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.

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.

\text{proj}: \mathbb{R} &\to \text{circle} \subset \mathbb{C} \\ x &\mapsto e^{i x}

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


hi-res

41 notes