In a recent e-mail, Dan again brought up the subject of the image referred to in the following sci.math post (emphasis added). As I needed something to try out my latest programming language (PLT Scheme), I decided to have a look.

From: Dan (30pack@sbcglobal.net) Subject: Re: Thoughts on the Collatz conjecture Newsgroups: sci.math Date: 2003-08-15 18:14:02 PST Thanks to all for your responses to my original post. I am still convinced that the conjecture is true, because after hastily throwing together a computer program that I believe makes the case. My argument here is, as each level of the Collatz tree grows new branches are born derived from a smaller integer start number and so does the density "bands" (see below) and their symmetry. This causes a symmetrical squeeze play so to speak for other possible counter examples start numbers and trees. I did a computer program where integer start numbers are entered in order where start number n = 1,2,3,4,5,6..n with there associated sequences. Each start # turns on a corresponding numbered pixel and all the path members turn on their respected pixels. These pixels stay on. Naturally some of these pixels are on when they get hit again for an "on" because of the branching tree affect and also returning back to 4,2,1 and terminating on 1. What happens as each new starting (n) the (5) density bands out ahead of the all white band become more apparent after about 32 "y" rows of all "on" (white) pixels. This would be an integer start number the size of n =32*640 = 20480. I set this thing up for reading across --- 640 pixels where x ( n) (start #)= 1 to 640 and y=1 Then where x = 1 to 640 and n=641 to 1280 and y=2 etc. Y is set for a max. of 350 pixels. It bypasses any on pixel that is not viewable on screen where y >350. It creates an interesting effect with 6 distinct bands with each of the 5 bands having distinct symmetrical density patterns out ahead of (n) the seed that is the all white band behind the seed. This density band effect could be do to certain delta factors out ahead of the seed or something to do with 2^n? The short basic program is listed below with plenty of documentation so someone can translate to Java, c++ or any other language. ‘ = Docs 4 ‘ A Collatz conjecture pixel evaluation 5 ' This program turns on the appropriate pixel for each starting integer and all its sequence members. Pixels once on, stay on. 10 CLS 12 Screen 9: ‘ Set graphics screen mode to 350 X 640 pixels 15 DEFDBL A: ‘Double precision for any variable beginning with A 20 A=1:A3=A:A4=1:Y=1:A5=640:A6=640: PSET(A,Y): ‘ "A" is starting integer (seed) and turns on pixel x(A) = 1 and Y = 1: This line never used again. 30 A1$=STR$(A): ‘ Line 30-54 checks to see if integer is odd or even. 40 A2=LEN(A1$) 50 J$=MID$(A1$,A2,1): IF J$= "1" THEN GOTO 200 51 IF J$= "3" THEN GOTO 200 52 IF J$= "5" THEN GOTO 200 53 IF J$= "7" THEN GOTO 200 54 IF J$= "9" THEN GOTO 200 60 A3=A3/2:A=A3:GOSUB 500: IF A =< 1 THEN 320 ELSE 30:' This line handles even integers and goes to subroutine that evaluates the correct x and y pixel to turn on. 200 A3=(A*3) +1:A=A3:GOSUB 500:GOTO 60: ' Handles odd integers of seed and its sequence. Ect. 320 A4=A4+1:A=A4:A3=A4:GOSUB 500:GOTO 30: 'Retrieves the next seed and repeats the whole process creating a new sequence from that seed. 500 IF A>A6 THEN A6=A6+A5:Y=Y+1:ELSE 530: ' Sets Up "A" for right row (Y) 510 If Y>350 THEN Y=1:GOTO 540: ' If integer value in any sequence is > (350*640) then this line bypasses the pixel command (PSET) because pixel will not be in a viewable area of the screen. 515 IF A>A6 THEN 500: ' Go back to line 500 and add another 640 to variable A6 520 IF A=< A6 AND Y>1 THEN Y=Y-1:A7=Y*A5:A8=A-A7:Y=Y+1:PSET(A8,Y):Y=1:GOTO 540: ' Sets up x(A8) value when y>1 and thus the correct x,y coordinates for any applicable integer with a value > 640. 530 If Y=1 THEN PSET(A,Y): GOTO 540:' A branch from 500 where Else means Y=1 540 A6=A5:Y=1:RETURN: ' Resets variables and returns for next integer. 600 END Please excuse the hastily thrown together code. Should have done a renumber also! You have to think of each row of 640 pixels as rows cut off at that point and then stacked on each other where you can easily view how these density patterns out ahead of the seed number are formed. This would probably go unnoticed if the line stayed continuous as in the number line.This creates 5 distinct and fascinating density band patterns that grow in width as the first solid white band or seed band grows in width. Please note, when first starting out the bottom (last) density pattern starts to show a checkerboard pattern on an angle.If nothing more, its interesting! As always, any evaluations or comments are welcome. Dan

This image looks consistent with Dan's description. The origin (0,0) is in the upper left corner. Adjectives such as "above" or "greater" will mean downwards or to the right in the image. Let's see what we can make of it.

First, some terminology. Collatz sequences are sometimes called Hailstone numbers for the way the value of x rises and falls through the sequence. Keeping the weather metaphor:

- HAILSTONE
- A number that is larger than its predecessor. A HAILSTONE results from the application of 3x+1.
- TROPOSPHERE
- For any sequence starting at n, the TROPOSPHERE is the area of the image of all points less than n. All points less than n have already been plotted and appear as the solid white band at the top of the image. A HAILSTONE disappears from view when it drops into the TROPOSPHERE.
- STRATOSPHERE
- The region of the image above the TROPOSPHERE. The STRATOSPHERE is the Collatz sequence's excursion, the part of the sequence where x is greater than its initial value of n.
- CONTRAIL
- The track of pixels left by a sequence as it traverses the STRATOSPHERE. A CONTRAIL starts when the HAILSTONE first enters the STRATOSPHERE and ends when it a) drops into the TROPOSPHERE or b) becomes an odd number at which point it becomes a new HAILSTONE initiating a new CONTRAIL.

