Posts tagged with neighbourhood

Topology gets appropriate for qualitative rather than quantitative properties, since it deals with closeness and not distance.

It is also appropriate where distances exist, but are ill-motivated.

These approaches have already been used successfully, for analyzing:

  • • physiological properties in Diabetes patients
  • • neural firing patterns in the visual cortex of Macaques
  • • dense regions in ℝ⁹ of 3×3 pixel patches from natural [black-and-white] images
  • • screening for CO₂ adsorbative materials
Michi Johanssons (@michiexile)


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.

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.

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)

"They don’t have money for a gym membership. They don’t have money for a 24-hour gym pass. This is a ghetto pass. They work out in the ‘hood. A lot of these guys are creative, because they’ve been incarcerated. They know how to work out with [whatever’s around]. And you know, these guys are just as toned, just as ripped. They look better than some of the cats at any fitness club around the world.”