Katabatak Math-An Exploration In Pure Number Theory

[Turtle]

Hello again everyone. I hope you're all well rested. Rather than going on from here in small leaps, I've decided to take you on a jump over the edge. If you feel your not up to it, just make yourself comfortable here & do some exploring on your own.
Those of you more adventurous, come on over here & get harnessed up. Now before you drop down the list below, I want to remind you how we came here. By addition, repetition, & comparison we came, & using these operations on themselves & in combinations. The K function is made of them, then we used the K on single numbers, then sets of numbers, & now we compound to sets of sets.
Without further ado, into the abyss!

[Turtle]

Back to the top of the chasm the rest of you. Well, after a leap earlier, we took a simple rest and now I think after an abysal plunge you need a ceremony. See here to your lefts I have authorized a banquet. Help yourselves to comfortable seating and food and drink, while I announce your awards & present your gifts.
First, to all of you who have come this far, I award you the Katabatak Black Label Of Action. This award recognizes your interest in persuing knowledge, & bravery in the face of a turtle. To those three of you who asked questions, I also award The Katabatak Red Label Of Reaction in recognition of your thinking outside of the box.
Now to the gifts for you all. You may have noticed by now I'm not above a little theatrics when it helps the cause along. Just so, if you remember when I first brought you into the Cave, I likened the Katabatak function to a UV light, and then to a label machine. Now these analogies have a distinct impression of "clunkiness" to them; who want's to carry around a clunky machine?
Without further ado, there next to you each, in that small wrapped box, find your new Katabatak Spectacles. Virtually weightless, virtually invisible, absolutely & completely indestructable. You see, the Katabatak function is more truly a filter for integers, & a simple lens suffices. The clunky analogies I issued you at first, were an artifice to make you work harder in order that you might attend this ceremony.
Go ahead & slip them on. Look around at some integers here in the cave. 137? Why that's just K(137) = 1+3+7=11 = 1+1 =2 There's 17, but with your glasses you see it's label 8 right off.
Well then; it's time to break up the party now & I leave you with this. Put on those spectacles & leave them on a week. Every integer you see this week, look at through your new Katabatak spectacles. Good job all & a fine good even.

[Turtle]

Now back to the exploration work. We resume with the table of squares & their K labels, & here we have some operational details to attend. As long as we keep our labels together with their progenator in a table, there is no ambiguity of the labels meaning. Working with the labels outside the table, we must adhere to using additional coding so as not to confuse K function output with non-output. An underline, whether curved or straight indicates output from the K function, as well as the color coding below.
In constructing the tables & tables of tables, I have adopted the convention of left to right & top to bottom. This is of course arbitrary & the Key below uses all four possible orientations of a table on a square lattice.
We now are not far from the original destination I promised when I brought you all in.

[Turtle]

Referring now to the list in post#21 which we descended, you see the top Katabatak Table is just an extension of the Table I drew in the dirt earlier. Descending another level, what logically else but the Katabatak Table of Cubes. And what logically else in accordance with our foundational principles, than to repeat everything we did to squares with the cubes. And as you see in the list of tables we reapply our reapplication on the ordered sets of Powers down to powers of ten.
While you consider that description, see below my tiling from the Katabatak Flexigon Of Base Ten Cubes. A further note here on using the flexigons; stiff cardboard works as well as the balsa wood I suggested. If you manage your flexigon into a shape that appears to tile, it may not truly tile. Strictly speaking, a geometric proof is required to affirm "tile-hood".
Onward & downward! See you soon.

[Turtle]

___You see how the topic expands, just as further exploration of a cave reveals more & more mysteries. A word here about the title of our little expedition, & my use of the word "Pure". By this I mean that there is no purpose to all this except exploration; math for math's sake. We look with open eye & no expectations except to find something new. Any applications or "practicality" to a discovery is secondary, but often such applications emerge. In post#23, the attachment illustrating the key, has the labels in the upper-left quadrant blank. Here you may attach your own label relations as Tormod is seeking to do with musical notation.
___Some of the relations I have considered for Katabatak Patterns: -polygon side length-tesselation design tool-weaving-music-multipliers for scaling-folds-knots-cellular automato inititial conditions-geometric vertices-encryption-turn sequences-determinants for unknown function-filter-efficiency model-cosmology-chemistry-graphs-trees-finite state machines-relations-combinatorics-sequences(arithmetic& geometric)-series-matrices-sieve...

Next post, a tiling from the Katabatak flexigon for powers of 4. Good even.

[Turtle]

Here now is the ordered list of Katabatak Patterns taken from the list we descended. The braces specifically group a Katabatak Pattern , & so the underlines are not necessary inside braces.

