Quantcast

Posted on Sunday, 21 November 2010

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

40 notes

  1. nameleswonderlove reblogged this from proofmathisbeautiful
  2. thehypocriticoaf reblogged this from proofmathisbeautiful
  3. shane4ster reblogged this from proofmathisbeautiful
  4. snarlyneon reblogged this from proofmathisbeautiful
  5. videodroner reblogged this from proofmathisbeautiful
  6. socinematic reblogged this from proofmathisbeautiful
  7. imthebadass reblogged this from isomorphismes and added:
    understand this. Probably what I’ll do tonight.
  8. t3hsiggy reblogged this from proofmathisbeautiful and added:
    Huh, I learned about stereographic projection in terms of the complex plane, but never really considered that it was...
  9. earthsea reblogged this from proofmathisbeautiful
  10. tejemaneje reblogged this from proofmathisbeautiful
  11. proofmathisbeautiful reblogged this from isomorphismes
  12. isomorphismes posted this