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Author Topic: a multiplication table based on walking along a triangle  (Read 1440 times)
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kram1032
Fractal Senior
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Posts: 1863


« Reply #15 on: February 08, 2017, 03:01:57 AM »

I understand if you are discouraged by these results and rather don't want to spend your time exploring these further. That's fine.
Nevertheless, let me try to address your concerns about the particular form these formulas take.

The all-positive nature of these is correct. It's a consequence of how these tables arise. It's more of an extension of split-complex numbers than complex ones:

(a1 + aj j)(b1 + bj j) = a1 b1 + aj bj + j (a1 bj + aj b1) - this, to my knowledge, doesn't give a circle or egg, but rather, which of course is also boring, a square. Its Buddhabrot take looks somewhat interesting though. - all the orbits are along 45░ angles of the axis but the densities vary in non-trivial ways.
I didn't expect super high detail interesting results from these but simply an "egg", and for all three of these no less, is surprising to me.

I'm not entirely sure if there is an obvious natural way to introduce negatives. I mean you could just randomly flip signs and define i▓ = -1 and/or k▓ = -1 or something but then the geometric meaning would be lost.
As for the lack of "cycling", I'm not quite sure what you mean by that, but the two that correspond to an edge pair that's open on one end certainly have a degeneracy to them:
The k-move turns this:
Code:
--.<-
into this:
Code:
->.--
and vice-versa, but if it's pointing outside:
Code:
<-.--
it will go around all the way and end up in the same position so then k does nothing.
The i-move, meanwhile, always works:
Code:
<-.--
turns into this:
Code:
->.--
and after that you can use the k-move as above.
Since k sometimes changes things and sometimes does not, that results in the degeneracies evident in the multiplication table / formula.
This is also why "Edges outfacing", i.e. calling either one of the following the "1-position",
Code:
<-.--
or
Code:
--.->
doesn't have k as its basis, but rather mixed moves like iki which does this:
Code:
<-.--
turns into
Code:
--.->
and vice versa.
(Basically, k rotates around the tip of a vector whereas iki turns around the shaft)

Putting numbers onto this is perhaps kind of weird but I was hoping something more interesting would happen. In principle, any group should be able to be made into this. For instance, here is a group with two elements, the integers modulo 2: (in the following, bold numbers aren't the same as your normal numbers. They will rather be the labels of the individual directions in a given system.)

0+0 = 0
0+1 = 1
1+0 = 1
1+1 = 0

aka the operation or.
This is slightly confusing in that the neutral element of addition is 0 and thus the neutral dimension that is equivalent to a scalar also is the one labeled 0. The corresponding multiplication table is:
01
10

so then you'd get a 2-dimensional system:
(a0+a1 1)(b0+b1 1) = (a0 b0 + a1 b1) 0 + (a0 b1 + a1 b0) 1
which, of course, is just the same as the split-complex numbers:
0 = 1
1 = j

There is also the and table, or multiplication over the integers modulo 2:
0*0 = 0
0*1 = 0
1*0 = 0
1*1 = 1

which actually can't quite be written in the same form as the other table. Gotta expand it to include a head column/row:
*01
000
101

with would, if used in the analogous way, give an incredibly degenerate number system:
(a0+a1 1)(b0+b1 1) = (a0 b0 + a0 b1 + a1 b0) 0 + (a1 b1) 1
which is just silly: In such a system, what ever is in the 1-direction, stays in the 1-direction.
I think this one isn't actually a group though: You cannot reach 1 from 0 so 0*1 does not have an inverse which would be a requirement for a group.

