see things differently

Posts tagged with manifolds

Once you’ve accepted that Pac Man takes place on a torus

you can extend the same trick to make higher-genus manifolds.

(Source: math.cornell.edu)

hi-res

When I prove the Poincaré conjecture using surgery theory and Hamilton-Ricci flow, I’m all like:

Branched Manifolds

Williams, Robert F.
Expanding attractors. Publications Mathématiques de l’IHÉS, 43 (1974), p. 169-203

(Source: heptagram)

A road map of mathematical objects by Max Tegmark, via intothecontinuum:

The arrows generally indicate addition of new symbols and/or axioms. Arrows that meet indicate the combination of structures.

For instance, an algebra is a vector space that is also a ring, and a Lie group is a group that is also a manifold.

hi-res

creator: Alberto Sevoso

This really happens, all the time:
Diffusion
$\frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla \cdot \big[ D(\phi,\mathbf{r}) \, \nabla\phi(\mathbf{r},t) \big],$
$\frac{\partial\phi(\mathbf{r},t)}{\partial t} = \sum_{i=1}^3\sum_{j=1}^3 \frac{\partial}{\partial x_i}\left[D_{ij}(\phi,\mathbf{r})\frac{\partial \phi(\mathbf{r},t)}{\partial x_j}\right]$
$\frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla\cdot \left[D(\phi,\mathbf{r})\right] \nabla \phi(\mathbf{r},t) + {\rm tr} \Big[ D(\phi,\mathbf{r})\big(\nabla\nabla^T \phi(\mathbf{r},t)\big)\Big]$
Schooly mathematics is all about rigid, blocky shapes. But since people realised that the infinite limit of a curve is straight, that dynamics are just another dimension (time), then all tame ploop-ploppulous and fandangulous shapes become fair game.
“Later” mathematics takes those simple forms—triangles, squares, circles—to the limit and comes up with these kinds of shapes.
A few of them have non-trivial topology — knotting within themselves or linking with other hoops of different colour.
Those are some nice 3-manifolds.

via chels

(Source: behance.net)

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