Ulam Spirals

Gaussian White Noise Wave Functions

2010-05-23T19:56:00.001-07:00

Various examples of ulam spirals using a continuous tube to plot the points in conjunction with a white noise wave function to plot the z-coordinate. Generator written in Python using the VPython library.

Some other shapes using the gaussian white noise to determine dimensions of plot objects.

Spiral Video Series

2009-04-06T14:21:00.000-07:00

Using smplayer/mencoder to convert PNG output to video. Formula and other control info is automatically inserted into the beginning of the video. This set is using a 2x2 pixel plotted out using the Ulam spiral "RULD" method. Pixel is black or red if the pixel number (pInt) passes the following test (e.g.): if round(((pInt)*pi)*pow(sqrt(pInt)/pi+(_multiplier/9), (1.2)) )%2==0: Iterating the value of _multiplier (or any other variable), we generate a new PNG file when the _range value is met. Stitching these images together into a video reveals motion within the overall pattern. For example, incrementing _multiplier very slightly for each image creates less disruption from one image to the next. Frame rate varies, generally within 2 to 15 FPS. Too bad blogger is overcompressing these, they are quite nice before they hit the FLV codec. C'est la vie.

Here's one of the first videos I cobbled together, a very quick prototype...

Random Grids

2009-04-02T11:01:00.000-07:00

Detail:

I was bored today so I modified the base routine to select a cardinal direction randomly. These images are all built pulling from a list of 1,296 possible 2x2 pixel images. The original image is...

Marilyn Hex

2009-03-12T15:54:00.000-07:00

Jaco Pastorius, Meet Stanislaw Ulam

2008-11-01T22:08:00.000-07:00

These are all using a very simple test for n, round( pow(pInt,sqrt(1))/pi )%2==0

Enter Jeff Buckley

...and Jimi

Respliced Satellite Maps

2008-10-31T15:04:00.000-07:00

Some tinkering with PIL and Google Maps produced these cartographic miscarriages:

And yes, we're still doing this with basic Ulam spiral algorithm.

Test: round( pow(sqrt(pInt)/e, 1.2) )%3==0

test: round( pow(sqrt(pInt)/e, 1.2) )%5==0

test: round( pInt/e )%3==0

Land-o-Lakes

Borrowing from Myka's number testing algorithm, I've introduced the x and y coordinates into the mix. The test round((pi*pX)*(pY)*pInt)%2==0
appears in Myka's Square series. Thanks for the math, mofo. Of course, the resulting imagery differs greatly. My functions seem to lack the additional voodoo required to create the interference patterns.

And then I studied the Myka algorithms a bit more carefully. Behold:

This is using round( (pi*pX)*(pY)*.0025) % 2==0, producing the same pattern as this Myka square.

But here we use 2500 unique 4x4 pixel images to iteratively construct the same pattern:

This is the core image that was used to create the above spiral:

This is not to say we simple overlay the image into the positive shape of the spiral. The core image is "ripped" into small square pieces, and a list of these images is referenced in as we step around the center point. The result is a color palette borrowed from the image.

Distilling the Lake Water

round( (pi*pX)*(pY)*(distillNumber(pInt)*.00025))%2==0

Pulling back on the stick we fly higher. This is at altitude .0000025 where round( (pi*pX)*(pY)*(distillNumber(pInt)*.0000025))%2==0

Higher:
round( (pi*pX)*(pY)*(distillNumber(pInt)*.00000025))%2==0

And yet higher still. Using .000000025 is absolutely rediculous, but it works.
round( (pi*pX)*(pY)*(distillNumber(pInt)*.000000025))%2==0

It appears to be a recursive fractal image, for the same pattern is re-emerging as we go deeper into the mathematical abyss.

A curious blend of pi and e...
round( ((pi*e)*pX)*(pY)*(distillNumber(pInt)* (sin(pInt)*.000000025) ))%2==0

round( (pX*pY)* pow(sqrt(pInt)/e, 1.2)*.0000025 )%3==0

Lord have mercy.
round( (pX*pY)* pow(sqrt(pInt)/e, 1.2)*.00000025 )%2==0

round( (pX*pY)* pow(sqrt(pInt)/e, 1.2)*.000000025 )%2==0

round( (pX*pY)* pow(sqrt(pInt)/e, 1.2)*.0000000125 )%2==0

round( (pX*pY)* pow(sqrt(pInt)/pi, 1.23)*.000000125 )%5==0

The Aboriginal Series

round( (pX*pY)* pow(sqrt(pInt)/pi, 1.23456789)*.000123456789 )%5==0

round( (pX*pY)* pow(sqrt(pInt)/pi, 1.23456789)*.0000123456789 )%5==0

round( (pX*pY)* pow(sqrt(pInt)/pi, 1.23456789)*.00005 )%3==0

Colorized version:

I'm pretty sure this overall pattern is a map of the universe, reality, and everything.

round( (pX*pY)* pow( sqrt(pInt), 1.23456789)*.00005 )%3==0

Image-Based Spiral Incantations

2008-10-30T20:24:00.001-07:00

Exchange Building, Seattle, Washington.

The goldfinch was placed dynamically by the Python script. You can see that some of the tiles overlay the bird. The tiles are composed of a Letraset sheet for Helvetica and were dynamically ripped into 40 pixel squares before being rearranged using the Ulam algorithm.

This is basically a brand new alphabet for crazy people. Here you go. It is my gift to the world.

Feeling strange?

Strange chains...