All these groups are not originally about numbers and so there is no reason for them to ever "be negative" as it were, so they will inherently never "accidentally" mimic the complex numbers. There might be something to change that though. Adding in a negative term is bound to make things more interesting seeing as to how j▓=1 is the least interesting of the three major cases (the other two being i▓=-1 and ε▓=0 with (a1 + aε ε)(b1 + bε ε) = a1 b1 + (a1 bε + aε b1) ε and the aε bε term just vanishing), but I wonder if there is a way to do this "reasonably mathematically" rather than by throwing it in there randomly.
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clacker
Forums Freshman
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Posts: 16


« Reply #16 on: April 06, 2017, 01:41:51 AM »

I ran the Fragemtarium code claude posted and it did give me an image.  Using the default settings but changing iterations to 100 I got an image that looks something like a clipped mandelbrot.  I am running Fragemtarium version 1.0.0 ("Cologne") on windows 10.  I can't say if that code is what kram1032 was asking for, but it did render an image.


* triangular-mandelbrot-2d.png (59.66 KB, 640x480 - viewed 147 times.)
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kram1032
Fractal Senior
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Posts: 1863


« Reply #17 on: April 06, 2017, 02:08:51 AM »

huh, that's weird and interesting. Where did claude link actual code though? I only saw the link to Groups of Order 6?
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clacker
Forums Freshman
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Posts: 16


« Reply #18 on: April 06, 2017, 02:13:36 AM »

it's at the bottom of that post:

http://www.fractalforums.com/fragmentarium/a-multiplication-table-based-on-walking-along-a-triangle/msg98920/#msg98920

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kram1032
Fractal Senior
******
Posts: 1863


« Reply #19 on: April 06, 2017, 09:20:01 AM »

ah, right, I'm just blind
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clacker
Forums Freshman
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Posts: 16


« Reply #20 on: April 08, 2017, 06:38:09 PM »

I collected all of the slices above the diagonal.  The slices on the diagonal are just diagonal bands, and the ones below the diagonal are 90 degree rotations of those above.  I ran 1000 iterations and changed the escape radius to 10.  I also added a cross hair at 0,0.  Slices 0-3, 0-4, and 3-4 look very much like Mandelbrot sets to me.


* triangular slices.png (105.98 KB, 1500x1500 - viewed 166 times.)
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kram1032
Fractal Senior
******
Posts: 1863


« Reply #21 on: April 08, 2017, 06:46:28 PM »

The slices 03 04 and 34 look the most interesting to me. Though they are "just" weirdly cut off MSets. Can you try zooming in a bit into a bunch of spots? It'll probably just be more of the same seemingly incomplete, sharply cut MSet zooms, but maybe something interesting is hiding down there...
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clacker
Forums Freshman
**
Posts: 16


« Reply #22 on: April 09, 2017, 03:17:25 AM »

Here is a short low resolution animation zooming in on the 0-3 slice.  I had to really narrow down the size and number of frames down to fit within the forums maximum file size allowed.  It really looks like a sheared Mandelbrot to me.

* 0-3 animation.avi (237.53 KB - downloaded 85 times.)
« Last Edit: April 09, 2017, 03:24:06 AM by clacker, Reason: typo: "narrow" != "know" » Logged
kram1032
Fractal Senior
******
Posts: 1863


« Reply #23 on: April 09, 2017, 11:18:35 AM »

Yeah, pretty normal-looking... thanks
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clacker
Forums Freshman
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Posts: 16


« Reply #24 on: April 10, 2017, 02:10:04 AM »

I looked at a 3D slice through dimensions 0, 3, and 4 and got an image that looked like a Mandelbrot that had been tilted and then extruded.  I can see why the slices look the way the do know.  It surprises me that the ends are flat.  There are no subtractions in the math.  I can't wrap my head around why the numbers don't constantly expand.


* triangle-3d.jpg (199.88 KB, 1000x1000 - viewed 166 times.)
* triangular-mandelbrot-3d.frag (3 KB - downloaded 79 times.)
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kram1032
Fractal Senior
******
Posts: 1863


« Reply #25 on: April 10, 2017, 12:39:40 PM »

So, at least in these three dimensions, it's simply an extruded/tilted MSet. That's weird. There ought to be some explanation for that. Something like a fairly straight forward transform to complex numbers.

There are a total of 20 such possible 3D slices. I don't expect all of them to be interesting, and some of them will be the same or just mirrored/rotated, judging by your 2D slices though.
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