Mathematics Feed

Reading Nassim Nicholas Taleb's The Black Swan: The Impact of the Highly Improbable


I realized with some sheepishness, after writing about Bruce Sterling's story "The Black Swan," that I had only read second-hand accounts of Nassim Nicholas Taleb's Black Swan idea, but not his actual book of that title. So I ordered it, and now have read the book itself. A "black swan" as defined by Taleb, is an event of "rarity, extreme impact, and retrospective (though not prospective) predictability." Taleb is an ex-quant turned iconoclastic mathematician. This book was a bestseller, and like many best-sellers on arcane topics, it is chatty, opinionated, and a bit on the loose and sloppy side.

As I expected, I enjoyed the book a lot. His subject matter appeals both to my taste in mathematics and my aesthetic of fiction. (My favorite section of David Hartwell's The Dark Descent is The Fabulous Formless Darkenss, which is the section for what he and I call "nature of reality" horror -- stories in which the parameters of what one thought was possible usually change for the worse.)

Other than having it confirm my belief system (always a pleasure), there were two basic things I got out of reading this. 

As I told my husband (David Hartwell) while reading the book, for me this is partly a book about how David's way of doing things is right. The way David publishes books as an editor, the way he runs our small press, the way he collects books, the way he runs the family finances, etc. all tend to limit downside risks while at the same time leaving him exposed to positive Black Swans -- serendipity. (Aren't I a good wife for blogging about how I just read a book that tells me how right my husband is?)


In the second half of the book Taleb spends a lot of wordage and emotion on condemning the "intellectual fraud" of the Bell Curve. I don't really need to be able to judge whether he is correct in terms of all of his examples. But it set me thinking about standardized testing.

Last week and this week, the school district is doing extensive testing of my son, which they do every three years. (This is separate from the group standardized testing.) When my husband I meet with the district in June, we will be presented with a sheaf of paper in which, page after page, our son will be raked in terms of percentiles and numbers of standard deviations from the norm. And we will try to make sense of all this data. Again.

Years ago, I read Stephen J. Gould's The Mismeasure of Man (another book which confirmed my belief systems) and I have a deep distrust of standardized testing anyway. And yet these tests mean something, and it is of crucial importance that I understand what. In past cycles of this, I've wasted a lot of time reading up on what is meant by "processing speed" and how to understand things like the results on subtests of IQ tests. 

My son tends to produce an anomalous pattern of results on such tests, and so far no one has yet to be able to explain what this actually means, though -- who knows -- this year's tests may be different. (We expect to be having our independent expert read the results for us this time.)

So I'm reading along through Taleb's attack on the uses of the Bell Curve, and I realize that of course all of these tests are normed to Bell Curves. They wouldn't give them if they weren't. But all around me I see human attributes that do not fit that model: the ability to draw a recognizable portrait, the ability to play the piano, gender, eye color, etc. Many human differences are not meaningfully normalized to the Bell Curve.

 And so I realized there was one further aspect of what these tests measure that I need to keep in mind: that the imposition of percentiles and standard deviations on what attributes are measured may be no more realistic than contemplating a spherical cow. We somehow need to try to consider whatever we're told each of these these tests are measuring is distributed Bell-Curve-like over the population, or whether it is a different kind of thing. I find this a bit daunting, though probably worth trying.

Say it with Math: Mathematica 7 Home Edition released!


Just in time for Valentines Day, Wolfram Research releases Mathematica Home Edition for $295 (chocolates not included), a small fraction of the full professional edition price: Get it now your own and make your valentines with Mathematica! Find the true mathematical expressions of your love! (Or test out that new sexual geometry without injuring yourself -- just kidding.)

Read all about it at MacWorld, MacNN, and Business Week.

My son Peter is really thrilled that decorating Easter eggs with Mathematica, something that he thought up, is used on the announcement page as an example. 


(See also my post Who Among You are Geek Enough to Decorate Your Easter Eggs in Mathematica?)

Using Mathematica to Analyze the US Election

Jeff Hamrick of the Wolfram Research Special Projects Group has done a blog post giving instructions for using Mathematica to analyze the US Presidential Election. He shows how to pull polling data into Mathematica and how to use some of the Mathematica 6 data libraries to create your own Red State/BlueState maps.

I think this is very interesting stuff because, for those of you with Mathematica out there, you don't have to rely on how other people choose to analyze and map this data. If you have questions of your own you can introduce your own data and draw your own maps. I will be very curious to see what people come up with.

What are the three forms of a number?

The question "What are the three forms of a number?" came home on a worksheet in my son's homework. I have a math degree and work with real math in my daily life. I can think of lots of different forms a number can take, but was unable to come up with the trinity of number forms on my own.

I have a mind like a steel sieve, so I thought maybe I'd learned this but had forgotten. So I Googled it.

Google was no help. The only Google hit for "three forms of a number" is an oblique reference to a Powerpoint presentation for teachers.

And so I consulted a Prentice Hall book entitled Preparing for the New York State 6th Grade Math Test which my son had brought home from school, and in that book, there is an answer to this question.

Am I to conclude that the idea that there are "three forms of a number" is an invention either of the authors of the New York State 6th grade math test, or of text book writers? Or is there out there in the world, some other provenance for this coinage?

Help me out here. I want to know. Are they teaching to the test? Or what?

I am unsettled by the idea that a reason that it is necessary to teach to the test is that the test authors are making up new mathematical coinages of their own that students are responsible for regurgitating.

UPDATE 9/24: So I had a meeting with the teacher yesterday. In summary, the phrase I reacted to comes from the Everyday Math curriculum that was in use in the elementary school when Peter was there, and so there was an expectation that students would recognize it in that form. Other answers differing from the Everyday Mathematics formulation would have been acceptable and were accepted with full credit. (The district has since replaced the U. of Chicago's Everyday Mathematics with a different curriculum, at least for my daughter's first grade class.) Peter's teacher is not using that curriculum (thank God!) at this point. The use of NY State practice tests was as a tool to assess what gaps in knowledge were there to be addressed, but the day-to-day classroom curriculum is not particularly oriented toward the test except as necessary to meet state standards.

I'd say I over-reacted to the assignments given by the middle-school teacher based on experience with the attitudes of the more testing-oriented elementary school teachers. Based on our conference, I'd say that preliminarily, I think the situation with the math class is going to be at least fine, if not better.

