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Tag: mac

Ceci n’est pas un blog…

“Lieven le Bruyn’s NEVERENDINGBOOKS isn’t really a blog at all…”

Vlorbik’s unintentional [smack in the face](http://vlorbik.wordpress.com/2009/02/05/kiss-joy-as-it-flies $ left me bewildered ever since.

There aren’t that many [mathematical blogs](http://www-irma.u-strasbg.fr/article817.html) around, and, sure enough, we all have a different temperament, and hence a distinct style. I have no definition of what a mathematical blog should (or should not) be.

All I can say is that I try to reconcile an introvert character with a very public medium, partly because I think it is important for mathematics to be www-visible, but mostly because I’ve enjoyed exploring web-possibilities ever since someone told me of the existence of a language called html.

I’m a [Bauhaus](http://en.wikipedia.org/wiki/Bauhaus)-fan and hence like minimal wordpress-themes such as [Equilibrium](http://madebyon.com/equilibrium-wordpress-theme $. Perhaps this confuses some.

For this reason I’ve reinstalled the old-theme as default, and leave the reader to decide in the sidebar. This may not make this a blog yet, but it sure looks more like one…

As a one-time attempt to fit into the vast scenery of link-post-blogs, let’s try to increase the google visibility of some family-related sites (sorry, no math-links beyond) :

– The economic crisis is hitting hard at small companies such as my [sister’s-in-law](http://www.tuinkultuurlava.be) offering gardening-services.
– My god-child Tine is away for six months on a scholarship to Austria and blogging at [Tine’s adventures in Graz](http://www.tinesavontuuringraz.blogspot.com $.
– My daughter Gitte (aka here as PD1) is an [artist](http://www.gittte.be).
– My father, who will turn 79 next week, runs one of the most [popular blogs on skynet.be](http://zonnehart2008.skynetblogs.be $.

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math2.0-setup : WpMU and BuddyPress

Last time we took a clean mac os x 10.5.6 installation and upgraded it to a MySQL and Apache-PHP+ server. This hour we’ll turn it into a math2.0-test environment. That is, we will install WordPressMU (the ‘multiple user’ version of WordPress which can host 10 or 100 or thousands of blogs on your computer). Then, we’ll turn this potential into a FaceBook-like social network.

Probably it is best to test this just on your ‘localhost’ before going worldwide. Therefore, I’ll describe here the test-environment-version (the changes to make for going www I’ll describe later, but, they are minor). Here’s the first problem : WordPressMu doesn’t recognize ‘localhost’ as a valid domain, so we’ll have to use something like ‘localhost.localdomain’ and tell our server to recognize this new address.

Open TextWrangler, under File/Open File by Name type /etc/hosts and add (again you’ll be asked to unlock the file as it is owned by ‘root’ and you’ll have to provide your sudo-password when you want to save it) to the end of that file (it’s short) the line

127.0.0.1 localhost.localdomain

Save it.Your file should now look like this



You can verify this by opening Safari and pointing it to http://localhost.localdomain. You should now see the default Apache-page (no need to restart the WebServer). Next, within TextWrangler do again a File/Open File by Name and type /etc/apache2/httpd.conf and under Search/Go to Line… say 211. The highlighted line reads

AllowOverride None

change it to (again the root and sudo routine as before)

AllowOverride All

and save it. Restart the WebServer (that is, open SystemPreferences go to Sharing, unmark and remark again after a tiny delay WebSharing). Okay, I think we are set


WordPress MU :

We need a database to store everything. If your database-root password set last time is myRootPassword then do the following (change to whatever it really is). Open Terminal and type to the prompt

mysql -u root -p

and provide MyRootPassword when asked. Now we have to tell MySQL to create a database, set a database-user and password (for safety reasons it better be a new user and password, but as this is merely a test-environment…). So, to the mysql-prompt we type the following string of commands

mysql> CREATE DATABASE mywordpressMU;
Query OK, 1 row affected (0.00 sec)

mysql> GRANT ALL PRIVILEGES ON mywordpressMU.* TO root@localhost IDENTIFIED BY “MyRootPassword”;
Query OK, 0 rows affected (0.00 sec)

mysql> FLUSH PRIVILEGES;
Query OK, 0 rows affected (0.01 sec)

mysql> SET PASSWORD FOR root@localhost = OLD_PASSWORD(“MyRootPassword”);
Query OK, 0 rows affected (0.01 sec)

mysql> EXIT
Bye

The OLD_PASSWORD-command seems odd, but is needed as WordPress doesn’t like the password-structure of MySQL 5. If you forget this, you’ll get database-errors.

