Picture of Integer Filling Through 3 Generations of Left Descent Assemblages

In my early days of exploring the Collatz 3n+1 conjecture,  I dreamed of looking for a pattern by constructing some sort of picture which might show up a pattern to the eye which simply never becomes evident in the numbers and the mathematics.  One such picture was constructed by Andrew Shapira and is available on the Internet.  Unfortunately,  it reveals almost nothing in its 230400 pixels.

I never got around to making a picture before developing the structure presented in these pages.  Now that I understand the structure,  it is easy to conceive of a pictorial resentation which would have been very helpful,  had one had the good fortune to happen upon this presentation or any of several close relatives.  Of course,  doing that by good luck would be virtually impossible.

My picture is a raster 648 wide by 101 high (thus only 65448 pixels).  The adjacent pixels in the even-numbered columns were filled also to make the entries for each integer a little more easily discerned than is afforded by a single pixel.  Thus actually only the 32723 odd numbers up to 65447 are included.  648 is selected as the raster width because it is 2^3*3^4.  The coefficients in the formulas for the elements of the left descent assemblies are of the form 2^a*3^b*n+c,  so this raster width is a simple multiple or a rational fraction of those coefficients.  This leads to a picture in which the instantiations of any given left descent element appear in a regular pattern in the raster.  The column of the first of two adjacent pixels representing an instantiation whose value is m was determined by m mod 648 and the row by m div 648.

I have used a different color (see the legend in the picture)  for each left descent assembly of 0,  1,  or 2 generations,  and light grey for all the ones in the third generation.  The formulas for the first 7 are included in the legend,  but the formulas for the ones in the third generation are omitted.  (They can be picked out of the file of complete left descent assemblages,  and their values of c are responsible for the entries at columns 29,  53,  229,  341,  373,  469,  533,  and 541 in the first (zeroth)  row.) 

The childless extensions have formula 24n+21,  and so appear at colums 21,  45,  69,  93,  ...  in every row of the raster.  The third generation left descent assemblages have 648 as the coefficient in their formulas,  so they appear as a single column at the location of their constant c.  An element with coefficient 864 appears at intervals of 4/3 of the raster width and so appears in three distinct columns as dots in every 4th row.  The most awkward coefficient among those included was 1536 which appears at intervals 64/27 of the raster width and so appears in 27 distinct columns in rows separated by 64.

The result looks strikingly like vertically striped wallpaper.  The picture is too small to show the detail accurately.  I believe the placement of dots to be accurate down through the first 7 left descent assemblages,  but there are probably errors among the light grey dots,  so the obscurity is probably a good thing for those.

The picture gives an impression of the rapidity with which the integers are covered by only the first 15 left descent assemblages.   The infinity still to come will occupy the relatively little white space left.

The picture also shows how neatly five different elements conspire to fill (incompletely)  some particular columns,  e.g.  81,  201,  321 ...  489,  609 before the pattern revisits 81.  They are 96n+81 (yellow),  192n+153 (blue),  384n+321 (dark grey),  768n+441 (light grey),  and 1536n+1281 (light grey).  The intervals for those are,  respectively,  4/27,  8/27,  16/27,  32/27,  and 64/27 of the raster width where the numerators give the rows between placement of the dots.

It is a pretty picture,  but not of much direct use because the small size make the details obscure and the complications introduced by taking mod(m,648)  confuse the picture and decrease its utility in clarifying the systematic way the integers are filled by the structure.

The picture might be of some use in indicating the rate at which successive generations of development of the abstract predecessor tree result in filling in the integers.  Fortunately,  other means of conveying an impression to that end are also presented.  The picture showing the successive subsetting of the sets in the abstract predecessor tree was drawn to horizontal scale,  which makes it evident how much progress is made in only 4 generations.  The table of cardinalities of formulas sharing {a,b}  of the 2^a*3^b coefficients now has a companion table in which each cardinality is accompanied with decimal fractions which indicate what contribution to all odd integers its cell makes.  Some cumulative contributions are also included in the latter table.


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