So my reaction was not baseless, but my experience as a Chappaqua elementary school parent left me with enough post-traumatic stress that I can be a bit quick on the trigger to judge the middle school which is, in the words of a local pediatric psychiatrist, a good bit more "touchy feely" in its academic stances than our elementary school.

Who Among You are Geek Enough to Decorate Your Easter Eggs in Mathematica?

I dare you. Send your Mathematica egg notebooks to me. This activity, begun last night and avidly pursued this morning, has proved wildly popular with my children who are threatening to run me out of toner in my color printer.

Here's mine:

Easter Egg made in Mathematica
(Mathematica notebook.)

Now show me yours!

UPDATE: Michael Croucher at Walking Randomly has risen to the challenge and put me in an egg. There will be a number of other Mathematica-generated eggs available via the Wolfram Demonstrations Project  next week (URLs TBA) including three fancy ones by my dad, John Cramer. There have also been forays by users of Maple and Sage.  Also, see some real math on real eggs.

URL UPDATE, 3/17/08: Michael Trott, whose book inspired my son to ask for Mathematica-generated patterns on eggs, has made an egg notebook of his own and added it to his Mathematica Guidebook website. He's made a very elaborate notebook which will be expanded and included in the Version 6 edition of the Mathematica Guidebook for Graphics volume.

(Those without Mathematica can view and play with these egg notebooks using the freely downloadable Mathematica Player.)


Also,  there are now a number of egg notebooks on the Wolfram Demonstrations site, with more on the way in the next couple of days.

Watch this space: there are more that aren't up yet.

A Visualization of the Effect of the Fed Rate Cut

Jason Cawley, whom I know through Wolfram Research, has created a neat visualization of the various effects of interest rate cuts by the Federal Reserve. I have discussed various financial news with Jason in the past, and he has written me long interesting letter. So I'm particularly pleased to see his analysis of this one on the Wolfram corporate blog.

Here's the link to his detailed blog post, "Analyzing the Fed Rate Cut in Mathematica," and here's a link to his interactive visualizations on the Wolfram Demonstrations site, entitled Macroeconomic Effects of Interest Rate Cuts.


UPDATE: Congrats to Jason, who made the Wall Street Journal blog!

An animation of how the I-35 Bridge could have collpased with the failure of only 3 pieces


Yifan Hu at Wolfram Research has come up with a computer model of the I-35 bridge that shows how the bridge could have collapsed with the failure of only 3 pieces. He explains:

The picture below shows the computed stresses in a simple 2D model of the I-35W bridge, with red meaning more stress. (I got the geometry from news pictures.) There are definitely aspects of the model that are not realistic. For example, the weight of the trusses themselves isn't included. And, of course, it's in 2D.

So what happens if one of the trusses weakens?

It's easy to include this in the computation by adding an upper bound on the stress in that truss. That just adds another inequality--which FindMinimum has no problem with.

One can actually compute all this in real time inside Manipulate. Here's an animation of the result:

One sees that when the truss with maximal stress weakens (shown in yellow), the stress spreads out to other parts of the bridge. If one weakens the next truss, then the stress propagates further. And when one weakens yet another truss, then the constraints can't be satisfied at all any more--so there is no static equilibrium for the bridge, and the bridge cannot stay standing.

See it HERE.

A New Kind of Bridgebuilding: Stephen Wolfram on How We Might Design Better Bridges in the Future

In a short essay "The Space of All Possible Bridge Shapes," composed in response to the Minneapolis bridge collapse, Stephen Wolfram suggests design principles that could lead to stronger bridges:

. . . it's been known for a while that the best networks don't have that kind of simple structure. In fact, they almost seem in some ways quite random.

Well, what about bridges? I strongly suspect that there are much better truss structures for bridges than the classic ones from the 1800s--but they won't look so simple.

I suspect one can do quite well by using simple rules to generate the structure. But as we know from NKS, just because the rules to generate something are simple, it doesn't mean the thing itself will look simple at all.

Two students at our NKS Summer School (Rafal Kicinger and Tom Speller) have investigated creating practical truss structures this way--and the results seem very promising.

So what should the bridges of the future look like? Probably a lot less regular than today. Because I suspect the most robust structures will end up being ones with quite a lot of apparent randomness.

Conservapedia vs. Wikipedia

Did you know that there exists a "Conservapedia," a conservative reaction to Wikipedia? I discovered this fascinating fact via the Liberal Avenger, which was making fun of their entry on the Moon. My favorite page on Conservapedia is their entry entitled Examples of Bias in Wikipedia. Here are a few choice bits:

  • Wikipedia allows the use of B.C.E. instead of B.C. and C.E. instead of A.D. The dates are based on the birth of Jesus, so why pretend otherwise? Conservapedia is Christian-friendly and exposes the CE deception. . . .
  • Wikipedia often uses foreign spelling of words, even though most English speaking users are American. Look up "Most Favored Nation" on Wikipedia and it automatically converts the spelling to the British spelling "Most Favoured Nation", even there there are far more American than British users. Look up "Division of labor" on Wikipedia and it automatically converts to the British spelling "Division of labour," then insists on the British spelling for "specialization" also. Enter "Hapsburg" (the European ruling family) and Wikipedia automatically changes the spelling to Habsburg, even though the American spelling has always been "Hapsburg". Within entries British spellings appear in the silliest of places, even when the topic is American. Conservapedia favors American spellings of words. . . .
  • Wikipedia has many entries on mathematical concepts, but lacks any entry on the basic concept of an elementary proof. Elementary proofs require a rigor lacking in many mathematical claims promoted on Wikipedia. . . .
  • The Wikipedia entry for the Piltdown Man omits many key facts, such as how it was taught in schools for an entire generation and how the dating methodology used by evolutionists is fraudulent. . . .

Oh, goodness. I wonder which mathematical claims were felt to be unchristian or subject to liberal bias.

Conservapedia is aparently a project of Andy Schafly, son of Phyllis.

(Notably absent from the Conservapedia entry on algebra is a discussion of the Arabic origin of the word.)

Blog Traffic Volatility (Revised Iron Blogger Edition)

Until very recently, I thought all blogs had traffic patterns rather like mine: that there was a certain baseline level of traffic punctuated by huge spikes. For non-ego related reasons I happened to compare my traffic with that of some other sites, and discovered that my traffic pattern seems to be the exception rather than the rule. Given my baseline readership, I seem to have the largest volatility in blog traffic I can find. I think I know why my traffic behaves that way. I just don't understand why everyone elses's doesn't.