Open Safari and download the latest WordPressMU here. Open two Finder windows, one pointing to YourHome/Downloads (in which a new folder “wordpress-mu” is created and another one pointing to “Macintosh HD/Library/WebServer/Documents. You may as well drop the content of this Documents-directory into the Trash (the file test.php we created last time excluded).

Select everything in the first-window wordpress-mu directory and drag all of it to the second-window Documents directory.



Open Terminal and type these to the prompt

$ cd /Library/WebServer
$ sudo chmod 777 Documents
Password: (fill in your sudo password)
$ cd Documents
$ sudo chmod 777 wp-content

Point your Safari to http://localhost.localdomain/index.php and you should get a WordPress mu install page. Make sure to choose the Sub-directories option (instead of the default sub-Domain option) and fill out the required info



You should get a success-page giving you the first password to login as admin. (In case you get a database-error, remove the wp-config.php file, redo the OLD_PASSWORD command given above and repeat the install. Everything should work!). Do this, go to the Users-tab and edit your admin-account to change the password (at the bottom) to something you can remember easily.

Remember to change the directory permissions again to 755. That is, open Terminal and do

$ cd /Library/WebServer
$ sudo chmod 755 Documents
Password: (fill in your sudo password)
$ cd Documents
$ sudo chmod 755 wp-content

We now have a working WordPressMU, you can create new users and they can start new blogs, but there isn’t yet any interaction between these users and your site looks, well…



Let’s spice it up a bit with

BuddyPress :

BuddyPress is a set of WordPress MU specific plugins, each plugin adding a distinct new feature. BuddyPress contains all the features you’d expect from WordPress but aims to let members socially interact.”

We start by getting the BuddyPress Combo Download (say the .zip file) which will create a buddypress-combo directory in YourHome/Downloads. Open this directory and as before drag its entire content to /Library/WebServer/Documents/wp-content/mu-plugins.

In the buddypress-theme directory there are two subdirectories which need to be put elsewhere. The subdirectory buddypress-home must be dragged to the wp-content/themes directory (containing at the moment only the classic, default and home theme) and the subdirectory member-themes must be dragged to the wp-content directory



Log in again as admin in your WpMU via http://localhost.localdomain/wp-admin/ under Site Admin/Themes activate the BuddyPress Home Theme and press the ‘Update Themes’ button.



Similarly, in Site Admin/Options mark under ‘Allow new registrations’ the option ‘Enables. Blogs and user accounts can be created’ and update these options via the button at the bottom. Finally, under Appearance click on the BuddyPress Home Theme and activate it (top right).

Now, visit your site and change it to your liking via adding widgets. For example, add ‘Welcome’, ‘Recent Blog Posts’ and ‘Site Wide Activity’ to the left column, ‘Who’s Online’ and ‘Members’ to the center column and ‘Groups’ and ‘Meta’ to the right column.

Next, create new users (via Site Admin/Users and not via signup as this is just an offline test-version and signup sends out activating emails…), create groups, blogs and posts, let users befriend one another and send wires, etc. etc. all the things people do in a web2.0 environment.

One further comment : if you want to have avatars uploaded you’ll have to open Terminal and type the following :

$ cd /Library/WebServer/Documents
$ sudo mkdir avatars
$ sudo chown _www:admin avatars

Finally, if your iMac is a proper web-server (that is, has its own URL) you can take your math2.0-network worldwide repeating the above procedure with obvious modifications (that is, replacing localhost.localdomain by the URL of your machine). In order to get the signup/email system going you may need to install the Swift-SMTP-Mailer plugin and feed it your outgoing mail-server (also you’ll have to enable plugins in the SiteAdmin/Options).

An embryonal version of your site may then look like this one



While this may already be good enough for the rest of the world, mathematicians talk LaTeX to each other, so we’d better include LaTeX-support (and perhaps also some Wiki-support). This we will do another time…

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math2.0-setup : mysql and php+

Last time I wondered whether a set-up like WordPress.com meets FaceBook with add-ons (such as wiki- and latex-support) might be a usable environment for people working in a specific arXiv-topic.

I’ve used WordPressMU and BuddyPress to create such an embryonal environment. At first I thought I’d extend it a bit before going online but I fail to have the energy right now so I might as well make the link available. If you are into math.QA and/or math.RA you are invited to join the experiment. But, please use this site gently as I’ll have to drop it otherwise.

