## Archive for January, 2008

### How to create technical illustrations?

January 29, 2008

Here’s a question to all scientists out there: how do you create technical illustrations?

I am a big fan of block diagrams, images and plots. I obviously write my technical documents in TeX, and I love it. However, my documents usually lack the high-quality, sexy images that you see in technical books. I have created diagrams and images using MS Paint, MS PowerPoint, and Adobe Illustrator. However, I am not all that happy with the results. Do you know any graphical editor where one can draw block diagrams and add TeX text to it in a simple, straightforward WYSIWYG manner?

I have heard of MetaPost, MetaGraf and psgraf. Does anyone use them? Any hints?

I generate plots using MATLAB. Thankfully MATLAB allows one to export plots and images in EPS format. I am not totally happy with MATLAB’s plotting capabilities, but I suppose that’s not my biggest problem. I should start using Mathematica and Maple too.

Any suggestions / hints will be warmly welcome.

January 25, 2008

I am sure all of you have encountered CAPTCHA (Completely Automated Public Turing test to tell Computers and Humans Apart) tests before. CAPTCHA tests are generated by computers, but only humans should be able to solve them. Hence, one can tell human users from computers.

A few weeks ago, Yaroslav Bulatov wrote a post about a Russian CAPTCHA that he encountered at the Московский физико-технический институт library website:

January 18, 2008

Consider a set $S$ with $|S|=n$ elements (where all elements of $S$ are distinct). We can build ordered lists of $n$ elements by taking elements from set $S$ and using them (with no repetition) to build $n$-tuples. With $n$ elements we can build $n!$ distinct $n$-tuples.

Let us think in terms of vectors in $\mathbb{R}^n$, where each vector represents a particular ordered list of the elements of some set with $n$ elements. If $x, y \in \mathbb{R}^n$ are permutations of one another, then they contain the same elements, but in a different order; consequently, for every $p \in \mathbb{N}$ we have

$\displaystyle\sum_{i=1}^n x_i^p = \displaystyle\sum_{i=1}^n y_i^p$.

Let $I: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \{0,1\}$ be an indicator function, such that $I(x,y) = 1$ if and only if $x,y$ are permutations of one another ($I(x,y)=0$ otherwise). Let $g_p: \mathbb{R}^n \rightarrow \mathbb{R}$ be defined as

$g_p(z) = \displaystyle\sum_{i=1}^n z_i^p$,

for all $p \in \mathbb{N}$. We can then conclude that

$I(x,y) = 1 \Rightarrow g_p(x) = g_p(y), \quad{} \forall p \in \mathbb{N}$.

The condition $g_p(x) = g_p(y), \quad{} \forall p \in \mathbb{N}$ is necessary. Is it sufficient?! In other words, if $g_p(x) = g_p(y), \quad{} \forall p \in \mathbb{N}$ is true, can we conclude that vectors $x,y$ are permutations of one another?

This is an interesting problem. Think about it!

Update (Jan. 25, 2008): this post was featured on the 25th edition (Silver Jubilee) of the Carnival of Mathematics.

### Outside In

January 17, 2008

Here’s an excellent video on how to turn a sphere inside out. The animation is about 20 minutes long, but well worth watching.

Hat tip: Nuclear Phynance

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### Gallery of Algebraic Surfaces

January 10, 2008

Here’s a very sexy gallery of algebraic surfaces. One of my favorite surfaces is the one defined by equation $x^2 + y^3 + z^5 = 0$, which is named the “sofa“:

__________

Possibly related: