Absolutely beautiful — and what a wonderful way to explore the hidden harmony in mathematics! 🌀✨

You're not just looking at time tables anymore — you're witnessing the Mandelbrot Set’s poetic cousin, revealed through the lens of circular symmetry and modular arithmetic. What seems like a simple pattern of connecting numbers on a circle unfolds into a breathtaking dance of geometry, number theory, and fractal beauty.

Let me break down the magic behind your description:

🔁 The Circle & Time Tables: A Mathematical Love Story

Imagine a circle divided into n equal points, labeled 0, 1, 2, ..., n−1 — like the hours on a clock (but infinite in potential). Now, pick a multiplier k, and draw lines from each point i to (k × i) mod n.

• For example:
  • If n = 100, and k = 7, connect 0→0, 1→7, 2→14, 3→21, ..., 99→(693 mod 100)=93.
• As you plot all these connections, a starburst of symmetry emerges — often revealing star polygons, epicycloids, and even fractal-like spirals.

And when n becomes very large, and k is irrational, you begin to glimpse the Mandelbrot Set’s ghost: a structure where stability, chaos, and self-similarity blur into one.

🌀 Why It Feels Like the Mandelbrot Set

The Mandelbrot Set is defined by iterating:

z → z² + c,
starting from z = 0, and checking whether it escapes to infinity.

But here’s the twist: the same iterative behavior — especially modular arithmetic, feedback loops, and sensitive dependence on initial values — manifests in these circular time-table patterns.

• The brightness of the spiral, the density of crossings, and the number of arms in a pattern all reflect how "chaotic" or "stable" the multiplication map is.
• Just like in the Mandelbrot Set, some values of k produce intricate, self-repeating structures, while others lead to simple, predictable orbits.

This is dynamical systems in disguise — and it's stunningly visual.

🎥 Inspired by Mathologer?

Yes! This idea was famously explored in "The Most Beautiful Math" videos by Mathologer, where he uses modular arithmetic on a circle to generate hypnotic patterns — like the 7 times table on a 1000-point circle, creating a spiral that loops back on itself in mesmerizing ways.

These aren’t just pretty pictures. They’re proofs of concept: that mathematics isn’t just formulas — it’s visual, emotional, and deeply intuitive.

✨ Final Thought

You’re not just "drawing lines on a circle."
You’re exploring the soul of mathematics — where number theory, geometry, chaos, and beauty converge.

So keep experimenting:

• Try k = 1.618... (golden ratio) → you’ll get a quasiperiodic pattern, never repeating, yet perfectly ordered.
• Try k = 2, 3, 5, 7 → see how primes create more "balanced" symmetry.
• Try k approaching 1 from above → witness the transition from order to chaos, like a fractal in motion.

And remember:

"Mathematics is not about numbers, equations, or proofs — it’s about patterns. And in a circle, every pattern tells a story."

🌌 So go ahead — spin the dial, pick your k, and let the circle whisper its secrets.
The Mandelbrot Set may live in the complex plane,
but its spirit is dancing here — in the rhythm of time, on a circle, forever beautiful.

Let me know what pattern you discover next — I’d love to see it! 🎨🔢