Typography and Pop Art

2008-10-30T18:55:00.000-07:00

I love typography. It seems a natural thing to combine the beautiful Persian rug spiral with type. In the image above, a small letter "a" set in Helvetica Bold Condensed is substituted for each number.
[click images to enlarge]

Same idea, only with xes.

Where to go from here? The natural progression is obvious and swift. Behold:

Make sure you click open that puppy and count how many "full Marilyns" you see. I count 3 in the current image. Of course, the number of full Marilyns will increase as the perimeter of the spiral increases. It is a simple calculation. But the simple man will only require her lips, for these are enough to tell the entire story.

I guess I should stop now. It's not likely to get any better than this.

"Classic" 2d Spirals

2008-10-29T14:57:00.000-07:00

[The Squirrel]

Well, hey, getting an algorithm to generate the Manhattan skyline with 1 billion red balloons floating overhead like some nightmarish Macy's parade is certainly fun, and a splendid way to waste away a Friday night. But I prefer the basic red-on-black, 2d, no bullshit Sunday-school algorithms. To generate your own squirrel, use this here (Python) code when you test each number:


#global
fooCounter = 1

def testNumber(pInt):
    global fooCounter

    if round(  (sqrt(pInt)/pi)+log(fooCounter) )%2 == distillNumber(pInt):
      fooCounter += 1
    return true
  return false

And if you forgot (or still care) what distillNumber does, here it is:


"""
recursive function for boiling down multi-digit ints into single digit ints
"""
def distillNumber(rawNumber):
    distilledNumber = 0

    tmpList = split_len(str(rawNumber))

    for k in range (len(tmpList)):
        distilledNumber = distilledNumber + int(tmpList[k])

    if len(str(distilledNumber))>1:
        distilledNumber = distillNumber(distilledNumber)

    return distilledNumber


def split_len(seq):
    return [seq[i:i+1] for i in range(0, len(seq), 1)]

Let's see what happens when we test for round( (sqrt(pInt)/pi)+fooCounter )%2 == 0...

..or, swap out pi with e round( (sqrt(pInt)/e)+fooCounter )%2 == 0

Now if you kids have been following along at home you know that the goal of all this is to create the ultimate Persian rug design. That's it. I think we've already established said design, but this one isn't too bad. It's quite simply

round(  (sqrt(pInt)/e)+ pow(cos(fooCounter),2)  )%distillNumber(pInt) == 0 or round(  (sqrt(pInt)/e)+ pow(cos(fooCounter),3)  )%distillNumber(pInt) == 0

Yes! That's really all there is to it!

Check out those dangly fish-hooky spiral bits. It is the metamorphosis!!!


round(  (sqrt(pInt)/e)+ pow(cos(fooCounter),2)  )%distillNumber(pInt) == 0 or round(  (sqrt(pInt)/pi)+ pow(cos(fooCounter),2)  )%distillNumber(pInt) == 0

Now if you're a Persian rug kind of guy, you know, and you're looking to expand beyond the run of the mill patterns that have been standard for thousands of years, look no further. Let's work together and hot rod your loom. You could be doing something like this:

Notice how almost all letters of the Thai alphabet are presented. It's just eerie. อักษรไทย อักษรไทย อักษรไทย


#thai alphabet
round(  (sqrt(pInt)/pi)+ pow(sin(fooCounter),2)  )%distillNumber(pInt) == 0

Here's another simple method to mess things up good:


def modifyCoordPoint(pNum):
    return pNum + cos(pNum)

You call that from the main algorithm after you've set up your new X/Y positions. Pass in X or Y accordingly.

A lovely variation of this is accomplished by introducing our little circular friend:


def modifyCoordPoint(pNum):
    return pNum + (cos(pNum)*pi)

3D Spiral Interpretations Via Pointed Hat Operator

2008-10-23T19:19:00.000-07:00

For these images we're testing each number with:


if round(pInt/pi)%2==0^pInt%5==true:
return true

...and the space between dots is determined by:
distillNumber(pInt)*pow(sin(pInt),9)*pi

Red Light District As Seen From Above


distillNumber(pInt)*pow(sin(pInt),5)*pi

Red Nimbus:

Red Balloons Over Manhattan

Spacer: (sin(pInt)*100) + distillNumber(pInt)/pi

Old Ulam spirals yield new discovery

2008-10-21T22:49:00.000-07:00

Hidden information within 2D patterns?

The following pattern yields yet another familiar pattern when each pixel is elevated according to its derived value. Behold:

Base pattern generated with round(sqrt(pow(n, 2.71828 ))*2.71828)%2==0:

The same algorithm, but pixels are elevated along the Y axis according to the spiral counter's distilled value (e.g. 14 = 5, 11 = 2). This technique governs all numbers, no matter how large to be boiled down to no more than 9 units.

Now, the image is slightly tilted to reveal the depth we added above. Note those nested "U" shapes.

We injected new information into this pattern by taking a 2D image and making it 3D by imparting elevation to each pixel. Now let's see if these "U" shapes persist when other pattern algorithms are applied...

round(n/2.71828)*n%2==0

Secondary pattern emerges from tilted image:

So, is there hidden information with the 2D patterns here? The answer, of course (alas, hindsight), is no. We added the information to the pattern when we made it 3D.

See e - Mathematical Constant Mapped on a Ulam Spiral for the Flash-based images created from the algorithms above.

Visual Python - Ulam Variations

2008-10-21T19:49:00.000-07:00

Some mucking about with VPython to create 3D interpretations of the Ulam spiral algorithm. The above images are a variation of the following pagoda like structure:

Detail:

Manhattan Height Limit:

This is the same algorithm as above images, just a different extruded shape with the camera positioned merely above the skyscrapers.

