Posts Tagged ‘Klein bottle’
“A hole can itself have as much shape-meaning as a solid mass”*…
Holes. Caity Weaver wonders about them:
What is a hole?
A hole is a portion of something where something is not. Beyond that, holes are slippery. (As a concept — only some in reality.) Is a hole necessarily empty on both sides, like the gaps in a slice of Swiss cheese? Or need it only be empty on one side, like a pit dug into the earth? Is a hole with a bottom less of a hole than one without one? Can a slit be a hole, or must a hole be vaguely round? Does a straw have two holes, as one Reddit user pondered, or just one — a single thick hole, if you will?…
[She then proceeds to explore the concept etymologically…]
Wait — What Is a Hole?
The Stanford Encyclopedia of Philosophy goes right for the, well… philosophical:
Holes are an interesting case study for ontologists and epistemologists. Naive, untutored descriptions of the world treat holes as objects of reference, on a par with ordinary material objects. (‘There are as many holes in the cheese as there are cookies in the tin.’) And we often appeal to holes to account for causal interactions, or to explain the occurrence of certain events. (‘The water ran out because the bucket has a hole.’) Hence there is prima facie evidence for the existence of such entities. Yet it might be argued that reference to holes is just a façon de parler, that holes are mere entia representationis, as-if entities, fictions.
[There follows a fascinating account of the theories of holes…]
Holes
A whole lot about nothing…
*Henry Moore
###
As we hit ’em where they ain’t, we might spare a thought for mathematician Henri Cartan; he died on this date in 2008. A founding member (n 1934) of and active participant in the Bourbaki group, Cartan made contributions to math across algebra, geometry, and analysis, with a special focus on topology (that branch of math that plays with holes in toruses, Klein bottles, and other other-worldly shapes).
“Geometry is not true, it is advantageous”*…
The tesseract is a four dimensional cube. It has 16 edge points v=(a,b,c,d), with a,b,c,d either equal to +1 or -1. Two points are connected, if their distance is 2. Given a projection P(x,y,z,w)=(x,y,z) from four dimensional space to three dimensional space, we can visualize the cube as an object in familar space. The effect of a linear transformation like a rotation
| 1 0 0 0 | R(t) = | 0 1 0 0 | | 0 0 cos(t) sin(t) | | 0 0 -sin(t) cos(t) |in 4d space can be visualized in 3D by viewing the points v(t) = P R(t) v in R3.
* Henri Poincare
###
As we follow the bouncing ball, we might spare a thought for Felix Klein; he died on this date in 1925. A mathematician of broad gauge, he is best remembered for his work in non-Euclidean geometry, perhaps especially for his work on synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Program, which profoundly influenced mathematics. He created the Klein bottle, a one-sided closed surface–a non-orientable surface with no inside and no outside– that cannot be constructed in Euclidean space.
A Klein bottle
You must be logged in to post a comment.