UYal's blog

A regular triangle feels sharp.
A square feels stable.
A regular pentagon has the symmetry of a well-balanced star.

But what about a regular polygon with a thousand sides?

I found myself wondering: What lies at the end of the sequence—triangle, square, pentagon, hexagon, …, regular n-gon?
If the number of vertices in a polygon increases infinitely, is it still a polygon? Or something else?

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🧮 From Shapes to Infinity

In school, we learned about triangles, squares, pentagons—basic geometric shapes.
But rarely did we ask: What happens beyond that?

One question led me down this path:

“If a regular polygon has an infinite number of vertices, does it become a circle?”

It sounds simple. But to go beyond intuition and into proof, we need math.

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🤖 The Role of GPT

To explore this idea, I turned to GPT.

A vague curiosity began taking mathematical form.
GPT showed how the coordinates of a regular n-gon’s vertices could be expressed as:

Pₖ = (r cos(2πk/n), r sin(2πk/n))  for k = 0, 1, ..., n–1


At first glance, it looks complex. But in essence, it places n points evenly around a circle’s edge.
Connect the dots, and you have a regular n-gon.

Now imagine increasing n.
100 sides.
1,000 sides.
1,000,000 sides.

As the number of vertices increases, the space between them decreases—
and the straight lines begin to resemble a curve.

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➕ A Circle as a Mathematical Limit

Mathematically, the perimeter of a regular n-gon can be approximated as:

arduino 
limₙ→∞ (n/2) × 2r × sin(π/n) = 2πr

In other words:
As n approaches infinity, the polygon’s perimeter approaches the circumference of a circle, 2πr.

So yes—a regular polygon becomes a circle as n grows without bound.

Not just visually, but provably, mathematically.

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🔍 From Wonder to Insight

The idea that a triangle could eventually evolve into a circle, simply by increasing its sides—
that’s not just a curious observation.
It’s a profound insight into how number transforms shape.

And in this journey, GPT wasn’t just a tool.
It became a translator between my imagination and the language of mathematics—
a companion who helped turn intuition into expression.

That’s why I wrote this post.
To remember the moment a simple thought turned into a small discovery.

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