The closest match I can find for my traffic patterns is Making Light. (And I think this is attributable to a shared blogging philosophy.) But Patrick and Teresa Nielsen Hayden start with a much higher baseline readership and still don't have traffic spikes as high as mine. Is anyone measuring this sort of thing?

(Click on the graphic to play with the Alexa graph tool.)

Why does my traffic behave this way? Partly because I take the idea of the blogger as public intellectual seriously, and partly because I am interested in the problem of how to re-envision the data served up to us by the news media, and that on occasion, I have been very successful at that.

UPDATE: Here's a link to a page on the mathematics of volatility in the stock market. The difference between stock volatility and blog traffic volatility is that blog traffic mainly moves up from a baseline and then returns to traffic of a similar order of maginatude, whereas stock volatility can go in either direction, through the roof or through the floor.

SO I DECIDED TO PIT MY SPIKES AGAINST SOME STIFFER COMPETITION: I went to the Truth Laid Bear's Blog Ecosystem and pitted my spikes against those of some of the Ecosystem's top ranked sites: "Higer Being" 8. Stop The ACLU (2728 blogs linking in); and "Mortal Humans" 13. Mudville Gazette (1948 blogs linking in); 21. BLACKFIVE (1537 blogs linking in); and 23. the evangelical outpost (1375 blogs linking in). (The Truth Laid Bear is a bit confused about me, but here I am at #4008.)

So. Look at this:

Guess what? I still win. OK. So this is great for my sense of self importance in the world and all that, but what the heck is going on here? Am I unique, or am I representative of a specific type of blogger?

Food for Thought, Served in Several Courses

This is Year's Best season for us, and so I've been reading lots of short fiction, which ought to give me a lot to say in this space, but instead leaves me sort of stunned. If you have a favorite science fiction or fantasy short story published in 2006, now is the time to tell me about it.

Mathlove800_1 One thing I found out about this weekend is Rudy Rucker's podcast of his fine story, "Chu and the Nants" published in  Asimov's. I told David about it and he was pleased I liked it because it is part of Rudy's next novel, Postsingular, which David is publishing at Tor in a year. Meanwhile, Rudy's new novel, Mathematicians in Love, just came out. Put it on your Christmas list.

On a not entirely unrelated topic, I have been surveying the Nerdosphere for worthy math and computation-related blogs, and have noticed the interesting phenomenon that there is a new generation of math and physics grad students who blog. Most of these are very low traffic sites  that would raise hardly a blip on the Technorati rankings, but seem to me indicative of an interesting technological shift.

Historically, physicists were among the first to have web sites (my father had the second in the State of Washington), however scientists have been a bit slower to embrace blogs than they were the web as such. When I've had more time to survey them, I may provide more targeted links.

Here are some blogs that caught my attention while I was surfing for math and math-related blogs:

  • Cellular Automata in Bio-Medicine by Prof. Gershom Zajicek M.D.who is "exploring ways to boost healing processes in a diseases, and particularly in cancer."
  • Anima ex Machina: (has an interesting pic of my friend Kovas presenting a paper last week)
  • Gooseania: My friends invited me over for dinner on Tuesday, just after my theorem was pronounced dead and so I immediately rejected their offer, worried that I'd better get working on something new.
  • NeverEndingBooks: A very pretty blog. The author has two categories for his posts: On Topic and Off Topic, and . . .
  • Backreaction, the blog of Canadian theoretical physicist Sabine Hossenfelder and Stefan Scherer, a physicist who now works in scientific publishing in Germany.

Further to the subject of data, I've had a look at David Sifry's much discussed State of the Blogosphere post. The part I found most interesting  is the discussion of splogs and how Technorati deals with them.

. . . some of the new blogs in our index are Spam blogs or 'splogs'. The good news is Technorati has gotten much better at preventing these kinds of blogs from getting into our indexes in the first place, which may be a factor in the slight slowing in the average of new blogs created each day.

The spikes in red on the chart above shows the increased activity that occurs when spammers create massive numbers of fake blogs and try to get them into our indexes. As the chart shows, we’ve done a much better job over the last quarter at nearly eliminating those red spikes. While last quarter I reported about 8% of new blogs that get past our filters and make it into the index are splogs, I’m happy to report that that number is now more like 4%. As always, we’ll continue to be hyper-focused on making sure that new attacks are spotted and eliminated as quickly as possible.

This relates to something I have been worrying a bit about lately, which is astroturf blogs in quantity founded for an unpleasant and possibly illegal, though not commercial, purpose. I won't link to the example of the phenomenon I have in mind, because that would give them traffic.

If, for example, sometime in the near future, a cult were to order its members to all found blogs to attack a particular individual or institution, how would search engines like Technorati or Google react? Would this be seen as covered by freedom of expression, or would it be seen as analogous to to the founding of many erection-enhancement pill blogs? How will this be dealt with? Could offending an organization with fanatical members ruin your reputation on the web permanently? Or is this something that people like Dave Sifry will have to start monitoring? 

And, if it were happening now, and you were to know about it, what would you be able to do about it? 

And what if you were a target?  What would you do?

Write to Dave Sifry? Call the FBI? Hire an attorney? Hire a publicist? How relevant and enforceable are, for example, the cyberstalking laws?

Food for thought.

Pix from the Wolfram Technology Conference 2006

I've posted a bunch of photos from the Wolfram Technology Conference 2006. Enjoy!

The Emergence of Science Blogging

Declan ButlerDeclan Butler has written a fine article in Nature about blogs by scientists: Science in the web age: Joint efforts

When Tim Berners-Lee invented the World Wide Web in 1989, he saw it as a collaborative workspace for his fellow scientists at CERN, the European particle-physics lab near Geneva, and beyond. His creation went on to surpass his prediction that "the usefulness of the scheme would in turn encourage its increased use". But in the rush to develop the web as a flexible way to find information, the original concept of users interacting in real time was largely forgotten. Fifteen years later, the web seems to be returning to its roots.  . . .

Outside academia, blogs are taking off in a big way. A study published in October by the Guidewire Group, a research firm in new media, says that 90% of marketing communication companies have either launched, or intend to launch, internal blogs. There are now some 20 million blogs, permeating almost every sector of society. But science is a glaring exception, and today there are still only a few dozen scientific bloggers.

