Herng Yi Cheng
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Tomb raider: an analogy for quantum weirdness

5/11/2013

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Part 1 of a six-part series of adaptations from Terence Tao’s book “Structure and Randomness".

[1.1] Quantum mechanics and Tomb Raider

Tao uses the analogy of the game Tomb Raider as a model to give some intuition for the reasons behind the "weird" consequences of quantum mechanics (QM), in particular the so-called "many worlds interpretation". The game consists of two worlds:

  • Internal System: Lara Croft, the protagonist, needs to navigate tombs and solve the puzzles in them to survive. Suppose that she is intelligent and independent of the Player of the game.
  • External System: the Player of the game can view the game world and "save" the game to "restore" it when Lara dies. "Restoring" resets Lara's memory (well, she simply went back to an "earlier state", right?). Unfortunately, he cannot see inside the tombs, which limits his assistance.

(At this point, Tao apologizes for the violent analogy. Well, I've added a little drama... and fixed a loophole.)

Each save point before a lethal puzzle causes Lara's world (from her point of view) to split into many possible "developments",  some of which involve her failure to solve the puzzle and thus death, while others have her survive. Figure 1 illustrates a sample puzzle: A tomb whose only exit a wooden trapdoor on the floor that leads to an underground passage. The tomb will collapse in, say, five minutes, and Lara ("L") must escape through the passageway, but the trapdoor is locked. The Player (smiley face) helps by creating a save point.
Tomb Raider as an analogy for Quantum Mechanics. The three tries it takes for Lara Croft (L) to survive a puzzle cause a split into many timelines.
Figure 1(a): Playing out a puzzle in a cave. The Player (smiley face) creates a save point to give Lara Croft ("L") three attempts to clear the puzzle; she only survives the last attempt.

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Bird's-eye views of Structure and Randomness (Series)

5/3/2013

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The first posts that I read from Fields Medalist Terence Tao's research blog "What's new" were pieces of advice to aspiring mathematicians, such as mathematical writing tips or what it takes to do math. His blog helped me decide to create my own blog to talk about my own math research, among other things. But his technical posts put me off reading his blog until I recently read a compilation of some of his posts in his book Structure and Randomness (Tao, 2008) (See cover at right).

His expository articles on math and science were surprisingly nontechnical, as well as fun and enriching to read. They conveyed the big picture of the topics in question, imparting an intuition and wonder of the way that math and mathematicians work. I could understand only a few articles, but even so I decided to share my joy with other nontechnical readers like myself. I have extracted what I could understand, expanded on it and adapted it to minimize prerequisite math knowledge, into the following six-part article series:
Structure and Randomness, the book version of Terence Tao's mathematical blog. A compilation of several expository articles, lectures and open problems.
Expository articles, lectures and open problems.

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Amateur Math Research

3/26/2013

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Nearly two months ago, I stumbled across this question on Math.StackExchange:
Is it possible to do mathematics WITHOUT background knowledge?... Is it humanly possible to do mathematics on own without research or is the information content too much to discover identities, or methods of proofs on one's own?... But, how to get started then? How to discover mathematics on my own?
I wouldn't say that I've been "discovering mathematics", but I definitely have been playing around with many of my original math research ideas, some of which I am happy with. So I introspected a little about my amateur math explorations and how my experience can help people who ask this question. Here is my slightly edited response.
Picture
Playful construction is fundamental to research.

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Art is more than the sum of its parts.

3/7/2013

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I'm completely unfamiliar with art, but I've had this interesting thought lately. First of all,
Art is more than the sum of its parts.
(Statement 1)
There are many ways how an art piece can be more than its constituents. The visual arts give a few easy examples, where emotion can be injected into a medium (the "sum of its parts"). The emotion, message, or whatever special qualities that qualify something as art, was not present in the medium itself, yet it present in its final form.
Picture
A painting is more than colors on a canvas.

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Complex Matrix Expansions

12/25/2012

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I was proud of this idea, as it was probably the best early research idea I've come up with.

I became very interested in matrices and linear algebra after reading a paper on modelling origami using rotation and translation matrices (Belcastro and Hull, 2012). I began to play with expressing all kinds of things as matrices and seeing what "meanings" matrix operations had in those contexts.

Somehow complex numbers cropped up, and I decided that they were a good candidate for this "interpretation" because multiplying by a complex number meant a rotation and dilation of the complex plane - or an "amplitwist" (Needham, 1996). So I represented a complex number as a rotation matrix together with a scaling factor.

Given a complex number $z = r\mathrm{e}^{\mathrm{i}\theta}$, let $\mathbf{Z} = r\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}.$ Some interesting properties follow immediately from this correspondence:

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Determinant from Matrix Multiplication

12/23/2012

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Another old idea. Given a $2 \times 2$ matrix $\mathbf{A}$,

$$\det(\mathbf{A}) = \begin{bmatrix}1&0\end{bmatrix}\mathbf{A}^T\begin{bmatrix}0&1\\-1&0\end{bmatrix}\mathbf{A}\begin{bmatrix}0\\1\end{bmatrix}.$$
Prove this and extend it to general $n \times n$ matrices. Can you go further than that?
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  • Home
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