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Topology of the United States.

At a gross resolution, just considering the land area, the United States has three disconnected parts:

  • {Alaska, Hawai'i, mainland}.

The complement of the United States is a connected space with a genus of three.

At a finer resolution you would measure a much higher genus. (Does Lake Tahoe count as a “hole” in the mainland US? What about Lake Winnibigoshish?) The Aleutian islands would all register as separate from Alaska, as would the parts of Hawai’i and even Nantucket. So at a fine resolution the complement of the land area of the United States would have a genus well over 100.

http://upload.wikimedia.org/wikipedia/commons/b/b5/Hebridesmap.png

For the UK & Ireland, again it depends on resolution. At a gross scale we could simply talk about two islands but that would leave off Orkney, Man, Guernsey, Jersey, the Hebrides, Skelligs, Ione, Skye, Shetlands, and many more.

http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Wfm_shetland_map.png/780px-Wfm_shetland_map.png

According to various Ordnance Surveyors in the Daily Mail (1995):

  • Our 1:625,000 scale database shows Great Britain (England, Scotland and Wales) has a total 6,289 islands, mostly in Scotland. Of these, 803 are large enough to have been ‘digitised’ with a coastline by our map-makers. The rest are recorded as point features
  • The 1:250,000 scale map of Northern Ireland shows 160 islands; 57 offshore.
  • Our 1:250,000 map of the Republic of Ireland has 279 offshore islands.

So, at fine resolution, the genus of the complement

  • |∁ {UK}∪{Ireland}| = 6289

and at a coarser scale, the genus of the complement of the isles is 803.

(Source: Wikipedia)










Pictures of the 3-sphere, or should I say the 4-ball? It’s a 4-dimensional circle.
Even though these drawings of it look completely sweet, I have a hard time parsing them logically. They’re stereographic projections of the hypersphere. All they’re trying to show is the shell of {4-D points that sum to 1}. That’s lists of length 4, containing numbers, whose items add up to 100%. Some members of the shell are
∙ 10% — 30% — 30% — 30%∙ 60% — 20% — 15% — 5%∙ 0% — 80% — 0% — 20%∙ 13% — 47% — 17% — 23%∙ 47% — 17% — 23% — 13%∙ 17% — 23% — 13% — 47%∙ 0% — 100% — 0% — 0%∙ 5% — 5% — 5% — 85% 
The hypersphere is just made up of 4-lists like that.

The 3-sphere was the object of the Poincaré Conjecture (which is no longer a conjecture). Deformations of this shell — this set of lists — are the only simply-connected 3-manifolds. Any other 3-manifold which doesn’t look holey or disjoint must be just some version of the hypersphere.

Pictures of the 3-sphere, or should I say the 4-ball? It’s a 4-dimensional circle.

Even though these drawings of it look completely sweet, I have a hard time parsing them logically. They’re stereographic projections of the hypersphere. All they’re trying to show is the shell of {4-D points that sum to 1}. That’s lists of length 4, containing numbers, whose items add up to 100%. Some members of the shell are

∙ 10% — 30% — 30% — 30%
∙ 60% — 20% — 15% — 5%
∙ 0% — 80% — 0% — 20%
∙ 13% — 47% — 17% — 23%
∙ 47% — 17% — 23% — 13%
∙ 17% — 23% — 13% — 47%
∙ 0% — 100% — 0% — 0%
∙ 5% — 5% — 5% — 85% 

The hypersphere is just made up of 4-lists like that.

image

The 3-sphere was the object of the Poincaré Conjecture (which is no longer a conjecture). Deformations of this shell — this set of lists — are the only simply-connected 3-manifolds. Any other 3-manifold which doesn’t look holey or disjoint must be just some version of the hypersphere.




How come the planets aren’t arranged spherically around the sun?

the solar system

Instead of being spherical, as seems to be the default shape of things in the universe, the planets are arranged in a more-or-less flat disk — a series of concentric ellipses.

planets and planetessimals

What’s up with that?

Answer: angular momentum + gravity.

The nascent planets (or planetesimals) may have started out spherically arranged around the sun.

But then the spinning sun (or proto-sun) flung its satellites—including Earth—further out into space, like how a merry-go-round flings you off when it spins fast.  So the shape would have become an oblate spheroid, with the major axis being the direction of the sun’s spin.

[an oblate spheroid]

At the same time as centrifugal force pushed out in just one circle, gravity pulled in on the entire sphere.

So the minor axis of the oblate spheroid had nothing pushing it out, only something pulling in, while the major axis was being pushed and pulled.  Eventually the minor axis mostly collapsed and now we humans observe the “flat disk” shape.

SOURCE: Essentials of Geophysics, Part One