🍷 Part 3 — Hannah’s Twist
At first, Ivey’s product (n) and Hannah’s product (2) seemed like separate curiosities. But then Hannah noticed something surprising:
Example
With 4 seats, Ivey’s product = 4
With 8 seats, Ivey’s product = 8. Notice the new seats are colored in purple.
The growth factor from 4 → 8 is exactly 2. Where did the 2 originate? It must originate from the new chord distances added from n=4 to n=8, which are the purple chords.
But that “2” is the same as Hannah’s constant product.
from Hannah's perspective
In fact: If you take just the four new guests from the doubling and rotate the picture slightly, the geometry is identical to Hannah’s setup for n = 4.
Use the 4 "odd" chords from n=8 and rotate them 1/8 of a turn counter clockwise. This produces Hannah's setup for n=4.
That means every doubling step is powered by Hannah’s perspective.
✨ “So Ivey’s growing products and Hannah’s constant 2 turn out to be one story, just told from different seats. But the mystery remains: why does doubling always contribute exactly 2? To answer that, we’ll need to look closer at symmetry, and even uncover a family tree. That secret is what Part 4 reveals.”
Go to Part 4
