28-30.01.14 Geometric algebra

Here’s a link to a gallery of projects which relate mathematics and art.

On Tuesday January 28 Mr. Gunn continued his discussion of the geometric algebra of $\mathbb{R}^2$.  He showed that reflections and rotations can be written as sandwich operators.  For example, reflection of the vector $\mathbf{u}$ in the vector $\mathbf{v}$ can be written $\mathbf{v}\mathbf{u}\mathbf{v}$. This is a sandwich, whose bread is $\mathbf{v}$ and whose meat is $\mathbf{u}$. By concatenating two reflections one obtains another sandwich operator for a rotation around the origin by an angle $\alpha$.

To obtain this rotation formula, define $\mathbf{R} := \mathbf{v}_2 \mathbf{v}_1$ be the geometric product of two unit vectors which meet at an angle of $\dfrac{\alpha}{2}$, CCW from $\mathbf{v}_1$ to $\mathbf{v}_2$. Then we showed in the previous lecture that $\mathbf{R} = \cos{\dfrac{\alpha}{2}} + \sin{\dfrac{\alpha}{2}}\mathbf{I}$.  (Recall that $\mathbf{I} = \mathbf{e}_1 \mathbf{e}_2$ is the pseudoscalar of the algebra).   Recall  that the concatenation of two reflections is a rotation around the meeting point of the lines, through twice the angle between the lines. Then the desired rotation can be written as the concatenation of reflections in these two vectors: $\mathbf{v}_2(\mathbf{v}_1 \mathbf{u} \mathbf{v}_1)\mathbf{v}_2$.  Since the geometric product is associative we can reparenthesize to obtain $(\mathbf{v}_2 \mathbf{v}_1 ) \mathbf{u} (\mathbf{v}_1\mathbf{v}_2)$.  Finally,  let $\mathbf{\widetilde{R}} = \mathbf{v}_1 \mathbf{v}_2$ denote the reversal of $\mathbf{R}$, the element obtained by reversing the order of all geometric products in $\mathbf{R}$.   Then the formula for the rotation reads \mathbf{R}\mathbf{u}\mathbf{\widetilde{R}}$.  The element $\mathbf{R}$ is called a rotor.

21-23.01 Napier’s laws, project progress

On Tuesday 21.01 Mr. Gunn continued his discussion of spherical trigonometry with an overview of Napier’s laws (which arise when an angle or side is a right angle).  He then showed the mathematical animation Not Knot, which builds on the ideas of the video Shape of Space to delve into the hyperbolic structure on a knot complement.  This led to a mention of Scott Kim, who designed the logo for the video.  You can find lots more of his nifty calligraphic inversions on his web site.

On Thursday 23.01 Mr. Gunn complained about two forms of new technology used by the Deutsche Bahn.  He then discussed the evaluation forms he collected on Thursday.  This led him to share memories of another evaluation several decades ago when he first taught a college course; the results were convinced him to give up teaching and seek a programming job.  These memories led him — through the magic of free association — to recount his encounters with two basketball legends, Michael Jordan and Earl “the Pearl” Monroe.  He managed to pull himself out of Memory Lane, however, and spent the rest of the hour presenting an introduction to geometric algebra which he hopes to work up into a short script.

14-16.01 Spherical trigonometry, the shape of space, …

Attention: There is now a Doodle for the project presentations which will be held April 8-10 before lectures start for the summer semester.  Please register for a date.

Snow math art:  Check out this series of photos in the Süddeutsche Zeitung. For example: snowart

Coming up next week:  The course evaluation forms are available.  You can fill them out during the lecture next week either on Tuesday or Thursday, I’ll provide time at the end of the lecture for this purpose.

Tuesday 14.01: I continued the derivation of spherical trigonometry begun last week in connection with the presentation on the Schatz linkage.  The motivation was to be able to solve for the various angles which appear in the motion of this linkage.  The approach I took can also be found in these lecture notes for Geometrie I from Boris Springborn, in particular, pages 5 and 6.

Thursday 16.01: We had the pleasure of hearing a lecture from Jeff Weeks, who is visiting the math department this week.  Jeff talked about “The Shape of Space” and demonstrated his entertaining and educational topology software, which can be found here.