Manhattan primes:

Here are the primes distributed within the pagoda:

This looks a bit like a scud missile to me. Prime numbers positioned throughout the Z axis using (n/2.71828)*cos( distillNumber(n)/2.71828 )
where distillNumber is a function returning the base root of a given number.

Seems to be a theme here...

Base 9 Spiral

2008-07-13T16:22:00.001-07:00

The above image is a clockwise spiral beginning with 1 and ending with 250. This is a simplification of the base 18 spiral in the following post, as it was brought to my attention that not only do the digits of 18 add to 9, but ultimately a spiral with 9 arms would be the simplest representation of this mathematics. This is because now we have a spiral where each number within an arm is separated from its neighbor by exactly 9.

The following is the printable version sans legend:

Table version of the Base-9 Spiral with primes highlighted in red. Note that numbers in a given column ultimately add up to whatever number is in the top row. For example, 109 is found in column 1 underneath the base digit 1. 1+0+9 = 10 = 1+0 = 1. This type of sequence is well documented as "Numbers whose digital root is n" where n is a digit 1 through 9, attributed to Cino Hilliard. For more on integer sequences and a useful search tool go to http://www.research.att.com/~njas/sequences/Seis.html

1	2	3	4	5	6	7	8	9
10	11	12	13	14	15	16	17	18
19	20	21	22	23	24	25	26	27
28	29	30	31	32	33	34	35	36
37	38	39	40	41	42	43	44	45
46	47	48	49	50	51	52	53	54
55	56	57	58	59	60	61	62	63
64	65	66	67	68	69	70	71	72
73	74	75	76	77	78	79	80	81
82	83	84	85	86	87	88	89	90
91	92	93	94	95	96	97	98	99
100	101	102	103	104	105	106	107	108
109	110	111	112	113	114	115	116	117
118	119	120	121	122	123	124	125	126
127	128	129	130	131	132	133	134	135
136	137	138	139	140	141	142	143	144
145	146	147	148	149	150	151	152	153
154	155	156	157	158	159	160	161	162
163	164	165	166	167	168	169	170	171
172	173	174	175	176	177	178	179	180
181	182	183	184	185	186	187	188	189
190	191	192	193	194	195	196	197	198
199	200	201	202	203	204	205	206	207
208	209	210	211	212	213	214	215	216
217	218	219	220	221	222	223	224	225
226	227	228	229	230	231	232	233	234
235	236	237	238	239	240	241	242	243
244	245	246	247	248	249	250	251	252
253	254	255	256	257	258	259	260	261
262	263	264	265	266	267	268	269	270
271	272	273	274	275	276	277	278	279
280	281	282	283	284	285	286	287	288
289	290	291	292	293	294	295	296	297
298	299	300	301	302	303	304	305	306
307	308	309	310	311	312	313	314	315
316	317	318	319	320	321	322	323	324
325	326	327	328	329	330	331	332	333
334	335	336	337	338	339	340	341	342
343	344	345	346	347	348	349	350	351
352	353	354	355	356	357	358	359	360
361	362	363	364	365	366	367	368	369
370	371	372	373	374	375	376	377	378
379	380	381	382	383	384	385	386	387
388	389	390	391	392	393	394	395	396
397	398	399	400	401	402	403	404	405
406	407	408	409	410	411	412	413	414
415	416	417	418	419	420	421	422	423
424	425	426	427	428	429	430	431	432
433	434	435	436	437	438	439	440	441
442	443	444	445	446	447	448	449	450
451	452	453	454	455	456	457	458	459
460	461	462	463	464	465	466	467	468
469	470	471	472	473	474	475	476	477
478	479	480	481	482	483	484	485	486
487	488	489	490	491	492	493	494	495
496	497	498	499	500	501	502	503	504
505	506	507	508	509	510	511	512	513
514	515	516	517	518	519	520	521	522
523	524	525	526	527	528	529	530	531
532	533	534	535	536	537	538	539	540
541	542	543	544	545	546	547	548	549
550	551	552	553	554	555	556	557	558
559	560	561	562	563	564	565	566	567
568	569	570	571	572	573	574	575	576
577	578	579	580	581	582	583	584	585
586	587	588	589	590	591	592	593	594
595	596	597	598	599	600	601	602	603
604	605	606	607	608	609	610	611	612
613	614	615	616	617	618	619	620	621
622	623	624	625	626	627	628	629	630
631	632	633	634	635	636	637	638	639
640	641	642	643	644	645	646	647	648
649	650	651	652	653	654	655	656	657
658	659	660	661	662	663	664	665	666
667	668	669	670	671	672	673	674	675
676	677	678	679	680	681	682	683	684
685	686	687	688	689	690	691	692	693
694	695	696	697	698	699	700	701	702
703	704	705	706	707	708	709	710	711
712	713	714	715	716	717	718	719	720
721	722	723	724	725	726	727	728	729
730	731	732	733	734	735	736	737	738
739	740	741	742	743	744	745	746	747
748	749	750	751	752	753	754	755	756
757	758	759	760	761	762	763	764	765
766	767	768	769	770	771	772	773	774
775	776	777	778	779	780	781	782	783
784	785	786	787	788	789	790	791	792
793	794	795	796	797	798	799	800	801
802	803	804	805	806	807	808	809	810
811	812	813	814	815	816	817	818	819
820	821	822	823	824	825	826	827	828
829	830	831	832	833	834	835	836	837
838	839	840	841	842	843	844	845	846
847	848	849	850	851	852	853	854	855
856	857	858	859	860	861	862	863	864
865	866	867	868	869	870	871	872	873
874	875	876	877	878	879	880	881	882
883	884	885	886	887	888	889	890	891

A more comprehensive table with numbers listed to 17991 can be found at thedamagereport.com. Maybe you will be "the one" to find the pattern in the primes.