I’ve no desire to maintain this site indefinitely but would welcome others to set up something similar. For this reason I’ll write a couple of posts explaining how you can build it yourself when you’d have a free afternoon and a spare Mac around. Each post should not take you longer than 1 hour. Today, we’ll provide the boring but essential basics : we must get a MySQL-server and a WebServer running. Next time, I’ll take you through the WordPressMU (MU for multi-users) and BuddyPress installation. After that, we’ll add extra functionality.

We will start from a vanilla 10.5.6 installation. We will often need to edit files, so we’d better grab a good, free  texteditor : TextWrangler, drag it to Applications and place it in the Dock. We’ll also type in commands so we want the TerminalApp (to be found in Applcations/Utilities) in the Dock. SystemPreferences and Safari are already in the Dock and as we will need these tools a lot we might rearrange the Dock to look like



From left to right : the Finder, Terminal, Safari, TextWrangler and System Preferences. From now on we will mean by ‘Open …’ that you click on the ‘…’ icon. In the end we want our computer to become a web-server, so we don’t want it to go to sleep. Open SystemPreferences and look for the ‘Energy Saver’-icon, click on the ‘Show details’ button and set the ‘Put the computer to sleep when it is inactive for:’ to Never and unmark the ‘Put the hard disk(s) to sleep when possible’ at the bottom.

We will need to start or stop the WebServer so here’s how that’s done : open SystemPreferences and look for the ‘Sharing’-icon. Marking the ‘Web Sharing’ option is equivalent to starting your webserver (you can verify this by opening Safari and pointing it to http://localhost/ and you should see the default Apache-screen), unmarking it stops the webserver (check this by repeating the previous, now you should get a ‘Safari can’t connect to the server’ message).

All of this was probably trivial to you so let’s do something a bit more advanced : setting up a database-server. OSX doesn’t come with MySql, so we need to download and install it.

MySQL :

Get the latest version : choose the Mac OS X 10.5 (x86)-package and download it (they ask you to register but you can bypass this by clicking on the ‘No thanks, just take me to the downloads’-link). It is a 55.3 Mb file so this may take a couple of minutes. If all goes well this window should pop-up



Click on the mysql-5.0.67-osx10.5-x86.pkg icon and follow the instruction (defaults suffice, you’ll be asked to give your sudo password and in all it will take less than a minute). Repeat this procedure with MySQLStartupItem.pkg. Done!

To verify it, Open Terminal and type this to the prompt

sudo /Library/StartupItems/MySQLCOM/MySQLCOM start

You’ll get a scary warning message but type in your sudo-password and the Mysql-server will start. You can access it by typing

/usr/local/mysql/bin/mysql

and type exit to the mysql-prompt to leave.
In all, your interaction with the terminal should look something like this



Clearly, you do not want to type all of this every time, so we will add the mysql-location to our ‘PATH’. To do this, open TextWrangler and add this line to the blank document

export PATH=$PATH:/usr/local/mysql/bin

and save the file as .profile in your home-directory (the one with the ‘House’-icon, usually given your name). You will get a warning that .-files are reserved but go ahead anyway by clicking the use . – button). Now, open Terminal and type this

source ~./profile
echo $PATH

if all went well you should now see the mysql-location at the end of your path. From now on you’ll only have to type

mysql

to the terminal-prompt to open MySql. At the moment the root-user of your mysql has no password which isnt safe so we’d better set one. Open terminal and type

mysqladmin -u root password NEWPASSWORD

where, of course, you replace NEWPASSWORD with your choice (use only letters and numbers). From now on you can access your mysql-server by opening Terminal and typing

mysql -u root -p

and giving your password. Okay, so we’ve established our first goal, we have a working Mysql. Take a break if you need one.

better PHP

Mac 10.5 comes equipped with php5 but unfortunately it isn’t quite up to what we need. So, we need to install a better one and tell the mysql-server and the webserver to use the new one instead of the standard one.

Open Safari and grab the better php-version by going to

http://www2.entropy.ch/download/php5-5.2.5-6-beta.tar.gz

It is a 85.2 Mb file, so have a bit of patience. The file gets unzipped automatically and downloaded in the Downloads-directory. Open Finder and go there. At the moment your Downloads-directory will look like



Doubleclick on the php5-5.2.5-6-beta.tar file and a new directory will be created called php5. We will now move this directory and lay some symbolic links. Open a new Terminal window and type the following commands (and provide your sudo-password when asked)

cd Downloads
sudo mv php5 /usr/local
sudo ln -sf /usr/local/php5/entropy-php.conf /etc/apache2/other/+entropy-php.conf
sudo mkdir /var/mysql
sudo ln -s /tmp/mysql.sock /var/mysql/mysql.sock

