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 very informative. Part Two: Follow @JBraungardt Related posts: Klein Bottle Knot theory Topology Turning a Sphere inside-out 0 to 6 dimensions and back […]

Read More...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: Follow @JBraungardt Related posts: Hyperbolic Space 0 to 6 dimensions and back – simple rotation. Klein Bottle Tesseracts: from 3 to 4 Dimensions. Kojève – hole and ring […]

Read More...This is a simple computer-simulated rotation from a point, which has 0 dimensions, to a line (1 dimension), a square (2), a cube (3), all the way up to 6 dimensions, and then down. Multi-dimensional objects are much more complex than we can imagine. Follow @JBraungardt Related posts: Tesseracts: from 3 to 4 Dimensions. […]

Read More...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.

Read More...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 we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the […]

Read More...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 spaces, because we can think of a knot as a way in which different dimensions can be connected. Mathematicians are working on notation systems for […]

Read More...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 in three-dimensional space. Therefore, it has to intersect itself in order to be representable in three dimensions. It is a non-orientable surface with Euler characteristic […]

Read More...