Prime Spiral - Intervals of 18

2008-07-03T19:53:00.000-07:00

This circular spiral (click on the above image to enlarge it.) is a distant cousin of the Ulam spiral with the primes up to 499 mapped and indicated in red. First, this spiral is drawn in such a way that all primes fall on dedicated "prime arms" originating with 1, 5, 7, 11, 13, 17 and 19. Even numbers sit on their own arms. Each number on any given arm is separated from his neighbor by intervals of 18, 36, 72, etc. The arm originating with 2 contains no other primes (of course, because all the numbers on this arm are even). Strangely, the arm originating in 3 contains no other primes either.

The bizarre thing about this spiral is when you add the digits of any given number within an arm. For example, the arm terminating just to the right of 6 o'clock is the root 19 arm. See the following:

The first number in this arm is 19. Adding the digits in the integer 19 is a straight forward affair: 1+9=10.

The number following 19 is 37. Adding the digits produces 3+7=10.

It goes on like this, with some exceptions where the digits add up to 19. But it's always one or the other, either 10 or 19, or probably 28 too if kept going up the arm. And interesting the digits comprising 19 add up to 10. Are we in fact looking at a segment of Gary W. Adamson's number sequence A238140?

Here's the root 11 arm. You can see that each digit in a given number adds up to 11 or the occasional 20.

What is the significance of this organization of primes along these arms? It is more than a simple grouping of numbers, because remember, the spiral originates with 1 at the center then follows a very orderly counter-clockwise path, AND each number within a given arm is separated from its neighbor by 18 or a factor of 18.

Another interesting bit of math with this spiral concerns predicting which numbers will appear after a given number. Take for example this: 191x2-11=371. I believe it is a coincidence that 371 is 11 steps away from 191 within the arm (if you begin counting at 191). Anyway, the formula seems to work on all the numbers within this arm:

You can apply this formula to any number within the arm to predict that a number will appear. The formula n*2-armNumber is modified slightly per each arm, where armNumber is the root integer for a given arm.

Here is the root 5 arm:

In this example the formula n*2-armNumber takes 5 for armNumber. Let's take (131*2)-5 to get 257. It's just uncanny.

For those of you following along at home, I've created a cheat sheet spiral with each arm numbered along with the general sums for digits in a given arm.

e - Mathematical Constant Mapped on a Ulam Spiral

2008-07-02T21:49:00.000-07:00

Here's what you get when you map the product of the square root of n² to e along a Ulam spiral:

Algorithm: (Math.round(Math.sqrt(Math.pow(n, 2.71828 ))*2.71828)%2==0)

This unusual creation uses PI and e to display a weird hodge podge of fractal influenced spirals:

Algorithm: (Math.round(Math.sqrt(Math.pow(n, Math.PI ))*2.71828)%2==0)

A simplification of the algorithm leads to this:

Algorithm: (Math.round(n/2.71828)*n%2==0)

The Strangest, Most Beautiful Spiral

2008-06-18T22:03:00.000-07:00

When my spiral generator output the above image for the first time, my jaw dropped. I had previously generated some interesting spiral images using triangular and square numbers, images that others have already generated elsewhere. But I have not yet seen anything as complex or intricate as this image, at least using the Ulam spiral algorithm.

The base calculation for generating this image (using ActionScript) follows:
(Math.round(Math.sqrt(Math.pow(n,2))*(1.6180339887*3.14159265)/2)%2==0)

It's the kind of pattern that would indeed make for an excellent Persian rug design, where your eye gets lost trying to follow the paths through the maze.

The following image uses the same calculation to determine the color of the dot, but rather than starting at the center here we are starting at top left 0,0 and moving across _x, then down and repeating until the space is filled.

Now we can see that the pattern is incredibly simple. If we say gray dots are 0 and red dots are 1, then we get:
01010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000101010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101000000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000010101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010000000000000000000000000000000000001010101010101010101010100000000000000000000000000000000000000101010101010101010101010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The groups of zeros alternate between 36 zeros and 38 zeros. In between these alternations is a string of zeros and ones. The first set of mixed digits is a 12-digit-long string "010101010101". After that it's always a 23-digit-long string "10101010101010101010101".

Now I'm trying to fine tune the algorithm to see if I can focus the pattern, perhaps improving it. Here is the result of using (Math.round(Math.sqrt(Math.pow(n, (Math.sqrt(2)+1) ))*2.2)%2==0)...As you can see there is not a huge difference.

Again...(Math.round(Math.sqrt(Math.pow(pInt, (Math.sqrt(2)+1) ))*2.3)%2==0). It seems the general characteristic of the pattern is similar, yet if you study it the differences become gradually more apparent. The way the spiral patterns are visually iterating reminds me a little bit of M.C. Escher's fish transforming into birds and back again. [See Metamorphosis]

After mucking about with the multiplier I have discovered that numbers greater than PI*PHI produce tinier spiral patterns to the point of turning the whole image into a seemingly random block of static. Going the other direction improves the clarity of the spiral patterns. Here is the same algorithm using 1.5 for our multiplier:

Multiplier = 1.2

Multiplier = .9

Multiplier = .2
We've crossed the threshold and the spirals are gone.

Multiplier = .005

Why do spirals form fancy daisy chains when using PI*PHI as the multiplier? Who knows? It's just how the dots line up.