Scientists who blog see their activities as a useful adjunct to formal journals, not a replacement. "The standard scientific paper is irreplaceable as a fixed, archivable document that defines a checkpoint in a body of work, but it's static, it's very limited," says Paul Myers, a biologist at the University of Minnesota, who blogs at Pharyngula.

"Put a description of your paper on a weblog, though, and something very different happens," says Myers. "People who are very far afield from your usual circle start thinking about the subject. They bring up interesting perspectives." By sharing ideas online, you get feedback and new research ideas, he says.

A senior US epidemiologist who blogs once or twice a day under the pseudonym 'Revere' on his public-health blog Effect Measure, has attracted a diverse readership. "About 1,500 people visit each day," he says. "If someone told me that I could show up at a lecture hall every day and deliver a short opinion, and that 1,500 people would show up to hear me, I'd be pretty satisfied — 1,500 is twice the subscription of many specialty journals."

But for most scientists and academics, blogs and wikis remain unattractive distractions from their real work. Many consider them an online version of coffee-room chatter, background noise that goes against the very ethos of heavily filtered scholarly information.

This post has been appreciated on a number of science blogs: Pharangula, Scitech Library Question, Nodalpoint, Aetiology, and Effect Measure.

On the subject of science blogging, here's what I want for Christmas: I want Wolfram Research to arrive at an arrangement with SixApart to have some version of WebMathematica run inside blogging software. I've told both companies. I have no idea if anything will come of this Christmas wish. But I think the possibility of having the math out there in a hands-on kind of way would give a big boost to scientific blogging.

As "merciless" explains in the comment section of Effect Measure,

One reason the scientific, mathematical, and engineering community has yet to embrace the internet is because it is still very difficult to type and disseminate math and scientific notation. Most people just have a querty keyboard and one or two scientific typesetting programs, which may nor may not translate well onto another person's computer.

The best solution right now is to convert everything into a pdf file, which is fine for reading, but cannot be manipulated (so it's like reading a book anyway).

New technologies are being created right now that will allow for real-time, editable mathematical and scientific dialogue. Once that gets out (that is, once publishers or somebody decides it's worthwhile to buy it and distribute it), then the internet can be a new and powerful force for worldwide scientific communication.

Come on Stephen WolframMena; come on Stephen: You can make this happen.

UPDATE: Last night I happened across an ISP,, that specializes in hosting webMathematica sites. I am trying to work out the details of how their services can be integrated with my Typepad account.

How to Put Your Wolfram Tones in Your Sidebar on Typepad

ZiporpsTheme.jpgCurrently, if you save a bunch of Wolfram Tones, you can go back and listen to them, but if your friends go to the "My Tones" page, they get theirs, not yours. So lets say you want other people to be able to listen to them, and you want this accessible from your blog's sidebar along with the results from your latest Quizilla experience. Here's how you can do it on Typepad:

First, go to Wolfram Tones and create a bunch of compositions which you save as "My Tones." Then, using Typepad, create a Typelist which you might want to call something like "My Wolfram Tones." (You will want to select the "Link" kind of list. This means that you need to go to the Typelist's Configure page and tell it to "Display Notes as Text," but the advantage is that you can specify the number of tones to disply.)

Modify_1On the My Wolfram Tones page, click on the Modify button on the tone you would like to add to your sidebar.

This will take you to the page for the individual tone. Copy  the URL and then go to your My Wolfram Tones Typelist. Click on Add a New Item. Then paste the URL of your tone into the Box that labeled "Enter a URL to quickly add a new link." This will add your link to the Typelist, but it will come out with the name "Wolfram Tones: Generate a Composition." You want yours to have a better name, something evocative that is going to make your friends want to click there. So call it something like REVENGE OF THE ANDROIDS! and click "Save Changes."

Repeat this process until you've added all the tones you want in your sidebar.

Then, go to the "Edit Current Design" screen for your blog and click on "Change Content Selections." Check the box to add your new Typelist to the sidebar, and then click "Save Changes." This will take you back to the "Edit Current Design" screen. This time, click "Change Ordering" and drag your Wolfram Tones sidebar to where you want it. Then click "Save Changes." And when you are back to the "Edit Current Design" screen, republish your weblog.

Once you've go the Typelist placed, you can add your new Wolfram Tones whenever you want.

WtsidebarHere's a screen shot of my Typelist. How I added the graphics is left as an exercise for the reader.

Young Jonathan Post on Whether to Duck and Cover

Bert2In the comments section, in the context of whether one is more likely to survive the collapse of a building using the duck-and-cover technique or the triangle of life technique (taking refuge next to furniture, not under it), Jonathan Post tells the following story:

When we had "duck & cover" nuclear drills at my Robert Fulton elementary school (P.S. #8), in the late 1950s, I refused to get under my desk, and got sent to the principal's office. I explained to him that we were directly across the East River from downtown Manhattan, and that Wall Street could be ground zero. I explained that the radius of the fireball varied with the 2/3 power of megatonnage, and that the desk would not give even a microsecond of protection. He agreed, and told me not to tell all this to the other students, as it might frighten them. For that matter, he advised me not to scare the teachers. I did not leave my baby teeth for a "tooth fairy." Rather, I had my Mom snailmail them to someone who was researching Strontium-90 levels in teeth, for fallout research.


WolframTones: The Ziporps Gets a Soundtrack

WolframTonesHere's a game for a rainy afternoon: Have your child choose a drawing that the child feels could use a sound track. In this case, my son Peter and I decided he would use a picture he draw a few days ago, a Feel Better Creature he'd drawn for a boy named Ashar in Muzaffarabad:


Peter decided that the creature was called a Ziporps. (If you like Peter's creature, there are many more at

Then go to WolframTones and open up one variety of controls or another an let the child play with it until he or she has a satisfactory soundtrack to go with the drawing. Here is what Peter came up with (and here is a longer version). This is a screen shot of what Peter did to create his composition:


About the soundtrack for his creature, Peter said,

The soundtrack would be perfect for it because the Ziporps goes like the sound of it's tail clenching onto something. The cymbal drum things sound like that. Also he's scuttle-running. His ecosystem is usually either a pond or a river. He's swimming really fast after a bunch of pond fish and ducks. he's trying to figure out where they're going. Because he knows wherever ducks are going, there's food for them and there' s food for him. He eats duck weed and tiny fishes. He's mostly an omnivore. He's a happy creature and he makes people feel better by doing clown stuff and he jumps up on rock ledges and slides down like it was just sand and it looks as if it is moving. That would really amaze someone.