- every HAILSTONE is == 4 (mod 6)
- a CONTRAIL alternates between == 4 (mod 6) and == 2 (mod 6), except for the terminating pixel which is either == 1 (mod 6) or == 5 (mod 6).
- numbers == 0 (mod 6) or 3 (mod 6) are only found in the TROPOSPHERE.

- 0 (mod 6): red
- 1 (mod 6): green
- 2 (mod 6): blue
- 3 (mod 6): magenta
- 4 (mod 6): yellow
- 5 (mod 6): cyan

Now that all the HAILSTONES are nicely aligned, we still have those diagonal patterns. This indicates the presence of some meta-pattern on top of the mod 6 coloring. The displacement of adjacent points on the diagonal lines turns out to be 243 (3**5). But 243 is not a multple of 6, so we need to set our window width to 2*243 or 486 pixels to reveal the meta-pattern while maintaining the mod 6 alignment.

Voilà, order out of chaos:

In the following discussion, a HAILSTONE n will have image coordinates (h,v) such that

- h = remainder of n/486
- v = quotient of n/486

Note that the third pixel is green, thus ending the CONTRAIL (it will, of course, initiate another CONTRAIL somewhere else in the STRATOSPHERE). The HAILSTONE at (484,122) descends all the way into the TROPOSPHERE, bouncing back and forth between blue and yellow columns:

So we see how a prominent yellow column leads to a prominent (but smaller) blue column which leads to a prominent (but smaller) yellow column which leads to a prominent blue column, etc.

Note also that the prominent yellow column at 484 (mod 486) has a prominent cyan column adjacent to it. Not a coincidence. Every yellow pixel at h:484 whose v is odd, becomes congruent to 485 (mod 486) when divided by 2. Thus, half the CONTRAILS that start in the yellow column 484 terminate in column 485:

But how did column 484 (mod 486) get to be so prominent in the first place? Up until now, we haven't investigated what happens when a CONTRAIL terminates (always on a green or cyan pixel). Green and cyan pixels are odd numbers, so the next number will be a yellow pixel. Let's follow the CONTRAILS of pixel (484, 63):

Lo and behold! We've got a resonator! The yellow pixel at 484 (mod 486) becomes a cyan pixel at 485 (mod 486) which becomes a yellow pixel at 484 (mod 486) which becomes a cyan pixel at 485 (mod 486), and so on. No wonder the column 484 has so many pixels, it's self-generating.

So how come it doesn't run off to infinity? Because the property that sustains the resonance cannot be maintained indefinitely. The binary resonator 2**n - 1 stops when it runs out of 1 bits. Something happens to stop the (mod 486) resonance. A quick look at the actual Collatz sequence will reveal what. That particular sequence originated with n = 3830. At iteration 19, the sequence hits a binary resonator. Halfway through, the CONTRAIL hits column 484 and the resonance begins:

pixel pixel n v h n (binary) -------- ------- ------- ----------------- 2047 4 103 11111111111 <-- binary resonance starts 6142 12 310 1011111111110 3071 6 155 101111111111 9214 18 466 10001111111110 4607 9 233 1000111111111 13822 28 214 11010111111110 6911 14 107 1101011111111 20734 42 322 101000011111110 10367 21 161 10100001111111 31102 63 484 111100101111110 <-- (mod 486) resonance starts 15551 31 485 11110010111111 46654 95 484 1011011000111110 23327 47 485 101101100011111 69982 143 484 10001000101011110 34991 71 485 1000100010101111 104974 215 484 11001101000001110 52487 107 485 1100110100000111 157462 323 484 100110011100010110 78731 161 485 10011001110001011 236194 485 484 111001101010100010 118097 242 485 11100110101010001 354292 728 484 1010110011111110100 <-- (mod 486) resonance ends 177146 364 242 101011001111111010

Aha! The resonance stopped when we got an even pixel v coordinate. As seen earlier, when both v and h are even, the CONTRAIL goes to the blue column in the center of the image, breaking the column 484:485 resonance. But before dropping into the TROPOSPHERE, it hits column 484 a second time:

pixel pixel n v h n (binary) -------- ------- ------- ----------------- 336310 691 484 1010010000110110110 168155 345 485 101001000011011011 504466 1037 484 1111011001010010010 252233 518 485 111101100101001001 756700 1556 484 10111000101111011100 378350 778 242 1011100010111101110

And it hits it again after dropping into the TROPOSPHERE, but only manages to toss a single HAILSTONE into the STRATOSPHERE.

pixel pixel n v h n (binary) -------- ------- ------- ----------------- 1942 3 484 11110010110 971 1 485 1111001011 2914 5 484 101101100010 1457 2 485 10110110001 4372 8 484 1000100010100 2186 4 242 100010001010

Checking a couple other yellow columns, 322 (mod 486) and 376 (mod 486), showed that although they may get hit as many as four times in a single Collatz sequence, they don't "ring" when struck like 484 (mod 486) does.

I don't what significance this has in proving the Collatz Conjecture, but it is certainly one of those many items that make it endlessly fascinating.