I am curious as to why having PI as part of the algorithm leads to these spirals. Here is a linear matrix (L to R, T to B) using (Math.round( Math.pow(Math.sqrt(pInt)/Math.PI, 3.3) )%2==0) per dot.

Another kind of MaxiCode

Have you ever examined a package delivered by UPS and wondered what those curious looking labels containing a matrix of dots are used for? They are quite ingenious and beautiful in their own way. If you've gotten this far in this particular post you have probably noticed how you might generate your own "maxi-code" using the principles of a Ulam spiral, a left-to-right, top-to-bottom pattern, or any contained pattern. Encoding data into a pattern is a pretty simple affair. It's the encryption that is difficult. If we take the above example of converting the Ulam spiral to a series of ones and zeroes, you wind up with what could be interpreted as a set of data encoded as a binary set. But what algorithm should one use to convert these numbers back to human readable data? Or machine-readable data? Who knows! Such is the fun in creating your own patterns, meta-languages and encoding/decoding/codec algorithms.

Images of Phi Part II

2008-06-17T20:09:00.001-07:00

A Ulam spiral testing each number for if(Math.round((n * Math.PI)/1.618)%9==0)

Same idea, swapping out mod 9 with mod 5:

Calculation per dot: (Math.round(Math.sqrt(Math.pow(n,2))*1.618/2)%5==0)

Calculation per dot: (Math.round(Math.sqrt(n*1.618)/2)%5==0)

Calculation per dot: (Math.round(Math.sqrt(n*1.618))%5==0)

Calculation per dot: (Math.round(Math.sqrt(n*2))%5==0)

Here are the selected primes up to 13693 from the previous image.

Prime Numbers Ending in 1

2008-06-17T19:00:00.000-07:00

A Ulam spiral with only primes ending with the integer 1. We start off 1, 11, 31, 41, 61, 71, 101, 131, 151.....10,331!

This prime goes to 11. Here we have all primes ending with the digits 11 from 11 to 12011.

Primes Numbers Beginning With 7

2008-06-17T16:59:00.000-07:00

This is a Ulam spiral showing just the primes that begin with 7 (except for the yellow integer 1 at center). So, we have 7, 71, 73, 79 in the center group. The middle ring contains 701, 709, 719, 727, 733, 739, 743, 751, 757, 751, 761, 769, 773, 787, and 797. The outer ring contains 7151, 7159, 7193.... The reason we have squares appearing is fairly obvious. What I find interesting is the distribution of primes within each level. There are notable gaps in the outermost ring. Of course, that ring contains a few thousand numbers so the distribution could appear however we decide to fire off the spiral.

Same idea, this time primes beginning with 2. Notice the deficit of primes in the lower edge of the outer ring.

Corner Primes

2008-06-15T00:08:00.000-07:00

Primes on "corners" are connected with a line. A corner, in the case of a Ulam Spiral, is like corners in real life. It's where we turn to go in a different direction. The corner primes in this image are:

2,3,5,7,13,17,31,37,43,73,101,157,197,211,241,257,307,401,421,463,577,601,677,757,1123,1297, and 1483.

Now if you take the average of these numbers you get 242.074074074.... Subtract 1 because I said so, and you get 241.074074074..., which rounds to 241. Prime!

Do the same with 2,3,5,7,13,17. Averages to 7.8 - 1 = 6.8. Rounds to 7. Bingo. Prime!

Do it with 2,3,5,7,13,17, and 31. Averages to 11.142857. Don't subtract the 1 because then our rounded average would not be prime. Dang it.

Weird? Yes. Significant? Nope.

Connect the Prime Dots

2008-06-14T23:19:00.000-07:00

Here's the basic algorithm: Starting with 1, cycle through all numbers and test each for prime. If the number is prime, connect a line from the last prime number to the current prime number. The particular map stops at 337.

Zooming out to prime #14033, I give you this:

Primes Overlapping Fractals on a Ulam Spiral

2008-06-14T20:24:00.001-07:00

Here is a typical Ulam spiral. Each bright dot is a prime number.

And here is our "rug" with the lovely Celtic pattern generated by multiplying the golden mean with PI. Where do the primes fall upon this pattern?

Overlapped primes:
As you can see, it's a complete mess. Some of the primes fit into the pattern, some don't, where the primes sit in the negative space. Nothing significant is happening here.

Or is there? Take a look at this:
The primes tend to "favor" areas where the celtic pattern's positive space occupies. That is, the negative space of the pattern, the dark regions, by and large are not polluted with primes. Of course you will find exceptions and even a contradiction: There are plenty of positive space regions (masses of red dots) where the primes also avoid. Another way to say this is the primes favor the edges, and if we continue to draw this pattern outward we may gain resolution into the nature of this happy random accident.

Alien Signals

2008-06-14T01:32:00.000-07:00

if(Math.round(Math.sqrt(Math.pow(n,2))*(1.6180339887*3.14159265)/2)%5==0){//paint a dot}. I'm not sure what the significance is of multiplying PI with the Golden Mean. But it sure looks cool.

A little more number crunching: if(Math.round(Math.sqrt(Math.pow(pInt, (Math.sqrt(2)+1) ))*(1.6180339887*3.14159265)/2)%2==0){//paint weird pattern}

Tighten up the resolution and you approach a nice pattern for a wool rug.

Note that the picture gets clearer the farther from center you go. It seems there is simply more mathematical "space" for the pattern to be expressed.

