✨ Epilogue — Where Does This Beauty Come From?
This is the circle’s hidden balance. What began as Ivey’s growing products, Hannah’s constant 2, and a surprising twist, resolves into a single truth: messy numbers folding into perfect simplicity, powered by sine and cosine.
As we followed the doubling, we saw something remarkable: each new chord wasn’t random, but a child of the one before it. A family tree began to grow inside the circle. Parents split into children, children into grandchildren, each messier to look at but carrying forward the same inheritance. What seemed like chaos was really kinship — a structure where every generation, no matter how tangled, faithfully reproduced the beauty of its parent.
And because points in a circle take up no space, this family can grow without end. Doubling upon doubling, generation after generation, the pattern continues into infinity. That’s where John Wallis, back in 1655, discovered his infinite product for π. In the same year, he gave us the infinity symbol (∞) — a reminder that some families of thought never end.
Original artwork by Carolyn Kester
For me, this is more than math. Sometimes my own life feels like one of those messy radicals: layered, complicated, even confusing. But when I step back, I glimpse how the mess can fold into something beautiful — especially when shared in community, like we did with this puzzle.
As actuaries, we often work with formulas created to solve practical problems of risk and profit. But in this circle story, the formulas weren’t invented — only discovered. The beauty was already present in the circle, waiting for us to notice.
That contrast is striking. On one side, human-made systems built for utility. On the other, timeless patterns that belong to no one and yet invite everyone.
I like how author Paul Lockhart describes why he does math:
“The thing I want you especially to understand is this feeling of divine revelation. I feel that this structure was ‘out there’ all along; I just couldn’t see it. And now I can! This is really what keeps me in the math game—the chance that I might glimpse some kind of secret underlying truth.” [1]
Where the beauty of math comes from is still a mystery. But as I’ve participated in it, here’s my perspective. I share Lockhart’s sense that these patterns feel like a kind of revelation. I take it one step further: my worldview is that the world and its beauty were created by a single God, one who invites a personal relationship with His creation. From that framework, these beautiful patterns give me a glimpse of God’s character. They feel like a handwritten note. Not because I’m special — quite the opposite. I’m just another messy radical. Yet He reveals beautiful structures for me, and for us, to enjoy.
This story has always been about perspectives. I invite you to consider for yourself whether these patterns are beautiful — and if so, where do you think they come from? Your answer may be different than mine. If you’d like to share your thoughts, I’d love to hear from you on LinkedIn.
🔗 Want to go deeper?
If this story sparked your curiosity, we’ve created a couple of videos exploring the Wallis product itself — the mathematical gem behind these patterns.
They show how the Ivey's products, Hannah’s constant, and the family-tree doubling all connect to Wallis’s famous formula for π. What’s most striking is the contrast: messy roots and tangled numbers on the surface, but perfect harmony underneath — a harmony that was already present in the circle, just waiting for us to be awakened.
👉 A Fraction Away From Forever: The Wallis story wrapped in Romeo and Juliet's geometry.
👉 Noticing . A behind-the-scenes look at our team’s pursuit and what we learned.