Peter is 8 years old and lives in Pleasantville, NY. He is in the third grade.

Also, if your child would like to draw a picture to make a child in the area affected by the earthquake feel better, upload the picture to Flickr with the tag "earthquakefeelbettercards," and email me and I'll see that the  picture gets where where going. Ashar's dad has agreed to make sure pictures get printed out and given to children they would help.

Earthquake Story Problem

Cnn101105Pakistan has about 2 percent of the world's population living on less than 0.7 percent of the world's land.

Q: What portion of the world's population lives in areas affected by the earthquake? How can you tell? How many of those are under age 18?

Show your work.

Extra credit: What is the population density in the most severely affected areas?

ALSO, there is a fascinating piece by an Indian seismologist, Arun Bapat, about what is to be learned from this earthquake tragedy, including some risk factors to that population your trying to do math about that might not have occurred to you:

. . . let us examine the fate of conventional structures. Press reports and television coverage indicate that there has been extensive damage in the mountainous areas of this region. The area in the vicinity of earthquake epicentre is situated at an altitude of 2,000 to 3,000 meters. Seismic vibrations have more amplitude at higher elevations. For example, take a 30-storeyed building. It will have the least vibrations at the level of the ground floor but, as you go higher, the amplitude of the vibrations increase. The earthquake damage in Baramulla, Uri, Poonch, and so on, which are located at heights of about 1,500 to 2000 metres, and at a distance of about 60 to 90 km from the epicentre, was therefore more severe, as compared to the damage at Islamabad or Haripur, which are at a distance of about 60 to 90 km, but situated at an elevation of about 500 metres or so.

Follow the link to the Indian Express news story, "Is there anybody out there?" It is the first one I've seen to give any account of what I've suspected was going on in the quake-ravaged hills.

Even in the fuzzy Digital Globe satellite images from 1999 -- the best I could get of the region over the internet -- it is apparent on my nice large monitor that the mountainsides are terraced with farmlands, and their creases are dotted with small white rectangles suggestive of roofs. There were people down there.

My Kids Meet Wolfram Tones

WolframTonesGiven confidence at my own skills as a cinematographer by a look through the GoogleVideo, I opened my own account and I uploaded this bad little clip I shot on the 28th of September of my kids in their first encounter with Peter Overmann's Wolfram Tones. It took a while for the busy folks at Google Video to approve my clip, but now it has finally been unleashed upon the public.

Back on the 28th,when I first uploaded it, I wrote:

After dinner this evening, I sat my son Peter, who has just started 3rd grade, down at my computer and let him play with Wolfram Tones for the first time. The first interesting thing that happened was that my daughter Elizabeth, who turns 3 in October, started jamming to the Wolfram Tomes soundtrack on the toy piano in the living room. (I had gotten the video camera out to film Peter, and she started while I was getting set up.) After about 10 minutes of fiddling, Peter came up with something he really liked.

I got out the video camera for a kind of personal note-taking to watch how Peter used the program. What happened while I was getting the camera out and turned on, I find quite remarkable: Elizabeth's jamming along with the music coming from the computer is something she usually only does with live music, implying that somehow the music coming from WolframTones passed the Turning test for her.

Hi from Champaign, Illinois: Visiting Wolfram Research

I've uploaded a bunch of photos from the Wolfram Technology Conference, most from Theodore Gray's fascinating collection of the elements.

Here Gray poses with a copy of Michael Swanwick's Periodic Table of Science Fiction for which he wrote the introduction.

UPDATE: Gray has on his web site the story of the literal periodic table he's sitting on in this picture (many more pictures in my Flickr account).

For well over a hundred years the world has failed to take proper notice of the word "Table" clearly contained in the name of the famous Periodic Table of the Elements.

One evening while reading Uncle Tungsten by Oliver Sacks, I became momentarily confused. He begins a chapter with a description of a periodic table display he loved to visit in the Kensington Science Museum, and in mis-reading the paragraph, I thought it was a table, not the wall display it actually is. While my confusion only lasted a few seconds, when I found out there wasn't a Periodic Table in the British Museum, it left a hole I felt I had to fill.

Actually I would never had had this confusion, or built the table, if I hadn't been thinking for the previous month about the need for a new conference table in my group's common office area. I had already built the Triangle Table to be our coffee table, but we needed a conference table too, and I certainly wasn't going to buy one of those expensive ugly ones from the office supply catalogs.

And I would never have built the table if I didn't happen to have a nice pantograph engraving machine with a complete set of fonts from the closing out auction of a local hospital (new $1700, mine for $50 with fonts).

So really the table is a result of three unlikely and totally unrelated factors coming together at the same time, which probably explains why, to the best of my knowledge, no one else has ever built one like it.

HIs page also contains a discussion of how it was made and many photos of its construction. He was also interviewed on Science Friday about this in 2002 (mp3).

I found the table and its accompanying collection fascinating because, though I have met many collectors, I had never met anyone before who collected elements. It seems such a logical thing to collect, but it had never occurred to me that anyone would do that.

I like my pictures of it, but what that little niche in the Wolfram offices needs is Rosamund Purcell to do it justice. These are objects she would really understand; she is the poet laureate to collection photography.


Rudy Rucker's New Book is Out!

Lifeboxcoversmall_1I'm so pleased: Rudy Rucker's The Lifebox, the Seashell, and the Soul just came in the mail. (I had preordered it.)

Rudy gave a presentation on the material covered in the book at the Conference on the Fantastic in the Arts in Ft. Lauderdale, last spring. I was inspired by the presentation to turn my attention back to mathematics and begin to work out how to go about using the program Mathematica with children. 

So I have already been inspired by this book and I haven't even cracked it yet.

We're having a New York Review of Science Fiction work weekend, and already, several people have wandered by and, spying the book on top of the frog terrarium next to my desk, exclaimed, Hey wow, I didn't know this was out yet.

(Full disclosure: My husband is the editor of some of Rucker's recent fiction, though not of this book.)

PS: Apparently, you can download some of Rucker's recent talks, though I have not tried it.

Origami Tessellations

Playing with tags on Flickr, I discovered an account full of origami tessellations.

Apparently, this is a subgenre of origami I was previously unaware of.  If I was better at folding paper, I might be tempted to try this. (OK, I'll admit it: Even though I'm sure that I won't be very good at it, this is tempting enough that I suspect that I am going to try it even though mine may not come out exactly tessellated.)

