Posts tagged with space

Spring 2011 Lapham’s Quarterly, “Lines of Work”
Another interesting space.

hi-res




Another interesting space.




one can basically describe each of the classical geometries (Euclideanaffineprojective,sphericalhyperbolicMinkowski, etc.) as a homogeneous space for its structure group.

The structure group (or gauge group) of the class of geometric objects arises from isomorphisms of one geometric object to the standard object of its class.

For example,

  • • the structure group for lengths is ℝ⁺;
  • • the structure group for angles is ℤ/2ℤ;
  • • the structure group for lines is the affine group Aff(ℝ);
  • • the structure group for n-dimensional Euclidean geometry is the Euclidean group E(n);
  • • the structure group for oriented 2-spheres is the (special) orthogonal group SO(3).

Terence Tao

(I rearranged his text freely.)

(Source: terrytao.wordpress.com)




Economic geography of the eastern USA
circa 1999, median incomes by zip code
Code and data source to follow in a longer post.

Economic geography of the eastern USA

circa 1999, median incomes by zip code


Code and data source to follow in a longer post.


hi-res




Forestry is the province of variability.

From a spatial point of view this variability ranges from within-tree variation (e.g. modeling wood properties) to billions of trees growing in millions of hectares (e.g. forest inventory).

From a temporal point of view we can deal with daily variation in a physiological model to many decades in an empirical growth and yield model.

Luis Apiolaza

(Source: quantumforest.com)




  • Roshi: What are we surrounded by--on all four sides?
  • Steve: Walls.
  • Roshi: What do the walls make?
  • Steve: The Zendo.
  • Roshi: Do they? What is inside the Zendo?
  • Steve: You, me, the Keisaku stick.
  • Roshi: But what is the _essence_ of the zendo? Could it be right here?
  • Steve: There isn't anything there.
  • Roshi: Aha! Now we are getting somewhere. Let's look at our big mistakes. What makes the lines of the characters on the page?
  • Steve: The space around them?
  • Roshi: Good! And the zendo?
  • Steve: The space in it.
  • Roshi: Yes! It is both the forms that surround us and the spaces with no form at all -- and how they interact together. It is what is in space and what is not. It is how we experience the relationship.




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)










by @MrPrudence:

The Wave is a prominent geological feature located on the Colorado Plateau near to the Utah and Arizona border. Composed of striated waves of cross-bedded sandstone, the landscape appears to have the quality of a frozen liquid, its appearance not unlike cooled molten lava fields. The layers of ribboned red coloured rock are in fact generated by what could be described as one of Earth’s many endlessly long doWhile loops – millions of years of precipitation of water and deposition of oxidization minerals. These geological linear patterns of self-organisation generated by this repeating process are know as Liesegang rings and are commonly found in other sedimentary oscillation ‘computations’ – a good example being Banded Agates (There is also a cross-referencing here with the spatio-temporal output of the Belousov-Zhabotinsky reaction).

Both Leonardo Solaas’s ‘Linear Landscape’ set and Jared Tarbell’s ‘Happy Place’ applet have used algorithms to generate artefacts with notable similarities to the geological patterns found at The Wave. The former uses a particle system to create an illusion of three-dimensional organic surfaces, the latter a node system to give rise to broken sedimentary textures.

Further Viewing & Reading

(Source: dataisnature.com)













Brains sound like a wicked-hard space to think about.
It’s a tightly connected (but not totally connected) network (graph theory)
Each of the nodes’ 3-D location may be important as well (voxels)
The signals propagate through time (dynamical)

Brains sound like a wicked-hard space to think about.

  • It’s a tightly connected (but not totally connected) network (graph theory)
  • Each of the nodes’ 3-D location may be important as well (voxels)
  • The signals propagate through time (dynamical)

hi-res