Various Path Representations

Sources of Communication Difficulties

Why are we in such an uncomfortable position with respect to  communication amongst us?  In large part,  the difficulty arises from from the large variety of different notations which have grown up as each of us explored the Collatz trajectories from our individual viewpoints.

L.d.a.s

First,  let me explain the reasons for my adoption of the practice of considering odd numbers only and my focus on l.d.a.s as the central feature for describing the Collatz graph.

It was clear to me a priori that the Collatz conjecture could not be solved in the absence of some Escher-like construct -- some pattern or construct which would dove tail together to include all the positive integers.  Now,  since there are two roles played by even numbers  ((a) as the result of some 3n+1 operation or (b) as some power-of-two multiple of an odd number not the result of any 3n+1 operation),  it was clear that any such construct including even numbers would be excessively complicated.  So I focused on the odd numbers.

When MapleV4 solved the simultaneous equations defining the descent to 27, without any reference to numeric values or boundary conditions,  giving a family of solutions sharing the arbitrary parameter _N1, it was clear that a first instance had been found of such a pattern or construct.  It proved necessary to impose boundary conditions in the form of restrictions on the values of the header of the descent to membership in  {5[8]}  and of the terminal node in the descent to membership in   {0[3]}  in order to prevent the involved integers from recurring in minor variations of the descent.  Uniqueness in the positioning of every integer is also a requirement of any Escher-like gestalt.

Development from there of the abstract predecessor tree,  and its relationships to a general predecessor tree and the binary version of the predecessor tree,  led quickly to the nomenclature of left descents and right descents in the binary form of the tree , reinforced by the very natural a priori construction of all the left descents in the abstract predecessor tree.

Now,  the predecessor tree must contain exactly one instance of every positive integer.  Developing the residue classes {c[d]} of every element of every left descent in the abstract predecessor tree,  determining the cardinality of the 2^a*3^b*n+c  (i.e. {c[2^a*3^b]})  for each (a,b) pair of values led to an understanding that all those l.d.a.'s element's densities exactly accounted for the totality of the odd positive integers.  The totality of the even positive intgegers as powers-of-2 multiples follows simply.  This powerful result doesn't arise in any way other than via l.d.a.s as far as I am aware.

Another troublesome aspect of any notations employed is agreement upon the direction of specification.  Reading from left to right,  should we agree that we visit nodes of trajectories in predecessor order or successor order?

Since I was using the a priori development of l.d.a.s in the abstract tree from headers in   {5[8]}  to leaf nodes in  {0[3]}  it seemed natural that the left to right order for representation of the steps composing the l.d.a.  trajectory fragments proceeds from header to leaf node, i.e.  from Collatz successor to Collatz predecessor.  It follows from this that to trace the Collatz trajectory from the leaf node one must read the l.d.a. trace from right to left. 

This convention has been followed throughout, even into sequence vectors  (a generalization of l.d.a.s ignoring their restraints).  To follow any path from root to leaf node,  follow the sequence vector from left to right.  To follow the path from anywhere within the Collatz predecessor tree to its root,  follow the sequence vector from right to left.  Another way of remembering this is to recall that the predecessor tree has its root at the top  (computer science style)  and pick up the sequence vector at its left end,  allowing the right end to dangle.  Now,  the vector's orientation is the natural one -- to start at the root and move through the predecessors start at the top;  to start at some integer and follow the path to the root,  start at the bottom.  Both senses of traversal are equally valid,  requiring only that the appropriate set of  (predecessor or successor)  formulas are employed in each case.  This arbitrary convention and the ease of mentally confusing the direction of motion along a path certainly contributes to the difficulty of communication.

Now this complete inattention to the even numbers has caused some to feel that the even numbers have in some way been slighted,  that they do play essential roles in the trajectories,  that a deeper understanding of the trajectories would follow from a gestalt which includes them in the total picture.  This is the motivation for the "pyramid" viewpoint,  for the whole notion of the sequence vector,  and for drawing graphs of trajectories using npathtrk.    Their  (sometimes explicit and sometimes implicit)  consideration of both odd and even integers adds complexity while conferring the desired wider world view.

Sequence Vector

I am indebted to Mensanator for the term sequence vector,  a far more precise term than the "path" I had been carelessly using,  and a careful definition for it.  A sequence vector is simply a vector of   (I've been using extended hexadecimal)  digits,  each of which represents the number of consecutive divisions by 2 encountered at its turn in the described trajectory.  An odd number is assumed on the left of the sequence vector,  an odd number appears between every digit of the vector,  and the last digit is assumed to imply an odd number at the right.  (In this work I use the predecessor direction and don't allow for even numbers at the right end.) In accord with the conventions, if the sequence vector represents a complete trajectory,  the odd number at the left will be 1,  and the odd number at the right will be the integer whose Collatz trajectory is represented.  Sometimes only fragments of Collatz trajectories are represented in a sequence vector.