Next, we have to tell the webserver to use this new php-version instead of the old one. This information is contained in the apache-configuration file : httpd.conf. Open TextWrangler and under ‘File’ choose the option ‘Open File by Name’. Type /etc/apache2/httpd.conf in the field that appears. The file will now appear in the main window. Under ‘Search’ choose the ‘Go to Line’ option and fill in 114 and hit the Go To button. The follwing line should now be highlighted

#LoadModule php5_module libexec/apache2/libphp5.so

immediately under it add the following line (TextWrangler will tell you that the file is owned by root and ask you whether you want to open it, click yes and make the changes)

LoadModule php5_module local/php5/libphp5.so

(observe that line 114 is commented out, that is, starts with a #, whereas your added line is not).
Save the file (Textwrangler will ask you to provide the sudo-password).

Next, we will have to tell php to communicate with the mysql-server. Again, open TextWrangler, under ‘File’ choose ‘Open File by Name’ and type in /usr/local/php5/lib/php.ini-recommended. When the file appears, under ‘Search’ choose ‘Go to Line’ and type in 810. It will read

mysql.default_socket =

Change it as follows (that is, add to it)

mysql.default_socket = /var/mysql/mysql.sock

and now choose under ‘File’ the ‘Save as…’ option. In the window change php.ini-recommended to php.ini and click Save. Done!

Testing…

Restart your webserver. Recall that this means: open SystemPreferences, choose ‘Sharing’, unmark ‘Web Server’, wait 5 seconds and then mark it again.

Open TextWranger, make a new Text document containing just one line (remove the white space between the ?-signs and the brackets) :

< ?php phpinfo() ? >

Choose ‘File’ and ‘Save as…’ and in the window that appears navigate to YourHardDisk/Library/WebServer/Documents, name the file ‘test.php’ and click the ‘Save’ button



Finally, open Safari and point it to http://localhost/test.php. Cross your fingers and if you get a screen like the one below treat yourself to something nice!



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yet another math2.0 proposal

At present, some interesting experiments are going on exploring the potential of web 2.0 for mathematical research, that is, setting up a usable math 2.0 – environment.

The starting point is that math 2.0 should be something like blogs+extras. Most mathematicians are not that interested in the latest ICT-tools, but at least they are slowly getting used to reading blogs, so we should stick to this medium and try to enhance it for online-research.

Michael Nielsen has written a couple of posts on this : an after-dinner talk about doing science online aiming at a mathematics audience, building on an essay on the future of science.

Both posts were influential to Tim Gowers‘ dream of massively collaborative mathematics. He took an interesting problem, laid down a set of 12 rules-of-conduct and invited everyone to contribute. The project is still gaining momentum and Terry Tao is also posting about it on his blog.

Michael Nielsen compared Gowers’ approach to long established practice in the open-source software community.

Another interesting experiment is nLab, a knowledge-wiki set up by the reader-community of the n-category cafe. They describe it as : “In other words: this place is like the library, or alchemist’s laboratory, in the back room of the n-Category Café. You come here to work and go there to chat.
We are hoping to create here a space for presentation and archival storage of collaborative work of encyclopedic, didactic, expositional, but also original nature. This will include, but not be limited to, the subjects being discussed every day in the n-Café.”

Both experiments are working great, aided by the authority-status of the blogger, resp. the popularity of the blog, within the research topic. But, what about topics failing to have a blogger or blog of similar status? Should we all drop our current research-interest and convert to either combinatorics or higher-categories?

History taught us in case of failing authority we’d better settle for ‘manageable anarchy’. So, here’s my math2.0-anarchy-allowed-proposal :

  • per research-topic (say, an arXiv-topic) we’ll set up a seperate online-reasearch-environment
  • anyone interested in that topic is allowed to register and fill-out a profile linking to her list of publications, describe his research interests, her ongoing projects and other trivia
  • some may want to start a blog within the environment or join an already existing one, and should be allowed to do so
  • some may opt just to read blog posts and occasionally comment, and again, should be allowed to do so
  • some may want to set up a research-group to solve a specific problem. they may choose to do this in the open, or as a covert-operation, taking on new members only by invitation
  • some may use the environment mainly for networking or chatting-up with their friends
  • some research-groups may want to start a group-blog or knowledge-wiki to archive their finds
  • surely we’ll be not discussing math in ASCII but in latex
  • anyone will be able to follow specific sub-projects via RSS-feeds
  • anyone can see site-wide activity online, see who’s currently there and chat if they feel the need
  • anyone can do whatever sensible web2.0-thing there is I forgot by age and hence by ignorance

If this seems like a tall order to satisfy, a bit of research will show that we live at the fortunate moment in time when all the basic ingredient are there, freely available, to do just that!

