Tagged: Topology

Hyperbolic Space

Here are two computer-animated videos from Youtube that demonstrate some properties of knots, the relations between knots and space, and how we arrive at hyperbolic space. Easy to watch, and...

Turning a Sphere inside-out

How do you turn a sphere inside out, without punching a hole into it? It is possible,as you can see in the transformations in this fascinating video:

Tesseracts: from 3 to 4 Dimensions.

The short video clip below shows the 3D rotations of a 4D object. The deeper questions concern the nature of a “dimension”. How do we know what a dimension is, and if we live in a 3D universe, could we possible also exist in a higher-dimensional universe? it is best to start with simple examples in order to train the mind to think about these questions. The clip below shows a tesseract, which is the four-dimensional analog of a cube. (In geomery, it is called a regular octachoron or cubic prism.) The tesseract is to the cube as the cube is to the square.

Poincaré Conjecture

This explanation is quoted from the  The Clay Mathematics Institute, Poincaré Conjecture. (solved by: Grigoriy Perelman, 2002-3) “If we stretch a rubber band around the surface of an apple, then...

Knot theory

Knot theory is a very fast growing field of mathematics.  Knots are not natural phenomena, and there exists only a finite number of distinct knots in three-dimensional space. Knots define...

Klein Bottle

The Klein bottle is the next step up from a Moebius strip.  A Moebius Strip is a two-dimensional object in three-dimensional space, and a Klein Bottle is a one-dimensional object...

Topology

Definition Topology is a mathematical sub-discipline that studies the properties of objects and spaces. Topology is the modern version of geometry; it was used first in 1847 by the German...