In a sense the sequence vector is the complete opposite of the l.d.a.s.  L.d.a.s include only odd integers,  omitting entirely any explicit mention of the one or more even numbers which must exist between the explicitly denoted odd ones.  Sequence vectors indicate the number of consecutive even numbers between the implicitly positioned odd numbers.

Mensanator has made extensive investigations of the possibility of loops appearing in Collatz itinieraries employing the sequence vector (in successor order) as a useful notation.

In this work,  sequence vectors are used to denote the paths generated by nbigpath,  the paths traced by npathtrk,  and the path segments requested for production by makepath.

Plots from Npathtrk

Only those aspects of npathtrk's plots of Collatz trajectories which are germane to identifying their key features will be discussed here,  in particular identification of even and odd integers in those plots.  There are only two kinds of line segments in the plots.  One kind represents the 3n+1 operation.  It is sloped slightly upwards,  gaining a single unit of the log[2] scale when going up to the right,  and gaining two units of the log[2] scale when going up to the left.  As befits the 3n+i operation,  the lower end of the sloped line is at an even number,  and the higher end is at an odd number.  The other kind is a vertical segment representing one or more divisions by 2.  Each division by 2 results in loss of a unit on the log[2] scale.  The upper end of the vertical segment is at an even number,  and the lower end is at an odd number.

In the case that the log[2] scale is very compressed due to the occurence of very large numbers so the slope appears horizontal,  the even end of the 3n+1 segment may be identified by the attached downward segment.  In the case of small numeric magnitudes,  examination of the height of the vertical segment gives an immediate impression of the number of successive divisions by 2.

Pyramids

Joe Parranto (JP) proposes that "pyramids" provide an excellent medium in which to explore and characterize features of Collatz trajectories. The concept of pyramids arises from the number of 0-bits in the binary representation of the integers. Numbers like 2^j or 2^j +/-2^(j-1) have j or j-1 consecutive 0-bits while numbers like 2^j-1 have no 0-bits in their binary representation. Whence the number of consecutive divisions by two encountered during a Collatz trajectory at such integers correlates with the pyramid height.

In particular, JP defines two essential trajectory features, SUUI and SUDI encountered during Collatz trajectories.

SUUIs (Sequential Uninterrupted Upward Iterations) run from an odd integer to an even integer via a (3n+1)/2 operation. The value of the SUUI is the number of odd-even pairs encountered before the next integer encountered is even.

SUDIs (Sequential Uninterrupted Downward Iterations) run through a series of divisions by 2 of even integers before encountering a closing odd integer. The value of the SUDI is the number of even integers traversed before the odd integer appears.

Here's an example from JP split out into separate lines for greater ease in tracing the changes and with the punctuation replaced with '*' for the 3n+1 step and '/' for the division by 2 steps.

27*82/41*124/62/ (2 SUUI)(1 SUDI)
31*94/47*214/107*322/161*484/242/ (5 SUUI)(1 SUDI)
121*364/182/(1 SUUI)(1 SUDI)
91*274/137*412/206/(2 SUUI)(1 SUDI)
103*310/155*466/233*700/350/(3 SUUI)(1 SUDI)
175*526/263*790/395*1186/593*1780/890 (4 SUUI)(1 SUDI)
445*1336/668/(1 SUUI)
.....

These definitions raise two complications. They require explicit inclusion of both even and odd numbers and they run in the Collatz trajectory sense, the opposite sense to that I've adopted in my work. If one removes the SUUI and SUDI tags, removes the digits, substitutes 'b' for '*//' and 's' for '*/', reverses the b/s operator string, and concatenates all the parts of the l.d.a. from extension back to leaf node

27*82/41*124/62/                         */*//        sb        bs
31*94/47*142/71*214/107*322/161*484/242/ */*/*/*/*//  ssssb     bssss
121*364/182/                             *//          b         b
91*274/137*412/206/                      */*//        sb        bs
103*310/155*466/233*700/350/             */*/*//      ssb       bss
175*526/263*790/395*1186/593*1780/890/   */*/*/*//    sssb      bsss
445
one gets 'bsssbssbsbbssssbs' precisely the ebsssbssbsbbssssbst string representing the l.d.a. which culminates the descent to 27.

While we are talking about the same thing, it is hardly surprising that we cannot make the translation easily.


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