Katabatak Math-An Exploration In Pure Number Theory

[Racoon]

Go on Turtle-san!

 

We've Known for years that your work is indeed special! :Waldo:

 

You should let it be published in scholarly mathmatic journal,

And hopefully, let the accolades you so richly deserve come rolling in! :Waldo:

 

A Big Fan of Turtles Math, :Waldo:

 

Racoon

[TheBigDog]

I feel like the kid in "The Sixth Sense", only instead of dead people, I see triangles. I have had triangles in my head for tha past few weeks and have spent a few hours playing with the charts in my Excel file today. And I am back to report more of my findings....

 

I have identified 2 of the curves on the chart, although I am struggling to identify the formulas (algabra) for describing the underlying function of each curve. I so tired right now I don't even know if that is a correct statement.

 

To review...

 

The players here so far are CraigD, Pyrotec, Turtle and myself...

 

We are playing with PIRTs - Perfect Integer Right Triangles. The defining characteristic of a PIRT is that all three legs are integers, and the legs do not share any prime factors. Hypothetically, 2 of the three legs could share a common prime factor and it would still be a PIRT, but for all the ones I have examined this never happens. a = short side, b = long side, c = hypotenuse. i = b-a, j = c-b, m = c-a.

 

When the PIRTs are charted with one axis for a and the other axis for b, they show patterns of interlinking arcs.

 

In the attached chart the a values go across, and the b values go up. If you drew a line from each point to the bottom left corner it would be the hypotenuse of a PIRT. These are all unique triangles, they are not multiples of smaller trianges, such as 6,8,10 from 3,4,5.

 

I have drawn some lines to hi-lite some of the features. The purple line shows the j curves. All points along each of these curves have the same value of j. The magenta line is the m curve. All points along each of these curves have the same vaule of m. Every point is an intersection of a j curve and an m curve, but not all intersections result in a point - only where the intersection happens at integer values.

 

The light blue and green lines are the other two major curves on the chart. I have not yet identified the common characteristic of these curves in my data. I am wondering if they might be illusion, like when driving next to a corn field and looking into the rows of corn at various angles and seeing paths. This may be, but I suspect that there is math to be found in these... The orange dots illustrate the locations of some intersections. These are the centers of Pyrotec's spiderwebs.

 

The math is fascinating. It all starts with Pythagoras - a^2 + b^2 = c^2. In the posts by CraigD you will see the algebra that breaks down into the values i and j, and lead to the computer program that generated the list of PIRTs. The acronym PIRT came from Pyrotec who had years earlier done a similar investigation for a time. He had come up with similar algebra as CraigD, and had identified the value i as Delta in his work.

 

There are some patterns to be seen in the values of j, i and m.

 

(c-b = j) has a distinct list of values that it contains. 1, 2, 8, 9, 18, 25, 32, 49, 50... And as first identified by Pyrotec these values fall into two supersets. The first being the square of all odd integers - 1, 9, 25, 49, 81, 121... And the second being the twice the square of all integers - 2, 8, 18, 32, 50, 72, 98... (c-a = m) has exaclty the same set of values - except for 1.

 

(b-a = i) has a very bizzar set of values. They are 1, 7, 17, 23, 31, 41, 47, 49, 71, 73, 79, 89... As prime numbers appear on this list they can become factors for non-prime numbers that appear later on the list. But no number on this list is factorable by any prime number not on the list. So 2, 3, 5, 11, 13, 19 are never factors of numbers on this list. If you look at a chart of triangles with the same value of i they draw 45 degree angle lines moving away from the bottom left corner. These are not easy to see on the chart because the points are widely scattered and seemingly random.

 

On every j curve, every point has a different i value. Same on every m curve. There is also weirdness along each j and m curve, with varying repeating patterns in the spacing of the dots on each curve.

 

Another weird pattern emerges when you look at the number of PIRTs with a common value of a. In the lower numbers most commonly there is only one PIRT for each a. As the numbers get bigger the distribution changes. In the first 10000 values of a the distribution is...

 

Num Rep

1 1642

2 3053

3 936

4 1457

5 2

6 86

7 254

8 61

13 3

14 4

 

This shows that there are 1642 a values with a single PIRT, there are 3053 with 2 PIRTs, etc. A very strange distribution indeed.

 

I had suspected for a while that no PIRTs shared the same value for the sum of their sides. But this turned out to be false. In the first 19271 PIRTs there is 1 sum of sides shared by 3 PIRTS, 151 shared by 2 PIRTs, and all the rest are unique.

 

I have also started looking at the k(n) of PIRTs, although only cursoraly, and have already found repeating patterns in various directions there.

