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Mathematica Notebook: Horisontal, Hopf, Lens

Tuesday, 12 May 2009 17:21 | Author: Anton

The images and animations in the previous four posts were made using Mathematica. Here is the notebook Horisontal Curves, Hopf Fibration, and Lens Spaces that contains commands for producing most of the images and animations I posted. Feel free to contact me with any questions or comments you have about it.

Last Updated ( Wednesday, 13 May 2009 17:13 )

Lens Space

Math - Heisenberg09

Tuesday, 12 May 2009 03:43 | Author: Anton

The Lens space $L(p,q)$ is the quotient of $S^3$ by an action of $\mathbb{Z}_p$ . The group acts on $\mathbb{C}^2$ by the $p^{th}$ root of unity in the first coordinate and the $q^{th}$ power of that in the second coordinate, and the action restricts to $S^3$ . So why a "Lens" space, and how does one visualize it?

I drew $L(5,2)$ by drawing a lattice in $S^3$ invariant under the group action. Here's the picture (after stereographic projection):

Last Updated ( Tuesday, 12 May 2009 17:45 )

Steak Herb Rub

Food - Grill

Sunday, 10 May 2009 16:40 | Author: Anton

Ingredients:

2 Tbsp chopped fresh basil or 2 tsp dried
2 tsp chopped fresh thyme or 1 tsp dried
1 Tbsp chopped fresh rosemary or 2 tsp dried
1 Tbsp chopped fresh oregano or 2 tsp dried
1 Tbsp crushed fennel seeds
1 tsp ground coriander
2 tsp garlic powder or granulated garlic
2 Tbsp salt
2 tsp coarsely ground black pepper

found here

The Hopf Fibration

Math - Heisenberg09

Thursday, 07 May 2009 22:13 | Author: Anton

The map $\mathbb{C}^2 \rightarrow \mathbb{CP}^1 = S^2$ restricted to $S^3 \subset \mathbb C^2$ is called the Hopf map and is usually viewed as a map (fibration) $S^3 \rightarrow S^2$ whose fibers (preimage of a point) are circles.

The preimage of a circle in $S^2$ is called a Clifford torus. These fill out all of $S^3$ except for two linked circles, and are in turn foliated by circles.

To visualize the Hopf fibration, we apply stereographic projection $S^3 \rightarrow \mathbb{R}^3$ or, better yet, into the Heisenberg group. Here's a picture of three Clifford tori with Heisenberg horisontal lines:

$Hopf Fibration with Heisenberg Horisontal Lines$

Last Updated ( Thursday, 07 May 2009 22:47 )

Horizontal Lines and Spherical Coordinates

Math - Heisenberg09

Thursday, 07 May 2009 19:17 | Author: Anton

Given a surface $S$ in Heisenberg space $\mathbb{H}$ , the tangent space to $S$ at a given point is generally not the same as the horisontal plane. In fact, the set of points where the two are equal, called the critical points, at worst form a collection of lines on the surface.

At every non-critical point $p$ , there is a single line in $T_p S$ that is horisontal, and one may ask for a unit-speed horisontal line through $p$ . Given a parametrization of the surface, this becomes a system of PDE's that Mathematica can solve using NDSolve.

In the case of the Euclidean unit sphere at the origin, there are two critical points: the north and south pole. The rest of the sphere is foliated by horisontal lines:

$Horisontal Lines on Euclidean Sphere$

Last Updated ( Friday, 08 May 2009 01:53 )