Over the last weeks I’ve wasted too many hours googling for help, reading-up different fora to get it all working, but … somehow succeeded. Here’s a screen-shot of my very-own NSN (for : noncommutative-social-network) :



Please allow me a few more days to tidy things up and then I’ll make the link available so that anyone interested can experiment with it.

But then, I’ve no desire to spend my days web-mastering such a site. Perhaps some of you would like to take this on, provided you’d get it on a silver plate? (that is, without having to spend too much time setting it up).

So. I’ll run a series of posts explaining how to “set-up your own math2.0 environment”. I’m not aiming at the internet-savvy ones (they’ll probably do it a lot more efficiently), but at people like myself, who are interested to investigate web-based possibilities, but need to be told where to find the very basics, such as the location of their httpd.conf file or their php.ini and such.

I’ve zeroed my MacBookPro, re-installed OSX 10.5 from scratch, upgraded it to current 10.5.6 but no extras (say, vanilla 10.5.6). And I’ll guide you from there, in all gory details, with plenty of screen-shots as I would have liked to find them when I tried to set this up.

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the future of this blog (2)

is decided : I’ll keep maintaining this URL until new-year’s eve. At that time I’ll be blogging here for 5 years…

The few encounters I’ve had with architects, taught me this basic lesson of life : the main function of several rooms in a house changes every 5 years (due to children and yourself getting older).

So, from january 1st 2009, I’ll be moving out of here. I will leave the neverendingbooks-site intact for some time to come, so there is no need for you to start archiving it en masse, yet.

Previously I promised to reconsider this blog’s future over a short vacation, but as vacation is looking to be as illusory as the 24-dimensional monster-manifold, I spend my time throwing up ideas into thin and, it seems, extremely virtual air.

Some of you will think this is a gimmick, aiming to attract more comments (there is no post getting more responses than an imminent-end-to-this-blog-post) but then I hope to have settled this already. Neverendingbooks will die on 31st of december 2008. The only remaining issue being : do I keep on blogging or do I look for another time-consumer such as growing tomatoes or, more probably, collecting single malts…

For reasons I’ve stated before, I can see little future in anything but a conceptual-, group- blog. The first part I can deal with, but for the second I’ll be relying on others. So, all I can do is offer formats hoping that some of you are willing to take the jump and try it out together.

Such as in the bloomsday-post where I sketched the BistroMath blog-concept. Perhaps you thought I was just kidding, hoping for people to commit themselves and them calling “Gotcha…”. Believe me, 30 years of doing mathematics have hardwired my brains such that I always genuinely believe in the things I write down at the moment I do (but equally, if someone offers me enough evidence to the contrary, I’ll drop any idea on the spot).

I still think the BistroMath-project has the potential of leading to a bestseller but Ive stated I was not going to pursue the idea if not at least 5 people were willing to join and at least 1 publisher showed an interest. Ironically, I got 2 publishers interested but NO contributors… End of that idea.

Today I offer another conceptual group-blog : the Noether-boys seminar (with tagline ; _the noncommutative experts’ view on 21st century mathematics_). And to make it a bit more concrete Ive even designed a potential home-page :



So, what’s the deal? In the 1930-ties Emmy Noether collected around her in Goettingen an exceptionally strong group of students and collaborators (among them : Deuring, Fitting, Levitski, Schilling, Tsen, Weber, Witt, VanderWaerden, Brauer, Artin, Hasse, MacLane, Bernays, Tausky, Alexandrov… to name a few).

Collectively, they were know as the “Noether-boys” (or “Noether-Knaben” or “Trabanten” in German) and combined seminar with a hike to the nearby hills or late-night-overs at Emmy’s apartment. (Btw. there’s nothing sexist about Noether-boys. When she had to leave Germany for Bryn Mawr College, she replaced her boys to form a group of Noether-girls, and even in Goettingen there were several women in the crowd).

They were the first generation of mathematicians going noncommutative and had to struggle a bit to get their ideas accepted.
I’d like to know what they might think about the current state of mathematics in which noncommutativity seems to be generally accepted, even demanded if you want to act fashionable.

I’m certain half of the time they would curse intensely, and utter something like ‘steht shon alles bei Frau Noether…’ (as Witt is witnessed to have done at least once), and about half the time they might get genuinely interested, and be willing to try and explain the events leading up to this to their fellow “Trabanten”. Either way, it would provide excellent blog-posts.