 

This leaves us with a few strange paths yet to follow. CraigD is looking for a single formula or algorythm that predicts all PIRTs(and writing software that is worlds faster than mine!). Pyrotec is looking, among other things, at the weirdness of point spacings in the various curves. Turtle has been inside his shell with a flashlight and graph paper, and occationally poking out his head and smiling like a kid on Christmas morning. I suspect that he will be finding k(n) information and reporting that to us at very soon. And I am trying to find the commonalities in the light blue and green curves to see what causes the spiderweb effect, and I am toying with making an interactive analysis tool. And as dry as this may seem to the non-three sided, this is really great fun!

 

Bill

[TheBigDog]

GAAA!!!!! I just spent an hour typing a post and it errored when submitting! And it was pure genious! Probably a Nobel Prize winner! And now its gone. Dammit! I'll try again tomorrow. Good night.

 

Bill

[TheBigDog]

Never mind. Now its there. Don't tell me why, then we both would not understand.

 

Bill

[Pyrotex]

Never mind. Now its there. Don't tell me why, then we both would not understand.Bill

I have nominated you for the Pulitzer Award. We'll see! :hihi:

[TheBigDog]

Howdy Turtle. I was helping #3 with his homework tonight. On a lark I showed him Katabatak and we logged in and looked at your artwork. This stuff is right up his alley. He is quite fascinated by the colors and patterns, and while he claimed that it is all very boring, the glow in his eyes does not lie. I am going to be escorting him through some of the previously explored territory, and see if I can get him to take on the art too. #3 is the one holding my hands in the avatar. He is 9 now.

 

Bill

 

ps. He first noticed that Katabatak is a palendrome (although he didn't know the word)

[TheBigDog]

Back to pure Katabatak in regard to PIRTs...

 

If you look at the k(n) for the sum of the sides of any set with the same value i, they start out with a pattern. But the pattern erodes after a few iterations. The same appears true for k(n) for any of the individual sides. The pattern is there for the first few iterations, then stops. I have not yet determined if it becomes completely random, or if it moves into larger patterns that I have not yet recognized. More to come...

 

Bill

[TheBigDog]

1) Do I understand that PIRTS are not all the Pythagorean triples?

If not, I think we may need a list of all such ordered triples for a proper Katabatak purvue.

You are correct. 3x4x5 is a PIRT. 6x8x10 is not a PIRT. Do you want a file with all triples, and not just the PIRTs? I can make one. I will still base it on all values of 'a' up to 10000.

 

Let me know.

 

Bill

[TheBigDog]

2) Any time no Katabatak pattern is apparent in a particular base, it is time to try more bases. Since we have infinite choices to do that, I usually start with base two & incrementally go up from there to some arbitrary practical limit.

Getting out of base 10 is goig to take some adaptation of thinking on my part. As is typical of me I am going to try and build a computer gadget to do the heavy lifting for me. I'll let you know how it turns out. I may be askin the guide for guidance in such matters. I will limit myself to 1 question per day in any base.

 

Bill

[TheBigDog]

I am encouraging #3 to join us here in the Katabatak cave. This is right up his alley. I was thinking of making some color by numbers grids for him so he could solve the k(n) for each grid and then color them in. Wouldn't every kid want a dad like me?

 

Bill

[Queso]

yawwwwwn.

I want to incorporate math into my drawings.

I've been working with Mobius strips lately,

Can either of you point me in a direction?

i'm a mad artist capable of the abstract and dynamic.

[Racoon]

solve the k(n)

Wouldn't every kid want a dad like me?

 

Bill

 

Its a good thing kids don't know any better. :hyper:

[Queso]

___Wrong!:steering: Stop the presses...er....stop the photons. Katabatak graphs do not tile a mobious strip; my bad.:hyper:

 

Yeah I was wondering how the hell you were tying those two together!

[TheBigDog]

Staying at camp

the order of the day,

nothing too strenuous

to hinder our way.

 

Maybe a short list of say just the first 40 triples? Most Katabatak patterns occur in harmonics of one-less-than whatever base you express them in. A list of 40 triples in base ten covers a harmonic of four & gives an overview easily taken in at a glance.

:hyper: :gift: :steering:

Here are the first 40....

 

Bill

[anglepose]

can you tell me where i mght find out about the basics of katabatak well i mean like the easiest stuff there is bout katabatak.

ive read all of your work on it and its amazing but lol i dont get a single bit of it

[anglepose]

thnanks ill spend a few days trying it out and working out some of my own problmes

[anglepose]

wow ive seen a hole in the cave light is shining down i can see whats happening ive stopped tripping over rocks and can now identify them

:rolleyes: