Mystery Teacher Theatre 2000 - Episode 2

So when Dave and I filmed the first episode, we actually filmed two. Given where the whole thing has gone, more episodes seem unlikely from us. (Unless we could do something meta with it...)

Of course, this also means whatever you disliked about the first one you will also dislike at least as much in this one. Audio quality, snark, the guy on the left...

Editing this it struck me that really Sal Khan is a tutor. He's like the strong student in the class that is willing to share with other students how he sees it. So his mathematical viewpoint is a bit procedural, and his understanding of the ideas comes across as a novice. A skilled novice, but without depth. It was his discussion of matrices that really sealed this for me. Undoubtedly there are a lot of teachers who see a matrix as just a table of numbers, but there is so much more. It may be a teaching decision to not share that 'much more' in an introduction, or it may be he's just sharing his viewpoint.

If this is your first exposure to MTT2K, there's still time to look away. If you don't, you might want to read Dave's two posts on how it came to be (one and two) or watch the first episode.  I have a (first crack at a) storify with some of the brouhaha surrounding the first.



I'll just say here, I'm not against Khan Aademy, I'm not jealous, and I'm not against flipped classrooms. I am for quality materials, intentional teaching and learning, and open discussion of ideas. I do believe satire is an appropriate response to exaggerated publicity and overhype.  If I could sing, I'd explore a Tom Lehrer style song on the matter. It's gratifying that the first episode started a big discussion, but at some level this is two guys goofing around to make a point about good use of resources.

In contrast to many places discussing Khan Academy, the comments are open. I'll ask you to be as civil as you can, though, please. Snark, satire and sarcasm are strictly permitted.

A surprising vector norm

A vector norm of an n-dimensional vector $\mathbf{x}$ denoted by $\parallel\mathbf{x}\parallel$, is a real-valued function of $\mathbf{x}$ such that:
-$\parallel\mathbf{x}\parallel > 0$
-$\parallel k\mathbf{x}\parallel = k \parallel\mathbf{x}\parallel$
-$\parallel \mathbf{x+y}\parallel \le \parallel\mathbf{x}\parallel+ \parallel\mathbf{y}\parallel$

It can simply be verified that the Euclidean length is a vector norm:
$$\parallel\mathbf{x}\parallel = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^x} $$

There is an entire class of norms called the $l_p$-norms:
$$\parallel\mathbf{x}\parallel_p = (x_1^p + x_2^p + \cdots + x_n^p)^{\frac{1}{p}}, $$
of which the Euclidean length or Euclidean norm is a member for $p=2$.

A very interesting ( and surprising ) norm is the $l_\infty$-norm:
$$\parallel\mathbf{x}\parallel_{\infty} = \max{(|x_1|, |x_2|, \cdots, |x_n|)},$$
this behavior can be explained from the fact that the higher the $p$, the more $l_p$ is dominated by the element of the largest magnitude.

Magic squares of type 5-by-5

Still in 're-discovery mode' generating magic squares has become trivial, although the computational resources increase fast depending on the size of the square.
$$\left(
\begin{array}{ccccc}
14 & 32 & 6 & 8 & 22 \\
14 & 7 & 30 & 17 & 14 \\
0 & 26 & 24 & 22 & 10 \\
44 & 9 & 16 & 7 & 6 \\
10 & 8 & 6 & 28 & 30
\end{array}
\right)$$
$$\left(
\begin{array}{ccccc}
4 & 22 & 6 & 28 & 42 \\
4 & 37 & 30 & 17 & 14 \\
20 & 26 & 24 & 22 & 10 \\
64 & 9 & 16 & 7 & 6 \\
10 & 8 & 26 & 28 & 30
\end{array}
\right)$$
$$\left(
\begin{array}{ccccc}
4 & 6 & 22 & 26 & 64 \\
6 & 55 & 30 & 17 & 14 \\
16 & 34 & 24 & 42 & 6 \\
86 & 7 & 16 & 7 & 6 \\
10 & 20 & 30 & 30 & 32
\end{array}
\right)$$

About definitions.

Magic square

A magic square is a square matrix with elements in $\mathbf{Z}$ such that the totals of rows, columns and both diagonals are equal.

Normal magic square

A normal magic square is a magic square with elements $1,2, \cdots, n^2$ where $n$ is the size of the matrix.

Latin square

A latin square is a $n$ by $n$ square matrix containing $n$ times the first $n$ elements of the alphabet such that each row and each column contains each letter only once. (i.e. Cayley Table)

Further developments

Clearly, normal magic squares are the most desirable objects in the realm of matrices. I am making detailed notes about this work in the ( still? ) unpublished personal mathematics wiki I am setting up.

Linear Algebra Thoroughly Explained

M373 ( Optimization ) renewed my interest in Linear Algebra which always has been one of my favorite branches in mathematics. Unfortunately I never got any further with it then the standard course up to Eigenvectors. Benedict Gross said in one of his lectures "You can't learn too much Linear Algebra." I suppose that he meant that any investment in learning more Linear Algebra always pays off. Quite a few books on advanced linear algebra have been published, understandably most of them by algebraists which doesn't make these books very accessible, let alone of any practical value. I recently discovered a book called Linear Algebra Thoroughly Explained by Milan Vujicic and published by Springer in 2008. From the foreword: "There are a zillion books on linear algebra, yet this one finds its own unique place among them." The book introduces Complex Inner-product Vector Spaces, new methods for solving linear systems, Dual Spaces, Tensor Products and a lot more.

Linear Algebra Thoroughly Explained @ Springer.

My first 4D-experiment

What you see are the 3D and 2D projections of the rotation of a tesseract in the Z,W plane in 4D.

Programming 4D rotations, SO(4).

Just had a ( minor ) cognition. I was trying to figure out a formula for rotating around a 4-dimensional line. As will become clear soon I did not make much progress. But let's start from the beginning.

If you follow my blog you might have seen videos of a Mathematica program I wrote. You can select a cube or tetrahedron, then from a list of all the rotationaxes of that polyhedron and rotate the object with a rotation slider control. The rotation is visible as well as a projection of it in 2D. - I wanted to extend this to 4-D. So take a 4D polyhedron, i.e. a tesseract. Show the rotation-axes (...) rotate it and show projections in 3D and 2D. - In order to do so I would need 4D rotationformulas, I supposed that they would depend on an axis and an angle just like in 3-D.

The 4D space is simply created by adding an ( imaginary ) axis. Points in 4D have four coordinates. Concepts like distance and angle are similar to the defintions for 2D and 3D. In 2D we rotate around a centre of rotation, a point. But extend 2D to 3D and you see that a 2D rotation is actually a 3D rotation around the z-axis. Thinking along these lines there is no such thing as rotating around a line in 4D. In 4D and above we rotate around planes and the six standard rotational planes in 4D are the XY, XZ, YZ, UX, UY and UZ planes. - This makes perfect sense because if you rotate a cube in 3D along one of the standard rotation axes X, Y and Z the equivalent 4D rotations are those in the YZ, XZ and XY planes. The remaining three rotations seem to distort the cube or make it disappear all together, while in 4D-reality the cube would remain fixed of course.

It turns out that the formula for rotation are simple. Setting up a hypercube is not too difficult either. Haven't added it to the program though.

What might prove difficult though is finding ( computable ) data of the symmetry groups of the 4D-polyhedra.

This website 4D Euclidean Space was very helpful in learning and compiling this data.