And here's another nice Flickr origami photo set, this one with a very different aesthetic:


Did I say I'm going to the Wolfram Technology Conference?

Wtc_1I know that lots of other mommies foist their children on their husbands and go to conferences all by themselves, but I haven't done anything like that since my children were born, and Peter turns eight next month. So, yippeee, I'm going to the Wolfram Technology Conference and I won't have to interrupt a conversation even once to say, Get down from there! or Don't eat that! (Or at least, I don't think I will.)

Cellular Automata Caterpillar

My son found this caterpillar in our yard Friday. I don't know what species it is. (Anyone know?) I took its picture because of the cellular automata-style patterns on its back. I sent it to the folks at Wolfram, who I'm sure could tell me what pattern that is.

(Regarding the ichor it's sitting in, we didn't know if it was injured or whether this was some other kind of secretion.)

Future Shock & Third Grade Math

DonpostmutantI'm a bit unsure what I shold have said this morning when my son's new third grade teacher told me, in the context of a discussion of the math cirriculum, that in the third grade "we don't teach memorization; we teach concepts." Perhaps she was a little too candid.

Good thing that here in the Future we can all work out multiplication problems from first principles whenever we need to multiply, isn't it?


WolframtonesAND NOW FOR SOMETHING COMPLETELY DIFFERENT: In the midst of all this mess, I got an email from one of the folks at Wolfram telling me about their lovely new Internet widget WolframTones, which is essentially the aesthetics of A New Kind of Science rendered as sound. One of the potentially revolutionary strengths of Mathematica is it's ability to render mathematics as sound, allowing us to gain greater understanding of math using our faculties for appreciating and understanding music. I've been looking into this myself, reading up on the neurology of math. One interesting book on this subject I have in hand is Functional Melodies: Finding Mathematical Relationships in Music by Scott Beall.

But what is special about the Wolfram version, and sets it apart from other attempts to integrate mathematics and music, is that it takes on the gnarly natural mathematics derived from Wolfram's attempt to parse the complexity of the geometry of nature. The piano selections remind me of Philip Glass's "Closing," which I think of as the best Thinking Music I have in my iTunes.

Teaching Perfect Squares with Legos

Lego_pyramidOne of my summer projects is to teach Peter his multiplication tables before school starts. Our exceptionally fine school district had an extremely difficult time teaching him his basic addition and subtraction facts in the first and second grade, and I have no reason to believe that they will have more success in the third grade with multiplication. Peter has David's amazing associative memory, and while associative memory is great for learning about, say, red efts, since calling to mind all the information you have about surrounding concepts such as salamanders and newts gives you context and allows you to interpolate information you don't know. But for recalling information about, say, the number seven it is a disaster (as my Google search link handily illustrates: nearly a billion results for the numeral 7; only a hundred and sixteen million for the word seven spelled out).

I have arrived at this formulation: Memory is something I do; memory is something that happens to David and Peter. So in order to get Peter over certain academic hurdles, I need to teach him how to work at memory. Simple recitation does not do it for numbers. The public schools have a slightly more complex technique that boils down to repetition which has failed us utterly, so far.

So I have been looking for alternatives. One of the alternatives, has been assigning multiplication problems to particular places in a classically organized "memory palace" structured around the pool area and grounds of the hotel where the International Conference on the Fantastic in the Arts is held. Each memory place is a place where he found a memorable creature. (The iguana he spotted by the whirlpool is given the spot 7 X 7 and is names Fortunine to invoke 49; the place he found a favorite caterpillar is designated 8 X 8, and the caterpillar is named "Sticky Boy" to invoke 64.) This was succeeding up to a point, but lacked a structure that could be extrapolated upon.

So this afternoon, I hit upon the idea of building a Great Pyramid, complete with a Lego Pharaoh, to illustrate the concept of perfect squares in a way that could be generalized to other multiplication problems, and would also allow us to deduce the existence of prime numbers.

The hardest part was sorting his vast and diverse collection of Legos for the collection of 200-odd square Legos with four bumps on them. This allowed us the make the first 8 layers of the pyramid. Starting with one, I had him tell me what the product of each number was when multiplied by itself; then we collected the right number of square Legos,; then we built the next layer of the pyramid. (Because of the tightness of the fit needed for the Legos near the middle, I did the middle parts, and he did the perimeters.) Having verified that the square of each number indeed yielded a square, we moved on to rectangles; and then we demonstrated experimentally that there are some numbers of blocks that can't be made into rectangles (the example we tried was 19). Then I explained about prime numbers.

I am pleased with this Lego activity, but also think that it would not have worked if I had not first helped him memorize the perfect squares using the pool-side memory palace. I was taking things he had memorized as arbitrary concepts and giving them a more conceptually based architectural structure which can be extrapolated from.

PS: I must say I'm rather hardpressed to understand the intended mneumonics of the "Fact Triangles" in the Everyday Mathematics curriculum that Peter's school uses.

A Few Shiny Pebbles on the Infobeach

I wrote to Alice Flaherty, expert on the neurology of writing, for help with references on the neurology of math. She suggested some places to look and some search terms, so I've been playing with PubMed and discovering interesting things such as that a lot more seems to be known about the neurology of metaphor than about the neurology of math. I came across a couple of articles with interesting descriptions which I though I'd share:

Research into the origins and characteristics of unicorns: mental illness as the unicorn. Abstract:

Ethical Hum Sci Serv. 2000 Fall-Winter;2(3):181-92.
Research into the origins and characteristics of unicorns: mental illness as the unicorn.
Simon L.
Kingsborough Community College, City University of New York, USA.

    Basic research, particularly into the psychological and neurological underpinnings of schizophrenia and other "mental illnesses," is flawed because of its adherence to the ideology that unwanted, hard-to-understand behavior constitutes true medical illness. It is argued here that psychiatric diagnostic terms represent moral judgments rather than medical entities. By reducing experimental subjects to a moral label, and assuming that neurological differences associated with unwanted behavior are brain diseases, researchers fail to take into account the conscious experience, organization of self and self-image, patterns of motivation, history and social contexts of their patients. The failure to consider the psychology of their subjects renders the results of these studies ambiguous and irrelevant for any uses except bolstering the biomedical model of psychiatry.

    PMID: 15278984 [PubMed - indexed for MEDLINE]

(I had recently noticed that the literature associated with various conditions affecting the social skills is often contaminated by the researchers' dislike of the research subjects.)

