Roll Over Mercator: Awesome map shows the unreasonable effectiveness of mixture models

I’m not gonna link to all the great xkcd drawings cos if I did, I’d just be linking to xkcd every day, but today’s is just too good to pass by:

He could’ve thrown in some Pacific islands and Scandinavia too, but it’s amazing in any case.

The relevant statistical point here is how good the approximation is! Each continent is coded by just four degrees of freedom (size, rotation, and location in two dimensions).

Just another instance of Neumann’s elephant principle.

1. “Each continent is coded by just four degrees of freedom (size, rotation, and location in two dimensions).”

Some of the islands (e.g. Great Britain) are mirror-reversed — add one to the number of degrees of freedom! (But yes, it’s a great illustration.)

• Anonymous says:

Is it? GB looks like it’s rotated 180° to me. (Can’t speak for the other islands. Indonesia is a little too small to make out.)

• You’re right — I was thinking that if you rotate GB 180 degrees, the “bulge” of Brazil is to the left, not the right. However, I was mistaking the Brazil bulge for the Peru/Ecuador/Columbia bulge. In fact, staring at it now, I can’t see it the way I was seeing it before, and it seems obvious that it’s just a rotation!

• Zhou Fang says:

Mirror reversal can just be encoded by having negative size.

• Carlos Ungil says:

Can it? I think negative sizes gives you a 180° rotation (i.e. two mirror reversals along perpendicular axes, not one).

2. Joshua says:

Is it aslo reladet to ths phmonemon wehre we cna raed smehting lkie thsi?

(only slightly fewer typos than my typical comment)

Or something like

https://youtu.be/2k8fHR9jKVM

• jrkrideau says:

I never heard/saw the McGurk Effect before. Thank you.

• David J. Littleboy says:

Sheesh. That’s _incredibly_ irritating. No matter how much I try, I hear fa. I can’t turn the dratted effect off. Aaaaaaaaarg.

3. Anoneuoid says:

Really the question is why the continents tend to taper from North to South.

4. David Chorlian says:

On a different tangent, one could consider the possible recursion in the lower right hand panel of xkcd.com/2251.

5. jim says:

???

With the four parameters you’ve allowed you could construct any landmass from any other landmass in any position, since there’s no constraint on the space between them, how they fit together, or the number of iterations. The spaces between iterations in North America are arbitrary – they don’t correspond to any physical features. The very rough similarity of Eurasia is an artifact of viewing from the equator. Viewing from the approximate weighted center of the landmass the shape doesn’t look anything like that shown on the map above – at least not without *alot* of squinting.

I’d say it’s not surprising that the simplest shape (SA) is used to represent the most complex ones. NA Europe and Asia all have very fuzzy boundaries, with numerous islands, bulges and tails, so it’s pretty easy to fit just about any shape to them.

6. Mendel says:

The fifth, implicit degree of freedom is the choice of continent for this approximation.

7. Tomas says:

Meh. I can approximate arbitrarily well any distribution using a bunch of 2D gaussians with only position, variance and mixture proportion.

8. Mikhail Shubin says:

The fifth degree of freedom is freedom to have fun making maps like this

9. APR says:

I see 5 Africas of diferent sizes