Katabatak Math-An Exploration In Pure Number Theory

[Turtle]

___You have stood me on my head CraigD; again. I knew asking this time before saying might pop something up. To whit, I made the list by taking the last digit of each bast ten power; I did not see a single tree for the forest. It is non-trivialy trivial (or visa versa) that the last digit of any power written in base b notation is n^m modulo b.

___So how does this reflect perhaps CraigD on your list in post #89? I went there to see the tree for myself.

1: 3
2: 4 5 7 9 13 25
4: 6 11 16 17 21 31 41 49 61 81 121 241
6: 8 10 15 19 22 29 37 43 57 64 73 85 127 169 253 505
8: 33 97 161 481...

___I bolded the entries, modularity 4, base eleven, & also modularity 6 base ten. Does this draw our attention to the trees of mystery here? Don't think I know, just think I think.

[CraigD]

Originally Posted by Turtle

___You have stood me on my head CraigD; again. I knew asking this time before saying might pop something up. To whit, I made the list by taking the last digit of each bast ten power; I did not see a single tree for the forest. It is non-trivialy trivial (or visa versa) that the last digit of any power written in base b notation is n^m modulo b.

There is indeed a strong connection between how modern (sometime after 400BC to 400AD in India, shortly after in Persia, but not ‘til around the time of Fibonacci ca. 1200AD in the Europe) people represent numbers, and modular arithmetic – the modern “Arabic” positional number representation of a counting (Natural) number is actually an arrays/lists/string of modular numbers of the form {n modulo b^m, n modulo b^(m-1),… n modulo b^0}

The difference between the representation of a number and the number itself is philosophically profound, especially for people like me (and more obscure folk like Kurt Godel and Doug Hofstadter ) who embrace the idea of formalism as the proposition that all Math can be described as a collection of rules for manipulating the representation of numbers and arithmetic operations on them. This particular rabbit hole is very deep, so by your leave, I’ll step back from it now.

___So how does this reflect perhaps CraigD on your list in post #89? I went there to see the tree for myself.

1: 3
2: 4 5 7 9 13 25
4: 6 11 16 17 21 31 41 49 61 81 121 241
6: 8 10 15 19 22 29 37 43 57 64 73 85 127 169 253 505
8: 33 97 161 481...

___I bolded the entries, modularity 4, base eleven, & also modularity 6 base ten. Does this draw our attention to the trees of mystery here? Don't think I know, just think I think.

I don’t believe there’s a strong connection between my 2 posts. The table in post #89 seems to result is some fashion I’ve yet to fathom from the Prime Number Theorem, where #114 and this one have more to do with the systems for representing numbers.

I’m failing to follow the significance of the bolded numbers in your post. Can you explain, please?

[Turtle]

Originally Posted by CraigD

The difference between the representation of a number and the number itself is philosophically profound, especially for people like me (and more obscure folk like Kurt Godel and Doug Hofstadter ...This particular rabbit hole is very deep, so by your leave, I’ll step back from it now.I don’t believe there’s a strong connection between my 2 posts. The table in post #89 seems to result is some fashion I’ve yet to fathom from the Prime Number Theorem, where #114 and this one have more to do with the systems for representing numbers.
I’m failing to follow the significance of the bolded numbers in your post. Can you explain, please?

___I call Godel's theorem "Kurt's Hammer". You may recall I (Turtle) played a small role in Dougy's Metamagical Themas opposite Hercules in a number of expository dialogues. ; we helped him build a ladder for getting back out of that rabbit hole that Mr. Dodgson helped us into. I love taking the little potion & going down deep, where it opens into this amazing cave of mathematics.

____On the bolded entries:
1: 3
2: 4 5 7 9 13 25
4: 6 11 16 17 21 31 41 49 61 81 121 241
6: 8 10 15 19 22 29 37 43 57 64 73 85 127 169 253 505
8: 33 97 161 481...
___Your list is predicated on Katabatak function transforms of powers, & shows in the left column how many different K patterns occur before they begin to repeat (your "modularity"). In line to the right of each of these, is the list of bases which generate that particular modularity. From the Katabatak patterns of powers base 10, we have modularity 6, but if you take that same list of base ten powers (the full base ten "numbers" we input into the K function) & put their last digits into an array, you end up with a 10 by 4 table which is congruent to your modularity 4 base eleven entry.
___My point then is that from any base numeration system & a given number n (numeral string) you derive from the last digit of n the value of n mod b & from the K(n) you derive the value of n mod (b-1). This means the join of these sets of values has a length of b*(b-1). For example, the K pattern of squares is length 9 & the end-digit pattern is length 10, so the pattern of their unique association does not begin a repeat until after the 90'th element.
___Hope this doesn't throw anyone unwilling into the rabbit hole.

[Turtle]

___I have broached this view a couple of times in regard to constructing 3-dimensional Katabatak graphs. I left the topic intact somewhere above when I just hinted at it, & I deleted a number of posts that addressed it directly. Essentially I thought I introduced it too early.
___Now we have it back & it is timely so I intend to put up the single 3-d graph I have constructed. It is the base 6 Katabatak multiplication table, & as laid out above the ending digits repeat every 6 elements whereas the K transforms repeat every 5. This gives a 3-d volumetric graph that is (5*6)*(5*6)*6 =5400 cells. Of these, only 900 cells have entries (or non-empty, if you like).
___Constructing these 3-d Katabatak graphs is considerably difficult with the resources I have, as well as making for very large images that offer considerable visual difficulty in comprehending. Nonetheless, they offer some interesting insights into our exploration & their algebraic representations offer some facility for analysis.
___

[Turtle]

___More irons in the fire than you can shake a stick at! Just so, a fresh view of Katabatak multiplication is cooling in the Science Gallery; Katabatak multiplication brands (graphs) on a regular triangular matrix, bases two through seven. [pencil & color pencil on printed traingular grid)

[Turtle]

___Breaking camp for some serious spelunking.
___In re-exploring the square tables triangularly, I focussed on the cellular view of the matrices rather than an intersectional view. Taking this intersectional view today, the square matrix gives 2 lines crossing at every intersection; the familiar Cartesion co-ordinates. The triangular matrix however, has 3 lines at every intersection & is almost completely unfamiliar.
___I noticed that 67 explorers have made copies of my drawing of boxes in the dirt at the very start of this expedition, & so I intend to scrape a similar terra-graph soon that demonstrates how the unfamiliar view maps the familiar data.
___In the mean time, compare the base seven Katabatak multiplication square table here::

with the base seven Katabatak multiplication triangular table attached:
:cup

[Turtle]

___See here in the dirt I have drawn our cellular matrices, one square & one triangular.
___I point out again our bias to square thinking inasmuch as "square" is a special name for a regular quadrilateral & a regular triangle has no such special name. A shame in light of the fact that the triangle is the only polygon which if equiangular is also equilateral. I won't assume to coin a name for the regular triangle at this time, but when I refer to a "triangle" here, I mean "regular triangle" unless otherwise stated.

[Turtle]

___Perhaps this description isn't as difficult a belay as I make it, but I don't care for any slips in this passage. Without adding notation to the map yet , it already has some specifics of note. Each matrix has an orientation which places a side/face on a horizontal line & so we have top & bottom. Lines parallel to the horizon line divide both matrices into 5 "layers". After some more coffee I plan to start adding the map legend.

[Turtle]

___I have added dashed green lines to the drawing to indicate the horizontal layering:
___We must hasten slowly. Kaffee bitte?

[Turtle]

___The next addition to the drawing indicates right-descending layering in the regular triangular matrix; shown in red dashed lines

http://hypography.com/gallery/showimage.php?i=476&c=3