And Neural activity associated with metaphor comprehension: spatial analysis.

Neurosci Lett. 2005 Jan 3;373(1):5-9.
Neural activity associated with metaphor comprehension: spatial analysis.
Sotillo M, Carretie L, Hinojosa JA, Tapia M, Mercado F, Lopez-Martin S, Albert J.
Departamento de Psicologia Basica, Facultad de Psicologia, Universidad Autonoma de Madrid, 28049 Madrid, Spain.

    Though neuropsychological data indicate that the right hemisphere (RH) plays a major role in metaphor processing, other studies suggest that, at least during some phases of this processing, a RH advantage may not exist. The present study explores, through a temporally agile neural signal--the event-related potentials (ERPs)--, and through source-localization algorithms applied to ERP recordings, whether the crucial phase of metaphor comprehension presents or not a RH advantage. Participants (n=24) were submitted to a S1-S2 experimental paradigm. S1 consisted of visually presented metaphoric sentences (e.g., "Green lung of the city"), followed by S2, which consisted of words that could (i.e., "Park") or could not (i.e., "Semaphore") be defined by S1. ERPs elicited by S2 were analyzed using temporal principal component analysis (tPCA) and source-localization algorithms. These analyses revealed that metaphorically related S2 words showed significantly higher N400 amplitudes than non-related S2 words. Source-localization algorithms showed differential activity between the two S2 conditions in the right middle/superior temporal areas. These results support the existence of an important RH contribution to (at least) one phase of metaphor processing and, furthermore, implicate the temporal cortex with respect to that contribution.

    PMID: 15555767 [PubMed - indexed for MEDLINE]

and Neural correlates of metaphor processing.

Brain Res Cogn Brain Res. 2004 Aug;20(3):395-402.
Neural correlates of metaphor processing.
Rapp AM, Leube DT, Erb M, Grodd W, Kircher TT.
Department of Psychiatry, University of Tuebingen, Osianderstrasse 24, D-72076 Tuebingen, Germany.

    Metaphoric language is used to express meaning that is otherwise difficult to conceptualize elegantly. Beyond semantic analysis, understanding the figurative meaning of a metaphor requires mental linkage of different category domains normally not related to each other. We investigated processing of metaphoric sentences using event-related functional magnetic resonance imaging (fMRI). Stimuli consisted of 60 novel short German sentence pairs with either metaphoric or literal meaning. The pairs differed only in their last one to three words and were matched for syntax structure, word frequency, connotation and tense. Fifteen healthy subjects (six female, nine male, 19-51 years) read these sentences silently and judged by pressing one of two buttons whether they had a positive or negative connotation. Reading metaphors in contrast to literal sentences revealed signal changes in the left lateral inferior frontal (BA 45/47), inferior temporal (BA 20) and posterior middle/inferior temporal (BA 37) gyri. The activation in the left inferior frontal gyrus may reflect semantic inferencing processes during the understanding of a metaphor. This is in line with the results from other functional imaging studies showing an involvement of the left inferior frontal gyrus in integrating word and sentence meanings. Previous results of a right hemispheric involvement in metaphor processing might reflect understanding of complex sentences.

    PMID: 15268917 [PubMed - indexed for MEDLINE]

Pokémon Cards & Folk-Mathematics

Through most of my career as a mother, I have made it a point of aligning my interests with my children's interests. This has taken me to many interesting places, taught me many interesting things, and even gotten me published in the science magazine Nature (reprint on Fantastic Metropolis).

I have made an exception for annoying fads, especially the Pokémon thing. (See my May 18th, 2003 post, "Pokémon Infestations and Other Matters.")

I realized in the middle of the night, night before last, that there was something big I had been missing about the whole phenomenon. Here is an out-take from what I wrote about it:

One puzzling phenomenon I've observed watching 2nd graders is how kids, who are only just getting basic addition and subtraction of multidigit numbers by the tail, can spend literally hours trading Pokemon cards (by which I mean 2 or 3 hours at a time). The decisions of whether or not to trade are based on multiple factors, some of which are linear functions like how many hit points does a given card have (or is the sum of the hit points of the two cards you are offering me equal to or greater than the hit points of the card of mine you want), and some of which are binary (is it a "shiny", i.e. a holographic card).
. . .
I spot-checked Peter's sense of the relative value of cards back in February. I had him show me what he thought of as his three best cards. I priced them on The cheapest of them came in at $47.00. I then had him show me three of his cards that he thought of as "not-so-good." priced those between 75 cents and $3.00.

Given what I know of the scholastically measurable of the math skills of the kids in question, there has to be some kind of pre-verbal calculation going on. They seem to me to be carrying out complex calculations involving multiple variables of different types, and arriving at basically correct conclusions via some kind of folk-math.
. . .
One other implication of this phenomenon, it seems to me, is that the equals sign, as a piece of mathematical notation, is highly socially embedded. I remember something about a second grade playground bead market at Ravenna during recess that spontaneously emerged and then spread until teachers banned it after a few weeks. It may be that there is a developmental phase around 7 or 8 in which the social embedding of trade is explored.

I would be interested in your anecdotes about young kids and card trading. I've decided to investigate further.

I should also say that this realization was inspired partly by Munir Fasheh's essay "Can We Eradicate Illiteracy Without Eradicating Illiterates?", an expansion on a paper given at a UNESCO meeting in Paris, on 9-10 September, 2002, to celebrate the International Literacy Day. The meeting was entitled "Literacy as Freedom."

In it, he dscribes his realization of his illiterate mother's mathematical sophistication:

My 'discovery' of my illiterate mother's mathematics, and how my mathematics and knowledge could neither detect nor comprehend her mathematics and knowledge, mark the biggest turning point in my life, and have had the greatest impact on my perception of knowledge, language, and their relationship to reality. Later, I realized that the invisibility of my mother's mathematics was not an isolated matter but a reflection of a wide phenomenon related to the dominant Western worldview. In this sense, the challenge facing communities everywhere, is to reclaim and revalue the diverse ways of learning, teaching, knowing, relating, doing, and expressing. This reclaiming has been the pivotal theme of my thinking and work for the last two decades.

