Personal Reflection — Smallest Children and Surprising Connections
The day after I finished writing the blog, I went for a jog in the usual hot and humid weather. As often happens when I run, my mind drifted toward math. I thought I was done with the Wallis problem, but some new questions popped into my head.
At the beginning of the blog, I introduced regular polygons — squares, octagons, and beyond — but I didn’t yet see how they fit into the parent–child story. As a quick refresher, here is the image animation.
Back then, my focus was just on the two beautiful products: Ivey’s growing n and Hannah’s constant 2. It was only later, while trying to summarize the mechanics of doubling, that the parent/child metaphor originated and then came alive. That clarity was so striking that I expanded the blog from three parts into five.
But the polygon idea never left me. While jogging, I began to wonder: could the parent/child story also explain something as concrete as the side length of a polygon?
It clicked. Each time we double the seats, we’re really doubling the sides of a polygon inscribed in the circle: square → octagon → 16-gon → 32-gon. And what is each side of these polygons? Just one of the chords in our family tree. More specifically: the smallest child at that generation.
- For the square, the side is (the child of the diameter).
- For the octagon, the side is , the smallest child of the square’s side.
- For the 16-gon, the side is , the smallest child of the octagon’s side.
And so it continues. Every polygon with sides is tied to our story. Its side is always the smallest child at generation .
That realization surprised me. A straight question (“what’s the perimeter of an octagon?”) was answered by looking at something round — the circle. At first glance, that seems backward: how can a round object help us measure a straight one? But the circle gave us the family tree of chords, and from that tree, the side lengths of polygons fall naturally into place.
It’s also delightful how this connects to everyday life. A stop sign isn’t just an octagon — it’s a reminder of a third-generation child in the family tree. A pizza cut into 8 or 16 equal slices isn’t just dinner — it’s a circle quietly teaching geometry. These shapes aren’t just practical; they’re living echoes of the harmony hidden in circles.
That’s why I love math. Often, I’m just playing, following a curiosity, and then — suddenly — something clicks. Worlds that seemed far apart connect. Once you see it, you can’t “unsee” it.
And this doesn’t just happen in math. You may have seen it in life: a random act of kindness that multiplies far beyond what you imagined. I remember when I was a new college freshman, unsure of my way, and an older student stopped just to talk with me. It wasn’t about solving a problem or giving advice. It was simply kindness. And yet, it shifted the way I wanted to live my own life.
Euclid, centuries ago, used arc bisection to construct polygons — the very process behind our parent/child chords. And in a much more ancient book, Jesus said: “Whatever you do for the least of these, you do also for me.” To me, that’s the same message: in the smallest things, whether in math or in life, surprising beauty and truth can be found.
✨ From polygons to pizzas, from stop signs to kindness — it’s often the smallest pieces that unlock the larger picture.
So the answer to our polygon question? It’s the “smallest child” in the family tree — a simple chord that carries the weight of the whole structure.
What does the Circle’s Hidden Balance tell me? That small things may look ordinary until you see them in context — then they shine with unexpected beauty.