In my last post, I shared how I used AI to explore the strange “mirror world” of negative integers in the Collatz Conjecture. Today, I want to talk about why I built this visualization app in the first place.
The Pen and Paper Struggle Back in 1996, and later during my PhD research, my “computer” was often just a notebook and a pen. I was obsessed with the Collatz problem. I wanted to share that excitement with my friends.
I would sit them down and start writing sequences on a piece of paper: “Okay, take 7. It’s odd, so 3n+1 makes 22. Even, so divide by 2…”
I would scribble number after number, filling pages with arrows and calculations. But more often than not, I’d look up and see their eyes glazing over. They would get lost in the arithmetic. They would forget the rules.
“Wait, do you multiply by 3 for even or odd?”
It was tedious. It was boring for them. And honestly? It was limiting for me. When you are stuck calculating the next step, you can’t see the bigger picture. You miss the forest because you are too busy counting the trees.
The Power of Visualization If I had the tool I built today back then, I truly believe I might have solved this problem—or at least found patterns no one else had seen.
Visualizations don’t just make things pretty; they make complex logic intuitive. Problems like the Collatz Conjecture are known in mathematics as incredibly difficult (some generalizations of these problems are even “undecidable” or comparable to NP-hard problems). They are hard because they look like chaotic noise.
But when you visualize them, the noise turns into structure. You stop thinking about division and multiplication, and you start seeing geometry, spirals, and loops.
Meet the Collatz Explorer I used “Vibe Coding” with Gemini to build a new interactive simulation for my website, Prayogashaala. It fixes the problems I had explaining this 20 years ago.
Here is how it makes the math come alive:
- No More Mental Math: You just type a number—no matter how big—and the app handles the crunching instantly.
- Visual Geometry: Instead of a list of numbers, you see a spiral. The numbers drift and swirl. You can literally see the chaos before it falls into order.
- Loop Detection: Remember the 4 loops I mentioned? The app detects them automatically. It lights up Red for the famous positive loop and Green for the negative loops.
- Audio Feedback: I added sound. As the numbers grow larger, the tone changes. You can hear the sequence rising and falling.
- Infinite Repetition: One thing I always struggled to show on paper was what happens after the loop. Does it just stop? No. In this app, you can see the sequence getting trapped, repeating the cycle over and over, stacking up on the screen.
Why This Matters We often think of math as something we do with our brains, but it helps to use our eyes and ears too. This app turns a dry, abstract puzzle into a sensory experience.
Whether you are a student trying to understand algorithms, or just a curious mind, playing with this tool bridges the gap. You can zoom in to see the specific integers or zoom out to see the massive chaotic web they weave.
Try It Yourself I invite you to experience the problem that has fascinated me for decades. Don’t just read the numbers—watch them dance. The link is: https://prayogashaala.com/collatz-conjecture/