Kn^2 {1 4 9 7 7 9 4 1 9} You Say: The Katabatak Pattern of Powers of 2 is one four nine seven seven nine four one nine.

Kn^3 {1 8 9 1 8 9 1 8 9} You Say: The Katabatak Pattern of Powers of 3 is one eight nine one eight nine one eight nine.

Kn^4 {1 7 9 4 4 9 7 1 9} You say: The Katbatak Pattern of Powers of 4 is one seven nine four four nine seven one nine.

Kn^5 {1 5 9 7 2 9 4 8 9} What do you say?
Kn^6 {1 1 9 1 1 9 1 1 9}
Kn^7 {1 2 9 4 5 9 7 8 9}
Kn^8 {1 4 9 7 7 9 4 1 9}
Kn^9 {1 8 9 1 8 9 1 8 9}
Kn^10{ 1 7 94 4 9 7 1 9}

While you contemplate this, the flexigon tiling I promised.

[Turtle]

(Always keeping on a Base Ten path still)
Now here it is best to push on & not loose our momentum. If you will look at the K patterns I just delineated above, notice that each successive set has a different K pattern; each repeating itself without bound. Now also notice that when arriving at the K Pattern for n^8, it is not different, but rather it is identical to the K Pattern for Powers of 2; moreover, K Pattern N^9 is identical to n^3. & K Pattern n^10 is identical to n^4...
Extrapolate the pattern of patterns as far as you please; drop into that abyss with your rope & then what? This, I hope, you see is an unexpected turn of events; an emergent property! A pattern of patterns by which you may sum up all powers. Draw back from the edge & observe.

[Turtle walks near the edge of the abyss & facing it he draws in his head & half closing his eyes, he begins a sudden up & down motion that looks for all the world like jumping except that his feet aren't leaving the ground. With each downward move his plastron pats the cave floor & little whorls of dust burst out from the edges of his shell. But just as quickly he rises up and all the dust is drawn back in under him. Pat, pat, pat... at first, it just looks odd, but now the whole floor is thrumming to his odd dance, & now it is throbbing & in a sudden tumult the entire abyss of powers closes in & fills. All that remains in its place is a shallow depression with steps going down. Silence...four last pats, & Turtle has stopped. His head reemerges from shell, he opens his eyes, & takes a few steps away. Left on the cave floor beneath where he stood, an impression from his shell of the Base Ten katabatak Pattern Of Powers.]

Before I see you all off again, you will find we are now where we started. In the upper left corner of the first attachment, of the first post.

[Turtle]

I never tire of seeing that... In magic we call that collapsing the cave;in math, adducing the induction. We're just going to spend some time here for awhile & as we have formed our spooling thread into a loop, it's easy enough now to find the Way.
You may want to retrace the path & assure yourself we are on solid ground & have not been misdirected.
You may see now why I call this thing Katabatak, that which brings down. We have only scratched the surface. When I return, we will explore the Base Ten Katabatak Table of Powers Graph from post#1 in detail.

[Turtle]

___I invite you to look at the Base Ten Katabatak Table of Powers Graph from post#1 in detail. First, notice the form in general of the entire page; we say it is a square lattice, or square matrix, or square grid, or cellular graph. The choice of this lattice is arbitrary inasmuch as a triangular lattice, or hexagonal lattice, or many other polygons that tile the plane would suffice. Now I'm not taking you anywhere near those forms as they have a very disorienting effect, but you should know they are here.
Now another reason I have chosen a square graph is that far far ahead I will show you that our 9x6 Katabatak Graph (Pattern, Glyph, Induction, etc.) is simply a 2 dimensional view of a 3 dimensional stucture. This means our grid must not only tile the plane but must also tile space, ie. a regular polyhedron. Well, I digress.
___Back to our graph you see at the left top, the title, a statement of congruence, & an array index expression. Next down going left to right, the ordinal index for n to whatever power as n grows without bound.
___Now along the left side going down, strange looking incantations. Observe the first, K(n^(6m-4)). This is simply indicating what I showed you earlier, that begining with powers of 2, that as you skip count your way down the infinite path of powers, every 6th power has the same K pattern as squares. Insert m=1 for the first in the set, ie squares, insert m=2 to find powers of 8, insert m=3 for powers of 14, etc.
Well, enough for now to look at; enjoy.

[Turtle]

___I have stepped on a splinter & require some treatment, however we can quickly cover this little bit. The remaining feature of the graph sheet are the separations; you see them by color, which of course is a label for a Katabatak residue, which of course you may label musically, etc.
___Over each seperation, with cell centers as vertices, is constructed a complete connected graph. While I'm away, perhaps you'll try gazing at the graph in order to hear what it may sound like.
___Finally, if you care to observe the entire graph has no orange or blue elements & speculate on what this means.