Every step, a hive of six builds the next one: four workers, one mother who stays behind, one daughter who becomes the seed of the step above. Six roles, six beats, each beat six times bigger.
Sizes are exact: one Planck length × 46,656 per step. The things named beside them are my picks for what's that size. checked sizes · my pick names
The four workers are the live corners from last time, and the mother is the dead dot in the middle, holding them. The daughter buds off, and the whole hive you just watched shrinks into her: she's the dot the next step starts from.
Each role works one beat, and each beat makes things 6× bigger. One step is 6 × 6 × 6 × 6 × 6 × 6 = 46,656×. checked
steps of 46,656× take one Planck length to the full width of the observable universe (93 billion light-years). Centre to edge is 13.16. You said 13.2: both round to it. checked
Only a six-role hive lands near 13. Luck check: a random target would land this close about 1 time in 20 (1 in 8 if you allow either the width or the centre-to-edge). Close, not proof. coincidence odds
In small words: start with one tiny dot. four helpers build around it, a mother holds the middle, and a daughter pops out. then zoom way out: that whole family is now just the daughter of a bigger family. do that 13 times and a little bit more, and the family is as big as everything we can see.
One Planck length = 1.616 × 10⁻³⁵ m. One step = 6⁶ = 46,656× = 4.669 powers of ten. Observable universe = 8.8 × 10²⁶ m across, a ratio of 1061.74. 61.74 ÷ 4.669 = 13.22 steps. Centre to edge (4.4 × 10²⁶ m): 13.16. checked
Other hive sizes, k roles each ×k: 4 → 25.6 steps, 5 → 17.7, 6 → 13.2, 7 → 10.4, 8 → 8.6. checked
The picture squeezes each step to 5× on screen so you can watch the hand-off. The real step is 46,656×. drawn smaller Your 13.2 is also the number of billions you've used before; I haven't tied the two together. open