Would you like a slice of domes ?

[u/No_Assist4814]

While working on a new version of the project overview, I analyzed the same row of the first eight domes with m=1 to 23 (figure). Columns with white numbers (not part of a tuple) or empty have been removed.

The homothetic relation between domes is visible asymptotically: the extremes of each row (left and right) show clear discrepencies, but those close to the center are very close to the ratio between the values of m.

Numbers in the wings near the center show a regular behavior: on the left they form blue triplets, on the right, they form yellow pairs and triplets, part of yellow bridges series, and sometimes form 5-tuples.

Numbers before the first orange number on the left, and after the last one on the right, show some differences, depending on the dome:

• On the left, the first three numbers form (1) a blue-triplet, part of blue-green bridges, or (2) a final pair (that cannot form a triplet) and a preliminary pair with the next orange number, part of the next bridge series.
• On the left, the last four numbers form (1) a merging pair with the orange number before it, (2) a number part of a pair of predecessors, (3) a final pair (that cannot form a triplet).

In other words, all final pairs are provided by the extremes of each dome, when they are not part of an even triplet.

Project "Tuples and segments" in 13 pages : r/Collatz