & three of length 9. Now join them end to end with tape hinges in the same order as the K pattern of squares. Finally, join the two free ends together.
Excellent! You now have in your hands what I call a "flexigon" (not to be confused with "flexagon"). Below see a scan of my Katabatak squares flexigon.
Your challenge, if you decide to accept it, is to push the little flexigon around until you get it arranged in such a way that it forms a tile (as in tiling the plane). I know for this pattern (the squares) it is possible & I will post my tile a little later. If you don't understand, please ask.

[Turtle]

Now we just took a rather big leap forward, so if I may have everyone take 1 step back with your flexigon & have a seat. If you remember, I brought us into the Base Ten Cavern of Integers, & now see to your lefts we sit at the mouth of the Base Ten Cavern of Squares. Make yourselves comfortable.
I have never brought more than one person at a time in here & they number fewer than a score. I must say I am humbled by your shared courtesy & trust; thank you for coming.

While we rest here in repose and flexing our 'gons, I just have a few comments on the squares & their K pattern. {1 4 9 7 7 9 4 1 9} My diagram of squares over there in the dirt, as far as it goes, is proof of what I'm claiming. ie. ALL squares have these, & only these, labels; moreover, the labels have this order & only this order.

"What does it mean?", I just heard from the back. First let me say I don't know what a lot of this means; that's why I brought you here. To discover all I have missed. I only know how to get here. So, I leave you with this: If you take any number, no matter how large or how small, & use the K function on it, if the label is 2, or 3, or 5, or 6, or 8, you know immediately that number is ABSOLUTELY not a square. Till the morrow.

Oh...the tiling I promised.