Sin mod n, Square roots and PI

2008-06-14T00:50:00.001-07:00

Calculation: if(Math.round(Math.sin(n))%6==0){//paint it!} As it turns out, it doesn't need to be 6 to generate this pattern because we round the SIN calculation first. So, if(Math.round(Math.sin(pInt)*Math.PI)%3==0){...} generates this:

And then there's this...


if(Math.round(Math.sqrt(pInt)*3.14159265)%2==0){//draw Tempest!}

Same formula as above but showing only the prime numbers.

And again with the primes blurred to enhance the pattern they create.

Same stuff, modding 3 this time:

Modulo 4:

Let's take the average of PI and the Golden Ratio and mod that by 2...Holy guacamole!

So if you were looking for a visual representation of what the relationship amongst primes and PI and PHI might look like, look no further.

Calculation per dot: (Math.round(Math.sqrt(Math.pow(pInt,2))*(1.6180339887*3.14159265)/2)%5==0)

Silver Ratio

2008-06-14T00:24:00.000-07:00

Here we go round and round Ulam's spiral, testing each number not for primality, but if(Math.round(n/1+1.41421)*n%5==0), paint circle. Basically another twisted formatter using 1+√2 modulo 5. Why 5? Why not?

Strange things happen around this number: Square root of 2

Same calc, swapping in modulo 7. Looks a bit like a cousin to the n mod 7 image.

Go modulo 8:

Calculation per dot: (Math.round(pInt/(Math.sqrt(2)+1))%5==0)

Images of Phi Part I

2008-06-14T00:08:00.000-07:00

if(Math.round(n/1.6180339887)*n%2==0){//paint the dot a color

Behold:

if(Math.round(n/1.6180339887)*n%2==0){//paint the dot a color}

Another good pattern for a rug.

3.14159265

2008-06-13T23:51:00.001-07:00

This is a fairly strange calculation, but here goes:

if( Math.round( n /)*n%2==0 ) {//color the dot}

A lot of fractal-like features with this formula. Here's another using the same calculation but using mod 3.

Using 5 as the modulo divisor delivers strange results. My favorite of the lot:

How about a nice, modern looking rug?

Calculation per dot: Math.round(Math.sqrt(pInt)*Math.PI)%5==0

Calculation per dot: Math.round(pInt*Math.PI)%5==0

Calculation per dot: Math.round(((Math.pow(pInt, 2))*Math.PI) * (Math.sqrt(2)+1))%5==0. Some of the patterns in here look like hiragana. If you look closely you can also find Kilroy. And no, I didn't put him in there.

Visualizing Modulo 1 through 6

2008-06-13T22:34:00.000-07:00

Modulo 1 and 2...Pretty much what you'd expect from dividing things individually and then by 2s.

There is no rule that states we can only plot primes with Ulam's spiral. This is what happens when you perform n mod 3 as you plot each dot. Hmmm...

Same idea, only mod 4:

And it just gets better. Mod 6:

Modulo 5

2008-06-13T22:28:00.000-07:00

n mod 5

Modulo 7, 8, 9, 10, 16, 21

2008-06-13T21:38:00.000-07:00

n mod 7

Mod 8. Like a scene from Space Invaders.

Mod 9.

Mod 10, in blue at positions 10 all the way to 3420.

Compare this to Mod 8.

Mod 21

Visualizing Prime Numbers

2008-06-13T19:21:00.000-07:00

This is the output from a Ulam spiral generator I built in Flash. The pattern is based on the Ulam spiral concept. I got the idea to play around with mapping prime numbers after reading a biography on Paul Erdos which talked about Stanislaw Ulam's now famous doodles. There's all kinds of patterns one can derive from this. Depending on how you draw out the pattern, you can indicate the interesting property where primes only fall on diagonals. But since that ground has been covered, I was hoping to discover the face of God, Elvis, or both.

Here I've colored each prime an independent color based on one of four directions the individual dot was plotted (the gray circles are composites, not primes). The cyan dots are also prime and are found at the end of each direction. Interestingly, there are no primes on the southeast quadrant between the red and green sections. There are similar prime DMZs, if you look. Apparently there isn't anything terribly profound about this pattern (some say), but I don't believe the naysayers. Hell, they just proved that 42 really is a significant number in the universe.

This blog contains many of the variants I have generated.