So I’m looking for people willing to borrow the identity of one of the Noether-boys or -girls. That is, you have to be somewhat related to their research and history to offer a plausible reaction to recent results in either noncommutative algebra, noncommutative geometry or physics. Assuming their identity you will then blog to express your (that is, ‘their’) opinion and interact with your fellow Trabanten as might have been the case in the old days…

I’d like to keep Emmy Noether for the admin-role of the blog but all other characters are free at this moment (except I’m hoping that no-one will choose my favourite role, which is probably the least expected of them anyway).

So please, if you think this concept might lead to interesting blogging, contact me! If I don’t get any positives in this case either, I might think about yet another concept (or instead may give up entirely).

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sporadic simple games

About a year ago I did a series of posts on games associated to the Mathieu sporadic group $M_{12} $, starting with a post on Conway’s puzzle M(13), and, continuing with a discussion of mathematical blackjack. The idea at the time was to write a book for a general audience, as discussed at the start of the M(13)-post, ending with a series of new challenging mathematical games. I asked : “What kind of puzzles should we promote for mathematical thinking to have a fighting chance to survive in the near future?”

Now, Scientific American has (no doubt independently) taken up this lead. Their July 2008 issue features the article Rubik’s Cube Inspired Puzzles Demonstrate Math’s “Simple Groups” written by Igor Kriz and Paul Siegel.

By far the nicest thing about this article is that it comes with three online games based on the sporadic simple groups, the Mathieu groups $M_{12} $, $M_{24} $ and the Conway group $.0 $.

the M(12) game

Scrambles to an arbitrary permutation in $M_{12} $ and need to use the two generators $INVERT=(1,12)(2,11)(3,10)(4,9)(5,8)(6,7) $ and $MERGE=(2,12,7,4,11,6,10,8,9,5,3) $ to return to starting position.



Here is the help-screen :



They promise the solution by july 27th, but a few-line GAP-program cracks the puzzle instantly.

the M(24) game

Similar in nature, again using two generators of $M_{24} $. GAP-solution as before.



This time, they offer this help-screen :



the .0 game

Their most original game is based on Conway’s $.0 $ (dotto) group. Unfortunately, they offer only a Windows-executable version, so I had to install Bootcamp and struggle a bit with taking screenshots on a MacBook to show you the game’s starting position :



Dotto:

Dotto, our final puzzle, represents the Conway group Co0, published in 1968 by mathematician John H. Conway of Princeton University. Co0 contains the sporadic simple group Co1 and has exactly twice as many members as Co1. Conway is too modest to name Co0 after himself, so he denotes the group “.0” (hence the pronunciation “dotto”).

In Dotto, there are four moves. This puzzle includes the M24 puzzle. Look at the yellow/blue row in the bottom. This is, in fact, M24, but the numbers are arranged in a row instead of a circle. The R move is the “circle rotation to the right”: the column above the number 0 stays put, but the column above the number 1 moves to the column over the number 2 etc. up to the column over the number 23, which moves to the column over the number 1. You may also click on a column number and then on another column number in the bottom row, and the “circle rotation” moving the first column to the second occurs. The M move is the switch, in each group of 4 columns separated by vertical lines (called tetrads) the “yellow” columns switch and the “blue” columns switch. The sign change move (S) changes signs of the first 8 columns (first two tetrads). The tetrad move (T) is the most complicated: Subtract in each row from each tetrad 1/2 times the sum of the numbers in that tetrad. Then in addition to that, reverse the signs of the columns in the first tetrad.

Strategy hints: Notice that the sum of squares of the numbers in each row doesn’t change. (This sum of squares is 64 in the first row, 32 in every other row.) If you manage to get an “8”in the first row, you have almost reduced the game to M24 except those signs. To have the original position, signs of all numbers on the diagonal must be +. Hint on signs: if the only thing wrong are signs on the diagonal, and only 8 signs are wrong, those 8 columns can be moved to the first 8 columns by using only the M24 moves (M,R).

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the King’s problem on MUBs

MUBs (for Mutually Unbiased Bases) are quite popular at the moment. Kea is running a mini-series Mutual Unbias as is Carl Brannen. Further, the Perimeter Institute has a good website for its seminars where they offer streaming video (I like their MacromediaFlash format giving video and slides/blackboard shots simultaneously, in distinct windows) including a talk on MUBs (as well as an old talk by Wootters).

