Non-Figurate Numbers

Non-Figurate Numbers Rate Topic: 1 Votes

#1261 Turtle

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Posted 18 May 2012 - 09:50 PM

i have noticed that as i increase the number of sides, the spacing gets disproportionate which makes "seeing lines" a virtual exercise in futility. in the attached 1000x1000 13-gon, the cells near the origin overlap wheras those further away are spaced ok within a ring. in the attached 1000x1000 33-gon, the array looks pretty much like nested circles. not sure there's anything to be done about it; que sera sera. :turtle:

Attached thumbnail(s)

This post has been edited by Turtle: 18 May 2012 - 09:51 PM

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#1262 modest

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Posted Yesterday, 05:26 PM

Turtle, on 16 May 2012 - 06:43 PM, said:

your link takes me to the wrong page, so i'll quote the one that matches the url number, 317474.

If I recall there are post tags that should prevent that trouble, but I don't recall how to use them.

Turtle, on 16 May 2012 - 06:43 PM, said:

i have snipped a bunch and attached an edited version of the "centered" version to show the line of triangular numbers. ( i see now it's on the "overlapping sides" version too, but going right & up. :clue:

i can't imagine i saw this/these line(s) before & said nothing; prolly didn't spend enough time with the images. :kick:

anyway, if it's what you're trying to program, i say it looks promising. :thumbs_up

Yes, I am excited to see those work. The biggest issue may be trying to get lines like these,

Posted Image

to run straight. I laid that out by hand so they look a little wobbly, but if the size of the polygons were chosen specifically and correctly it looks to me like they could run in straight lines. An algorithm, especially if the size of the dots and the distance between them is variable, is one scary geometry proposition.

Turtle, on 16 May 2012 - 09:05 PM, said:

reporting back. :cap:

first, we may want to lighten the green of lone-odds as it looks black when zoomed out. they should stand out even against triplets because lone-odds are, well...lone. :lol:

Absolutely. ngon.zip

You can edit nGon.pl at these lines:

$orange = $im->colorAllocate(255,145,0);
$white = $im->colorAllocate(255,255,255);
$blue = $im->colorAllocate(0,0,255);
$black = $im->colorAllocate(0,0,0);
$red = $im->colorAllocate(255,0,0);
$darkblue = $im->colorAllocate(0,0,255);
$yellow = $im->colorAllocate(255,255,0);
$gray = $im->colorAllocate(145,145,145);
$lightgray = $im->colorAllocate(220,220,220);
$darkgreen = $im->colorAllocate(0,220,0);
$yellowgreen = $im->colorAllocate(186,255,0);
$lightblue = $im->colorAllocate(80,80,255);
$lightgreen = $im->colorAllocate(80,255,80);
$gold = $im->colorAllocate(240,198,0);

where the last three numbers are the rgb values (easy to get from photoshop's 'window -> color' window) for the color. To lighten the green I changed:

$darkgreen = $im->colorAllocate(0,109,0);

$darkgreen = $im->colorAllocate(0,220,0);

Turtle, on 16 May 2012 - 09:05 PM, said:

ok; to the 5-gonal edge-centered spiral array. as we agreed, the horizontal- bottom-line-extending right is the set of 5-gonal numbers. in my detail drawing i have joined them with yellow-dashed line.

5-gonal: (3n^2-n)/2 for n={3,4,5...}

12 1 [3, 5]
22 1 [4, 5]
35 1 [5, 5]
51 2 [3, 18, 6, 5]
70 2 [4, 13, 7, 5]
92 1 [8, 5]
...

then, starting from 9 in layer 2, a line of all-polygonals; joined by green-dashed line in detail. these are all n=3 as s advances by {3, 4, 5...} starting from 4.

9 1 [3, 4]
18 1 [3, 7]
30 1 [3, 11]
45 3 [3, 16, 5, 6, 9, 3]
63 1 [3, 22]
84 1 [3, 29]
...

then just above & parallel to that, another all-polygonal line, joined by pink-dashed line in the detail. these are n=3 as s advances by the 3-sided numbers starting with 10.

27 1 [3, 10]
42 1 [3, 15]
60 1 [3, 21]
81 3 [3, 28, 6, 7, 9, 4]
...

there may be other's but the dark-green may be confusing my eye.

attached are the 5-sided 500x500 program generated arrays coded for poly/non-poly and even/odd, and my clip & edit from the standard coloring. . :clue:

Interesting. So, for the green dashed line let's say it would be...

$P= \frac{1}{2}(n^2s-2n^2-ns+4n)$

where n = 3

$P= \frac{1}{2}((3)^2s-2(3)^2-(3)s+4(3))$

and simplified a bit,

$P= \frac{3}{2}(2s-2)$

where s goes 4,7,11,

...ahh!!... Annoying phone call will take forever. I'm hitting post. Will be back tonight. Haven't proofed the incomplete math above.

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#1263 modest

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Posted Yesterday, 08:27 PM

Back. Took care of the phone with the table saw.

modest, on 21 May 2012 - 05:26 PM, said:

Yes, I am excited to see those work. The biggest issue may be trying to get lines like these,

Posted Image

to run straight.

Do you think they should run straight? I'm wondering if that thought may have been a bit silly.

modest, on 21 May 2012 - 05:26 PM, said:

So, for the green dashed line let's say it would be...

$P= \frac{1}{2}(n^2s-2n^2-ns+4n)$

where n = 3

$P= \frac{1}{2}((3)^2s-2(3)^2-(3)s+4(3))$

and simplified a bit,

$P= \frac{3}{2}(2s-2)$

where s goes 4,7,11,

Ok, I think we could have simplified a little further...

$P= 3(s-1)$

for s = [4,7,11,16,22,29...]

So, an expression that gives [4,7,11,16...] would be... uh...

$s = \frac{1}{2}t^2+\frac{1}{2}t+1$

that gives

$P= 3( \left[ \frac{1}{2}t^2+\frac{1}{2}t+1 \right] -1)$

and that simplifies to...

$P= \frac{3}{2}(t^2+t)$

And that hopefully should give the green-dashed line sequence, which was, 9,18,30,45,63... for t > 1 I guess we'd have to say. Yikes.

Reading on...

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#1264 modest

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Posted Yesterday, 08:44 PM

Turtle, on 18 May 2012 - 09:50 PM, said:

Yeah, I didn't consider that. I think you're right—not sure what could be done. What dot size are you using by the way? Yours are rendering very smooth.