1	2	3	4	5	6	7	8	9
10	11	12	13	14	15	16	17	18
19	20	21	22	23	24	25	26	27
28	29	30	31	32	33	34	35	36
37	38	39	40	41	42	43	44	45
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64	65	66	67	68	69	70	71	72
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91	92	93	94	95	96	97	98	99
100	101	102	103	104	105	106	107	108
109	110	111	112	113	114	115	116	117
118	119	120	121	122	123	124	125	126
127	128	129	130	131	132	133	134	135
136	137	138	139	140	141	142	143	144
145	146	147	148	149	150	151	152	153
154	155	156	157	158	159	160	161	162
163	164	165	166	167	168	169	170	171
172	173	174	175	176	177	178	179	180
181	182	183	184	185	186	187	188	189
190	191	192	193	194	195	196	197	198
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208	209	210	211	212	213	214	215	216
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235	236	237	238	239	240	241	242	243
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253	254	255	256	257	258	259	260	261
262	263	264	265	266	267	268	269	270
271	272	273	274	275	276	277	278	279
280	281	282	283	284	285	286	287	288
289	290	291	292	293	294	295	296	297
298	299	300	301	302	303	304	305	306
307	308	309	310	311	312	313	314	315
316	317	318	319	320	321	322	323	324
325	326	327	328	329	330	331	332	333
334	335	336	337	338	339	340	341	342
343	344	345	346	347	348	349	350	351
352	353	354	355	356	357	358	359	360
361	362	363	364	365	366	367	368	369
370	371	372	373	374	375	376	377	378
379	380	381	382	383	384	385	386	387
388	389	390	391	392	393	394	395	396
397	398	399	400	401	402	403	404	405
406	407	408	409	410	411	412	413	414
415	416	417	418	419	420	421	422	423
424	425	426	427	428	429	430	431	432
433	434	435	436	437	438	439	440	441
442	443	444	445	446	447	448	449	450
451	452	453	454	455	456	457	458	459
460	461	462	463	464	465	466	467	468
469	470	471	472	473	474	475	476	477
478	479	480	481	482	483	484	485	486
487	488	489	490	491	492	493	494	495
496	497	498	499	500	501	502	503	504
505	506	507	508	509	510	511	512	513
514	515	516	517	518	519	520	521	522
523	524	525	526	527	528	529	530	531
532	533	534	535	536	537	538	539	540
541	542	543	544	545	546	547	548	549
550	551	552	553	554	555	556	557	558
559	560	561	562	563	564	565	566	567
568	569	570	571	572	573	574	575	576
577	578	579	580	581	582	583	584	585
586	587	588	589	590	591	592	593	594
595	596	597	598	599	600	601	602	603
604	605	606	607	608	609	610	611	612
613	614	615	616	617	618	619	620	621
622	623	624	625	626	627	628	629	630
631	632	633	634	635	636	637	638	639
640	641	642	643	644	645	646	647	648
649	650	651	652	653	654	655	656	657
658	659	660	661	662	663	664	665	666
667	668	669	670	671	672	673	674	675
676	677	678	679	680	681	682	683	684
685	686	687	688	689	690	691	692	693
694	695	696	697	698	699	700	701	702
703	704	705	706	707	708	709	710	711
712	713	714	715	716	717	718	719	720
721	722	723	724	725	726	727	728	729
730	731	732	733	734	735	736	737	738
739	740	741	742	743	744	745	746	747
748	749	750	751	752	753	754	755	756
757	758	759	760	761	762	763	764	765
766	767	768	769	770	771	772	773	774
775	776	777	778	779	780	781	782	783
784	785	786	787	788	789	790	791	792
793	794	795	796	797	798	799	800	801
802	803	804	805	806	807	808	809	810
811	812	813	814	815	816	817	818	819
820	821	822	823	824	825	826	827	828
829	830	831	832	833	834	835	836	837
838	839	840	841	842	843	844	845	846
847	848	849	850	851	852	853	854	855
856	857	858	859	860	861	862	863	864
865	866	867	868	869	870	871	872	873
874	875	876	877	878	879	880	881	882
883	884	885	886	887	888	889	890	891

1	2	3	4	5	6	7	8	9
10	11	12	13	14	15	16	17	18
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28	29	30	31	32	33	34	35	36
37	38	39	40	41	42	43	44	45
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55	56	57	58	59	60	61	62	63
64	65	66	67	68	69	70	71	72
73	74	75	76	77	78	79	80	81
82	83	84	85	86	87	88	89	90
91	92	93	94	95	96	97	98	99
100	101	102	103	104	105	106	107	108
109	110	111	112	113	114	115	116	117
118	119	120	121	122	123	124	125	126
127	128	129	130	131	132	133	134	135
136	137	138	139	140	141	142	143	144
145	146	147	148	149	150	151	152	153
154	155	156	157	158	159	160	161	162
163	164	165	166	167	168	169	170	171
172	173	174	175	176	177	178	179	180
181	182	183	184	185	186	187	188	189
190	191	192	193	194	195	196	197	198
199	200	201	202	203	204	205	206	207
208	209	210	211	212	213	214	215	216
217	218	219	220	221	222	223	224	225
226	227	228	229	230	231	232	233	234
235	236	237	238	239	240	241	242	243
244	245	246	247	248	249	250	251	252
253	254	255	256	257	258	259	260	261
262	263	264	265	266	267	268	269	270
271	272	273	274	275	276	277	278	279
280	281	282	283	284	285	286	287	288
289	290	291	292	293	294	295	296	297
298	299	300	301	302	303	304	305	306
307	308	309	310	311	312	313	314	315
316	317	318	319	320	321	322	323	324
325	326	327	328	329	330	331	332	333
334	335	336	337	338	339	340	341	342
343	344	345	346	347	348	349	350	351
352	353	354	355	356	357	358	359	360
361	362	363	364	365	366	367	368	369
370	371	372	373	374	375	376	377	378
379	380	381	382	383	384	385	386	387
388	389	390	391	392	393	394	395	396
397	398	399	400	401	402	403	404	405
406	407	408	409	410	411	412	413	414
415	416	417	418	419	420	421	422	423
424	425	426	427	428	429	430	431	432
433	434	435	436	437	438	439	440	441
442	443	444	445	446	447	448	449	450
451	452	453	454	455	456	457	458	459
460	461	462	463	464	465	466	467	468
469	470	471	472	473	474	475	476	477
478	479	480	481	482	483	484	485	486
487	488	489	490	491	492	493	494	495
496	497	498	499	500	501	502	503	504
505	506	507	508	509	510	511	512	513
514	515	516	517	518	519	520	521	522
523	524	525	526	527	528	529	530	531
532	533	534	535	536	537	538	539	540
541	542	543	544	545	546	547	548	549
550	551	552	553	554	555	556	557	558
559	560	561	562	563	564	565	566	567
568	569	570	571	572	573	574	575	576
577	578	579	580	581	582	583	584	585
586	587	588	589	590	591	592	593	594
595	596	597	598	599	600	601	602	603
604	605	606	607	608	609	610	611	612
613	614	615	616	617	618	619	620	621
622	623	624	625	626	627	628	629	630
631	632	633	634	635	636	637	638	639
640	641	642	643	644	645	646	647	648
649	650	651	652	653	654	655	656	657
658	659	660	661	662	663	664	665	666
667	668	669	670	671	672	673	674	675
676	677	678	679	680	681	682	683	684
685	686	687	688	689	690	691	692	693
694	695	696	697	698	699	700	701	702
703	704	705	706	707	708	709	710	711
712	713	714	715	716	717	718	719	720
721	722	723	724	725	726	727	728	729
730	731	732	733	734	735	736	737	738
739	740	741	742	743	744	745	746	747
748	749	750	751	752	753	754	755	756
757	758	759	760	761	762	763	764	765
766	767	768	769	770	771	772	773	774
775	776	777	778	779	780	781	782	783
784	785	786	787	788	789	790	791	792
793	794	795	796	797	798	799	800	801
802	803	804	805	806	807	808	809	810
811	812	813	814	815	816	817	818	819
820	821	822	823	824	825	826	827	828
829	830	831	832	833	834	835	836	837
838	839	840	841	842	843	844	845	846
847	848	849	850	851	852	853	854	855
856	857	858	859	860	861	862	863	864
865	866	867	868	869	870	871	872	873
874	875	876	877	878	879	880	881	882
883	884	885	886	887	888	889	890	891