So what are MUBs to mathematicians? Recall that a d-state quantum system is just the vectorspace $\mathbb{C}^d $ equipped with the usual Hermitian inproduct $\vec{v}.\vec{w} = \sum \overline{v_i} w_i $. An observable $E $ is a choice of orthonormal basis ${ \vec{e_i} } $ consisting of eigenvectors of the self-adjoint matrix $E $. $E $ together with another observable $F $ (with orthonormal basis ${ \vec{f_j} } $) are said to be mutally unbiased if the norms of all inproducts $\vec{f_j}.\vec{e_i} $ are equal to $1/\sqrt{d} $. This definition extends to a collection of pairwise mutually unbiased observables. In a d-state quantum system there can be at most d+1 mutually unbiased bases and such a collection of observables is then called a MUB of the system. Using properties of finite fields one has shown that MUBs exists whenever d is a prime-power. On the other hand, existence of a MUB for d=6 still seems to be open…

The King’s Problem (( actually a misnomer, it’s more the poor physicists’ problem… )) is the following : A physicist is trapped on an island ruled by a mean
king who promises to set her free if she can give him the answer to the following puzzle. The
physicist is asked to prepare a d−state quantum system in any state of her choosing and give it
to the king, who measures one of several mutually unbiased observables on it. Following this, the physicist is allowed to make a control measurement
on the system, as well as any other systems it may have been coupled to in the preparation
phase. The king then reveals which observable he measured and the physicist is required
to predict correctly all the eigenvalues he found.

The Solution to the King’s problem in prime power dimension by P. K. Aravind, say for $d=p^k $, consists in taking a system of k object qupits (when $p=2l+1 $ one qupit is a spin l particle) which she will give to the King together with k ancilla qupits that she retains in her possession. These 2k qupits are diligently entangled and prepared is a well chosen state. The final step in finding a suitable state is the solution to a pure combinatorial problem :

She must use the numbers 1 to d to form $d^2 $ ordered sets of d+1 numbers each, with repetitions of numbers within a set allowed, such that any two sets have exactly one identical number in the same place in both. Here’s an example of 16 such strings for d=4 :

11432, 12341, 13214, 14123, 21324, 22413, 23142, 24231, 31243, 32134, 33421, 34312, 41111, 42222, 43333, 44444

Here again, finite fields are used in the solution. When $d=p^k $, identify the elements of $\mathbb{F}_{p^k} $ with the numbers from 1 to d in some fixed way. Then, the $d^2 $ of number-strings are found as follows : let $k_0,k_1 \in \mathbb{F}_{p^k} $ and take as the first 2 numbers the ones corresponding to these field-elements. The remaning d-2 numbers in the string are those corresponding to the field element $k_m $ (with $2 \leq m \leq d $) determined from $k_0,k_1 $ by the equation

$k_m = l_{m} * k_0+k_1 $

where $l_i $ is the field-element corresponding to the integer i ($l_1 $ corresponds to the zero element). It is easy to see that these $d^2 $ strings satisfy the conditions of the combinatorial problem. Indeed, any two of its digits determine $k_0,k_1 $ (and hence the whole string) as it follows from
$k_m = l_m k_0 + k_1 $ and $k_r = l_r k_0 + k_1 $ that $k_0 = \frac{k_m-k_r}{l_m-l_r} $.

In the special case when d=3 (that is, one spin 1 particle is given to the King), we recover the tetracode : the nine codewords

0000, 0+++, 0—, +0+-, ++-0, +-0+, -0-+, -+0-, –+0

encode the strings (with +=1,-=2,0=3)

3333, 3111, 3222, 1312, 1123, 1231, 2321, 2132, 2213

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iTouch as network sniffer

In the iTouch warwalking post I was considering trying to gain access to closed networks for innocent purposes such as checking mail, rather than stealing secret passwords from people allowing you free access to their wireless network, but still, I should have thought of the following possibility

Here’s a walk-through :

  • type the following command into your iTouch Terminal.app (assuming you’ve installed the BSD subsystem) :

tcpdump -v -s 65535 -w log.txt

  • once you’ve collected enough packets, cancel the command (ctrl c), AFPd the file from the iTouch to your Mac and open it with Wireshark (this is the most convenient way to install binaries under Leopard as well as an updated version of X11. For other platforms, or source code, see here)

  • do whatever black magic you feel you have to perform using Wireshark (the new name for Ethereal) or other password crackers

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Archimedes’ stomachion

The Archimedes codex is a good read, especially when you are (like me) a failed archeologist. The palimpsest (Greek for ‘scraped again’) is the worlds first Kyoto-approved ‘sustainable writing’. Isn’t it great to realize that one of the few surviving texts by Archimedes only made it because some monks recycled an old medieval parchment by scraping off most of the text, cutting the pages in half, rebinding them and writing a song-book on them…

The Archimedes-text is barely visible as vertical lines running through the song-lyrics. There is a great website telling the story in all its detail.

