**Stereographic projection** is a way of mapping ℝ — which is infinitely long — onto a circle, which has a finite length (circumference).

In a sense, **the circle is larger than ℝ**. That is, you can map injectively every point from ℝ onto the circle, and then there’s still a point left over at the top of the circle. That point gets associated to ∞.

This mapping also generates the **Cauchy distribution from statistics**.

**An infinite thing is smaller than a finite thing.** Weird, right? Well on the other hand, you could also map ℝ to the unit circle and exchange “**x** inches ∈ ℝ” for “**x** degrees ∈ circle”, like wrapping an infinite string around the circle, and it would of course wrap around many times.

So** either thing could be considered bigger**. This is why the axiom of choice is messed up.

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Tags: ℝ, Cauchy distribution, education, geometry, math, mathematics, maths, statistics, stereographic projection

This entry was posted on November 21, 2010 at 3:38 pm and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed.
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