My concern is not about statistical measures - for example, how many learn the alphabet - but about our perception of the learner and what happens to her/him in the process of learning the alphabet. My concern is to make sure that the learner does not lose what s/he already has; that literacy does not replace other forms of learning, knowing, and expressing; that literacy is not considered superior to other forms; and that the learner uses the alphabet rather than be used by it. My concern is to make sure that in the process of eradicating illiteracy, we do not crush illiterates.

In the 1970s, while I was working in schools and universities in the West Bank region in Palestine and trying to make sense out of mathematics, science and knowledge, I discovered that what I was looking for has been next to me, in my own home: my mother's mathematics and knowledge. She was a seamstress. Women would bring to her rectangular pieces of cloth in the morning; she would take few measures with colored chalk; by noon each rectangular piece is cut into 30 small pieces; and by the evening these scattered pieces are connected to form a new and beautiful whole. If this is not mathematics, I do not know what mathematics is. The fact that I could not see it for 35 years made me realize the power of language in what we see and what we do not see.

Her knowledge was embedded in life, like salt in food, in a way that made it invisible to me as an educated and literate person. I was trained to see things through official language and professional categories. In a very true sense, I discovered that my mother was illiterate in relation to my type of knowledge, but I was illiterate in terms of her type of understanding and knowledge. Thus, to describe her as illiterate and me as literate, in some absolute sense, reflects a narrow and distorted view of the real world and of reality. A division, which I find more significant than literate and illiterate, would be between people whose words are rooted in the cultural-social soil in which they live - like real flowers - and people who use words that may look bright and shiny but without roots - just like plastic flowers.

(It's a neat essay. Read the whole thing.)

Rudy Rucker, Live, in Santa Cruz

Rudy Rucker announces an interesting event that I'd love to attend, except that I'm on the wrong coast. But if you live near Santa Cruz, go and then tell me about it. He says:

I’ll be performing with Phil Curtis at a small-scale event called ELSA, that is, ELectron SAlon #11, on Friday, June 3rd , starting at 8 PM .

I’ll read my story “Ain’t Paint” which appears in my forthcoming The Lifebox, the Seashell, and the Soul.  Phil will create some appropriate heuristic electronic music on the spot, and for video, we’ll use live demos of nine of my CAPOW software Zhabotinsky scrolls; the guys shown below. . . .

Phil and I will go first, so if you want to see us, you actually have to be there at 8. Usually at ELSA events there’s some free wine and food. It’s almost like a party.

Set 1: Rudy Rucker and Phil Curtis
Set 2: Run Return, an electronica duo with Kevin Dineen and Tommy Fugelsang
Set 3: The inimitable DJess and mixmaster, Ms Pinky, a. k. a. P. Minsky, with friends.

The venue is Next Door, 1207 Soquel Avenue, Santa Cruz, CA 95062, 831-429-1596. Next Door is next to the Rio Theater, see map.

Real Life Hard SF

Yesterday, I came across a really charming anecdote that reads like hard SF, but is in fact non-fiction. It is from the question and answer session following astronaut Michael Foale's keynote address at the 10th Anniversary Mathematica Conference, Friday, June 19, 1998, published in The Mathematica Journal in 1999. Foale took has own laptop with Mathematica on it with him to the Mir:

I had Mathematica with me; I owned it personally. It wasn't even a copy that NASA had bought for me. And I had intended to work on tensor calculus in all that free time that I was going to have. And I had it along with my music CDs in my CD pack that NASA nicely made for me, in the Spektr module. I also had it on the hard drive, installed on a laptop in the Spektr module.

But there was an accident. An experiment in which the crew was to try to dock a Progress convoy vehicle to the station didn't work out and caused severe damage to the Mir.

A simple TV image was used to measure the rate at which we were closing in. That's "black ground rush" to a parachutist. As you come in closer, the image gets bigger, and you can try to use that to calculate what the speed is while at the same time deriving a closing rate. Then you figure out the docking, using a little joystick to fire the thrusters.

As you know from the media, this was a terrible mistake. It left the station not mortally, but severely, wounded. The Progress basically impacted, we think now, on this part of the solar array on the Spektr module, and then it bounced and slowly floated away along the base block.

The Progress weighs seven tons. We think it collided at about three meters a second. I was in the base block; I didn't see it at all. Sasha Lazutkin saw it; he told me, in all haste, to go straight to the Soyuz escape craft, and as I was passing into the node region of the Mir, I heard a big thump. . . .

It had hit the Spektr module. If we'd been strapped in, we'd have all been shaken around. This is just the opposite of being on Earth, where you're in a car and you're always supposed to strap in. Bash the space station, and nothing happens to you because you're not in contact with the station -- an interesting backwards twist.

Like any good hard SF protagonist, Foale set out to do a bunch of calculations aimed at solving the problems encountered by the crew of the crippled space station. (For those who want to know all about the calculations, the keynote speech discusses them in detail.) He whipped out his trusty slide rule. Well, no, it was a little more complicated than that.

First, the problem he was trying to solve:

The task was this: when you lose attitude control on the station, what happens? The station, low-powered, starts to tumble; then the solar arrays are no longer pointed toward the sun; and then slowly the batteries of the station start to deplete, because the solar arrays aren't charging the batteries. And then in about two or three hours you have no power on the station.

Gyrodynes, the momentum wheels, are always acting, spinning at different rates to change the orientation of the station very slightly. Once the station's lost all its power, or the guidance and control system has failed, the gyrodynes start to spin down, and that momentum gets transferred back into the station. It spins in the opposite direction to the gyrodynes.

Lo and behold, because you have twelve gyrodynes all spinning and working really well to do a nice job at holding the station in attitude, as the space station loses control of those gyrodynes and the gyrodynes spin down, then the space station picks up all of the angular momentum that was in the gyrodynes and starts to spin in the opposite way -- and in an unpredictable way.

So my whole task was to basically try to figure out what the rotation was, null it, establish an orientation, and then spin. But the problem with the station is that it has unequal moments of inertia.

So, hard SF readers. You're in a damaged space station and you need to do some calculations on your computer. But the power keeps going out. Your install disks for the crucial program flew out will the escaping air when the station was damaged. People on the ground are trying their darnedest  to help. What else can possibly go wrong. Read it and find out! Actually solving the equtions seems to have been the least of the problems.

(Did you know that the IBM Thinkpad warranty does not cover exposing your laptop to hard vacuum?)

PS: Further to the subject of math, check out the amusing flame war in the reader review section of the Amazon page on Stephen Wolfram's A New Kind of Science; the book, published 3 years ago, has 318 reader reviews so far.