Carlos Shahbazi has presented the first two articles of a series that introduce the layman to the basics of String Theory.
link: http://mappingignorance.org/2015/05/06/the-geometry-of-string-theory-compactifications-i-the-basics/
"The sphere is also a simple example of a compact manifold, which is a particular class of manifolds of utmost importance in String Theory compactifications, as we will see in a moment. The compactness condition can be intuitively understood using Euclidean space. A manifold embedded in Euclidean space, as the sphere in figure 3, is compact if and only if it is bounded, namely it is contained in a finite size region of E and it is closed, namely it contains all its limiting points."
[The second article is "The geometry of String Theory compactifications (II): finding the Calabi-Yau manifold" -Ed.]
If that's too easy for you; the same author also has "Black hole solutions of N=2, d=4 super-gravity with a quantum correction, in the H-FGK formalism"
link: http://www.mathpubs.com/author/Carlos+S.+Shahbazi
(Score: 2) by Geezer on Monday July 27 2015, @11:24AM
"The natural world (e.g. Trees, animals etc.. ) is non-Euclidean."
Physical structures and shapes in the natural world, narrowly defined as stuff on this planet, can be quite complex and irregular taken as a whole. However, they can all have their component structures' shapes described perfectly well by simple solid geometry. It takes a lot of polygons and spheres to build an elephant.
(Score: 2) by opinionated_science on Monday July 27 2015, @11:30AM
I suggest reading about fractal geometry and the mathematics of non-Euclidean space - it is truly surprising how little of the natural world is compactly described by Euclidean geometry.
I suppose the contrast might be illiustrated as that between a vector drawing and a raster image... one is an arbitrary precision representation and the other is an approximation.