Manifolds, Star Fox, and Self-versus-Other

Branes, D-branes, M-theory, K-theory … news articles about theoretical physics often mention “manifolds”.  Manifolds are also good tools for theoretical psychology and economics. Thinking about manifolds is guaranteed to make you sexy and interesting.

Fortunately, these fancy surfaces are already familiar to anyone who has played the original Star Fox—Super NES version.

In Star Fox, all of the interactive shapes are built up from polygons.  Manifolds are built up the same way!  You don’t have to use polygons per se, just stick flats together and you build up any surface you want, in the mathematical limit.

The point of doing it this way, is that you can use all the power of linear algebra and calculus on each of those flats, or “charts”.  Then as long as you’re clear on how to transition from chart to chart (from polygon to polygon), you know the whole surface—to precise mathematical detail.

Regarding curvature: the charts don’t need the Euclidean metric.  As long as distance is measured in a consistent way, the manifold is all good.  So you could use hyperbolic, elliptical, or quasimetric distance. Just a few options.

Manifolds are relevant because according to general relativity, spacetime itself is curved.  For example, a black hole or star or planet bends the “rigid rods” that Newton & Descartes supposed make up the fabric of space.

In fact, the same “curved-space” idea describes racism. Psychological experiments demonstrate that people are able to distinguish fine detail among their own ethnic group, whereas those outside the group are quickly & coarsely categorized as “other”.

This means a hyperbolic or other “negatively curved” metric, where the distance from 0 to 1 is less than the distance from 100 to 101.  Imagine longitude & latitude lines tightly packed together around “0”, one’s own perspective — and spread out where the “others” stand.  (I forget if this paradigm changes when kids are raised in multiracial environments.)

Experiments verify that people see “other races” like this. I think it applies also to any “othering” or “alienation” — in the postmodern / continental sense of those words.

The manifold concept extends rectilinear reasoning familiar from grade-school math into the more exciting, less restrictive world of the squibbulous, the bubbulous, and the flipflopflegabbulous.

Geotagging

Geotagged photos (e.g. flickr) and text (e.g. twitter) associate data to a particular point on the globe × time. In other words, a fibre bundle over S²×T.

Imagine the position future historians will be in — if they can synthesise the petabytes of digital data we generate these days. I would love to have crowd-sourced pictures or live-tweeting bystanders’ microblogs of the Yan tie lun 鹽鐵論 (81 B.C.) — Rashomon effect be damned.

In the loop quantum gravity approach, space-time is quantized by a procedure that encodes it in a discretized structure, consisting of spin networks and spin foams.

A spin network consists of an oriented embedded graph in a 3-dimensional manifold with edges labelled by SU(2) representations and edges labelled by intertwiners between the representations attached to incoming and outgoing vertices. These representations relate to gravity in terms of holonomies of connections, and the formulation of Einstein’s equations in terms of vierbein, or tetrads, and dual co-tetrads.

Thus, to a spin networks, or the 1-skeleton of a triangulation by tetrahedra, one assigns operators of quantized area and volume, coming from counting intersection points of a surface, or 3-dimensional regions, with the edges or vertices of the spin network with a multiplicity given in terms of the spin representation attached to the edges and the intertwiners attached to the vertices. 

SOURCE: Listening to Golem

Loop quantum gravity is one of a few general frameworks that may eventually form the basis of how physicists think of the smallest scales of time and distance.

(These frameworks are sometimes called Theories of Everything but they’re really just thoughts-on-the-way-to-theories of brief-and-tiny matter.)

 

GLOSSARY

• SU(2) is the group of 2×2 unitary matrices with determinant 1. They have the form
  
  SU(2) is just like the unit quaternions, which represent rotations in 3-D. Here are the pieces that make up SU(2) . 
  , ,  
• 3-dimensional manifold — any shape that can be made with dough. Including that if you stretch the dough, the outside and the inside stretch.
• oriented graph — things like this:
  
• embedded oriented graph — oriented graphs sculpted in 3-D
• edge — arrows in the above pictures
• spin network — a graph like above and each circle has value +½ or −½
• representation — every group can be represented as a matrix
• holonomy — how to move things in parallel within the dough (curved 3-manifold)
• connection — parallel transport
• tetrad or vierbein — a 4-D spacetime frame of reference

I love randomly throwing out dense mathematical statements from theoretical physicists like this. Sometimes math people sound like wizards canting magical runes.

(Source: http://listeningtogolem.blogspot.com/2010/01/raft-of-medusa.html)