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100	101	102	103	104	105	106	107	108
109	110	111	112	113	114	115	116	117
118	119	120	121	122	123	124	125	126
127	128	129	130	131	132	133	134	135
136	137	138	139	140	141	142	143	144
145	146	147	148	149	150	151	152	153
154	155	156	157	158	159	160	161	162
163	164	165	166	167	168	169	170	171
172	173	174	175	176	177	178	179	180
181	182	183	184	185	186	187	188	189
190	191	192	193	194	195	196	197	198
199	200	201	202	203	204	205	206	207
208	209	210	211	212	213	214	215	216
217	218	219	220	221	222	223	224	225
226	227	228	229	230	231	232	233	234
235	236	237	238	239	240	241	242	243
244	245	246	247	248	249	250	251	252
253	254	255	256	257	258	259	260	261
262	263	264	265	266	267	268	269	270
271	272	273	274	275	276	277	278	279
280	281	282	283	284	285	286	287	288
289	290	291	292	293	294	295	296	297
298	299	300	301	302	303	304	305	306
307	308	309	310	311	312	313	314	315
316	317	318	319	320	321	322	323	324
325	326	327	328	329	330	331	332	333
334	335	336	337	338	339	340	341	342
343	344	345	346	347	348	349	350	351
352	353	354	355	356	357	358	359	360
361	362	363	364	365	366	367	368	369
370	371	372	373	374	375	376	377	378
379	380	381	382	383	384	385	386	387
388	389	390	391	392	393	394	395	396
397	398	399	400	401	402	403	404	405
406	407	408	409	410	411	412	413	414
415	416	417	418	419	420	421	422	423
424	425	426	427	428	429	430	431	432
433	434	435	436	437	438	439	440	441
442	443	444	445	446	447	448	449	450
451	452	453	454	455	456	457	458	459
460	461	462	463	464	465	466	467	468
469	470	471	472	473	474	475	476	477
478	479	480	481	482	483	484	485	486
487	488	489	490	491	492	493	494	495
496	497	498	499	500	501	502	503	504
505	506	507	508	509	510	511	512	513
514	515	516	517	518	519	520	521	522
523	524	525	526	527	528	529	530	531
532	533	534	535	536	537	538	539	540
541	542	543	544	545	546	547	548	549
550	551	552	553	554	555	556	557	558
559	560	561	562	563	564	565	566	567
568	569	570	571	572	573	574	575	576
577	578	579	580	581	582	583	584	585
586	587	588	589	590	591	592	593	594
595	596	597	598	599	600	601	602	603
604	605	606	607	608	609	610	611	612
613	614	615	616	617	618	619	620	621
622	623	624	625	626	627	628	629	630
631	632	633	634	635	636	637	638	639
640	641	642	643	644	645	646	647	648
649	650	651	652	653	654	655	656	657
658	659	660	661	662	663	664	665	666
667	668	669	670	671	672	673	674	675
676	677	678	679	680	681	682	683	684
685	686	687	688	689	690	691	692	693
694	695	696	697	698	699	700	701	702
703	704	705	706	707	708	709	710	711
712	713	714	715	716	717	718	719	720
721	722	723	724	725	726	727	728	729
730	731	732	733	734	735	736	737	738
739	740	741	742	743	744	745	746	747
748	749	750	751	752	753	754	755	756
757	758	759	760	761	762	763	764	765
766	767	768	769	770	771	772	773	774
775	776	777	778	779	780	781	782	783
784	785	786	787	788	789	790	791	792
793	794	795	796	797	798	799	800	801
802	803	804	805	806	807	808	809	810
811	812	813	814	815	816	817	818	819
820	821	822	823	824	825	826	827	828
829	830	831	832	833	834	835	836	837
838	839	840	841	842	843	844	845	846
847	848	849	850	851	852	853	854	855
856	857	858	859	860	861	862	863	864
865	866	867	868	869	870	871	872	873
874	875	876	877	878	879	880	881	882
883	884	885	886	887	888	889	890	891