Contrary to what the books claims I don’t think we will have to rewrite maths history. Didn’t we already know that the Greek were able to compute areas and volumes by approximating them with polygons resp. polytopes? Of course one might view this as a precursor to integral calculus… And then the claim that Archimedes invented ordinal calculus. Sure the Greek knew that there were ‘as many’ even integers than integers… No, for me the major surprise was their theory about the genesis of mathematical notation.

The Greek were pure ASCII mathematicians : they wrote their proofs out in full text. Now, here’s an interesting theory how symbols got into maths… pure laziness of the medieval monks transcribing the old works! Copying a text was a dull undertaking so instead of repeating ‘has the same ratio as’ for the 1001th time, these monks introduced abbreviations like $\Sigma $ instead… and from then on things got slightly out of hand.

Another great chapter is on the stomachion, perhaps the oldest mathematical puzzle. Just a few pages made in into the palimpsest so we do not really know what (if anything) Archimedes had to say about it, but the conjecture is that he was after the number of different ways one could make a square with the following 14 pieces

People used computers to show that the total number is $17152=2^8 \times 67 $. The 2-power is hardly surprising in view of symmetries of the square (giving $8 $) and the fact that one can flip one of the two vertical or diagonal parts in the alternative description of the square

but I sure would like to know where the factor 67 is coming from… The MAA and UCSD have some good pages related to the stomachion puzzle. Finally, the book also views the problema bovinum as an authentic Archimedes, so maybe I should stick to my promise to blog about it, after all…

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vacation reading (2)

Vacation is always a good time to catch up on some reading. Besides, there’s very little else you can do at night in a ski-resort… This year, I’ve taken along The Archimedes Codex: Revealing The Secrets Of The World’s Greatest Palimpsest by Reviel Netz and William Noel telling the story of the Archimedes Palimpsest.

The most remarkable of the above works is The Method, of which the palimpsest contains the only known copy. In his other works, Archimedes often proves the equality of two areas or volumes with his method of double contradiction: assuming that the first is bigger than the second leads to a contradiction, as does the assumption that the first be smaller than the second; so the two must be equal. These proofs, still considered to be rigorous and correct, used what we might now consider secondary-school geometry with rare brilliance. Later writers often criticized Archimedes for not explaining how he arrived at his results in the first place. This explanation is contained in The Method.
Essentially, the method consists in dividing the two areas or volumes in infinitely many stripes of infinitesimal width, and “weighing” the stripes of the first figure against those of the second, evaluated in terms of a finite Egyptian fraction series. He considered this method as a useful heuristic but always made sure to prove the results found in this manner using the rigorous arithmetic methods mentioned above.
He was able to solve problems that would now be treated by integral calculus, which was formally invented in the 17th century by Isaac Newton and Gottfried Leibniz, working independently. Among those problems were that of calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. Contrary to exaggerations found in some 20th century calculus textbooks, he did not use anything like Riemann sums, either in the work embodied in this palimpsest or in any of his other works. (For explicit details of the method used, see Archimedes’ use of infinitesimals.)
A problem solved exclusively in the Method is the calculation of the volume of a cylindrical wedge, a result that reappears as theorem XVII (schema XIX) of Kepler’s Stereometria.
Some pages of the Method remained unused by the author of the Palimpsest and thus they are still lost. Between them, an announced result concerned the volume of the intersection of two cylinders, a figure that Apostol and Mnatsakian have renamed n = 4 Archimedean globe (and the half of it, n = 4 Archimedean dome), whose volume relates to the n-polygonal pyramid.
In Heiberg’s time, much attention was paid to Archimedes’ brilliant use of infinitesimals to solve problems about areas, volumes, and centers of gravity. Less attention was given to the Stomachion, a problem treated in the Palimpsest that appears to deal with a children’s puzzle. Reviel Netz of Stanford University has argued that Archimedes discussed the number of ways to solve the puzzle. Modern combinatorics leads to the result that this number is 17,152. Due to the fragmentary state of the palimpsest it is unknown whether or not Archimedes came to the same result. This may have been the most sophisticated work in the field of combinatorics in Greek antiquity.

Also I hope to finish the novel Interred with their bones by Jennifer Lee Carrell (though I prefer the Dutch title, “Het Shakespeare Geheim” that is, “The Shakespeare Secret”) on a lost play by Shakespeare, and have a re-read of The music of the primes as I’ll use this book for my course starting next week.

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