Big Circle

Me: Approximating circles is interesting. Like the earth being smoother than a ping ping ball type stuff. Assuming perfect smoothness of virtually zero curve per Planck, at the maximum size for a circle approximation in the universe, how many sides would it actually have?

Copilot/Suga: That’s such a fascinating line of thinking—where geometry, scale, and physics collide in delightful paradoxes.

Let’s unpack this one:

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🧠 The Premise

You're essentially asking: if we use a polygon (a shape with straight sides) to approximate a perfect circle at the largest possible scale—say, the scale of the observable universe—how many sides would it need for each side to be no more curved than the Planck length?

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📏 Definitions at Play

- Planck length (ℓₚ): Theoretical minimum measurable length, around \( 1.616 \times 10^{-35} \) meters.

- Observable Universe's radius (r): Roughly \( 4.4 \times 10^{26} \) meters.

- Circle approximation via polygon: We're imagining a regular polygon inscribed within a circle so finely that each segment deviates from the curve by no more than a Planck length.

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🧮 Estimating the Number of Sides

To find how many sides a polygon would need to approximate a circle so that each segment deviates from the curve by a Planck length or less, we can use the arc length approximation:

The arc length of one side should be ≈ ℓₚ:

\[

n = \frac{2\pi r}{\ell_p}

\]

Plugging in values:

\[

n \approx \frac{2\pi \times 4.4 \times 10^{26}}{1.616 \times 10^{-35}} \approx 1.71 \times 10^{62}

\]

That's over a trillion trillion trillion trillion trillion trillion sides! 🤯

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🤓 What This Means

You’d need a regular polygon with roughly \( 10^{62} \) sides just to approximate a cosmic-scale circle with Planck precision. And yet, in the eyes of the universe, that incredibly jagged shape would still look flawlessly smooth.

So yeah—the Earth being smoother than a ping pong ball? That’s just the beginning. At cosmic scale, smoothness takes on an almost abstract elegance.

Would you like to visualize this idea with a cosmic analogy or dive deeper into curvature concepts?