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John Baez
Works at Centre for Quantum Technologies
Attended Massachusetts Institute of Technology
Lives in Riverside, California
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John Baez

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The small stellated dodecahedron

What if you let yourself make shapes that are as symmetrical as Platonic solids, but where all the faces are stars?   Then you get things like this.

If you look carefully, you'll see lots of 5-pointed stars.  Each one is a regular pentagram - a 5-pointed star whose corners are a regular pentagon.  Each one touches 5 others at each corner, in exactly the same way.  So, it's as regular as you might want. 

But it's funny in some ways.  First, the faces are stars instead of regular polyhedra.  Second, the faces intersect each other: that's why you don't see all of any star. 

There are 4 polyhedra whose faces are all regular stars, with each face just like every other and each vertex like every other.  They're called Kepler-Poinsot polyhedra

This particular one is called the small stellated dodecahedron because if you remove all the pyramid-shaped pieces you're left with a dodecahedron!  Each star lies in the same plane as one of this dodecahedron's faces.  So, there are 12 stars in this shape.

On the other hand, the sharp points of this shape form the corners of an invisible icosahedron!  So, there are 20 sharp points.

Puzzle: how many edges does this shape have?

This shape can be seen in a floor mosaic in the Basilica of St. Mark in Venice, built in 1430.  It was rediscovered by Kepler in his work Harmonice Mundi in 1619.  This book, about the "harmonies of the world", is an amazing mix of geometry, astronomy and music theory - a mystical warmup for his later breakthroughs on the orbits of the planets. 

Much later, Escher made himself a wood model of the small stellated dodecahedron, which he drew in two woodcuts called Order and Chaos.

While the Kepler-Poinsot polyhedra are beautiful, I've avoided studying them because I don't see how they fit into the theory of Coxeter groups - the study of discrete symmetries that connects Platonic solids, Archimedean solids and hyperbolic honeycombs to deeper strands of math like Lie theory, the study of continuous symmetries.  I've been afraid these shapes are merely cute, not deep.

Maybe it's time to find out.

For more, see:

The Mathworld page has a much better picture of the mosaic in the Basilica of St. Mark.

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+Layra Idarani - fun questions!  If we ignore the self-intersections and treat it as an abstract space, is it a topological manifold?  I.e.: is it locally homeomorphic to the plane?

Is it orientable?

If the answers to both these questions are "yes", and I momentarily take your word that it has Euler characteristic -6, then it's homeomorphic to a 4-holed torus.

And that makes it interesting to seek a 4-holed Riemann surface on which the symmetry group of this shape - the group you're calling H3, or more precisely its orientation-preserving part, with 60 elements - acts as conformal transformations!

For a similar example, see the picture here:
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Order in Chaos

That's what Taffgoch, the creator of this picture, calls it.  It's a computer-generated image, made to look nicely weathered... but it's based on an actual model, made by a monk named Father Magnus Wenninger.

Wenninger's story is interesting.  In the 1940s went to the Bahamas to teach at a Benedictine school there. He was asked whether he wanted to teach English or math.  He chose math. But not having taken many math courses in college, he struggled at first to stay a few pages ahead of the students!  He taught algebra, Euclidean geometry, trig and analytic geometry. 

In the 1950's he felt he was getting stale, so he went to Columbia Teachers College in the summer for 4 years.  He got interested in the “New Math"... and started studying polyhedra.

In 1966 he wrote a booklet called Polyhedron Models for the Classroom.  He wrote to H. S. M. Coxeter, the world's expert on polyhedra and higher-dimensional polytopes, sometimes called the 'king of geometry'.   Apparently Coxeter sent Wenninger a copy of his book Uniform Polyhedra

A uniform polyhedron is one that has regular polygons as faces and is symmetrical enough that there's a symmetry carrying any vertex to any other.  There are 75 uniform polyhedra - not counting the infinite list of prisms and 'antiprisms'... and a very weird thing called the 'great disnub dirhombidodecahedron'... which is a topic for another day.

After getting Coxeter's book, Magnus Wenninger spent a lot of time making models of uniform polyhedra. He made 65 of them and put them on display in his classroom. Then he decided to publish a book about them. He had the models photographed and wrote the accompanying text, which he sent to Cambridge University Press.

They said they'd be interested in the book only if Wenninger built all 75 of the uniform polyhedra!  And so he did...

It took him 10 years to finish the book, Polyhedron Models, which was published in 1971.   Mathematics is full of stories of amazing persistence, and this is one!  

The key, which not everyone realizes, is that math is immensely fun.   To leave behind this world of woe and lose yourself in a world of beauty and perfection - it's dangerously addictive.

Puzzle: this shape is covered with little pentagons, little hexagons and funny-looking nonconvex shapes.  How many of each are there?

It only takes a tiny bit of persistence to figure this out... at least compared to Wenninger's persistence.

I got my story from here:

and the image from here:

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This is one of the reasons I started creating fractals, #Fractaled is my tag if you want to see them.
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Do you know this man?

There are lots of wonderful faces poking out from the buildings of Paris, and I've gotten into the habit of photographing them.  This is one of the strangest.  It's in a large blank wall by the walkway on the southern  Bank of the Seine, and it seems modern.

Who is it?  Who made it?

I think this face is smiling even though its lips are turned down.  This seems to prove what experts say: the smile is not in the curve of the lips, it's in the eyes! 

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I agree with +Joe Friday .  It would be difficult to have downturned lips with 'smiling' cheeks.
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This cartoon made me laugh out loud.  Now I've become 'desensitized' and can't get myself to laugh even if I try.  That's how it usually works with jokes.  But why was it funny?

First, the surprise.  We see this woman's chance of getting the job suddenly drop to zero.   We don't know much about her except that she's blonde and has carefully curled hair and an expensive-looking cashmere sweater... and that's she's the sort of person who is looking for a job where personal skills matter.  All this makes her tactless and profane comment completely unexpected.

Second, the self-referentiality.  I admit to loving self-referential humor - it's a weakness of mine, surely related to studying mathematical logic.  But it's best when it's subtle.  Here is a kind of mutant version of the Liar Paradox, which is not a paradox but a joke: "My honesty is a weakness, and I don't give a shit what you think."  But it's been cleverly written into a dialogue, with the last line suddenly making the pieces fit together.

Third, the profanity.  Intellectually I don't like this because it confuses honesty with bluntness or even rudeness. If she'd said "I don't care what you think", that would still be honest.  She would still not get the job.  But would the cartoon still be as funny?  There is no way for me to go back in time and find out!  But profanity in a job interview transgresses social norms and shocks us... and that probably helps make us laugh... except for those of us who find the cartoon offensive for this reason.  (If you're one of those, I apologize.)

Fourth, the smiling faces of the interviewers.  They are crucial somehow.

Fifth, it makes us think.  It's made me think quite a bit about the advantages and dangers of honesty, or really forthrightness: saying what you think when it's not completely necessary. Sometimes it's almost suicidal.   Sometimes it's unpleasant.  Sometimes it's a great thing!   And sometimes it's unpleasant and a great thing.

What do you think?  Please be honest. But not too honest.

I got this cartoon from +Malin Christersson, who got it from Mahendra Kariya, who probably got it from someone else....
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personal 6th: ridiculing the cliche' of the candidate attempting to sell an actual strength as his/her only weakness (e.g. "my weakness is that I'm a perfectionist" etc.) - in this case, since the unexpected does turn out to be a real weakness, it's just all the funnier that she would even mention it as an answer.
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People are arguing about the new IPCC report, but what does it actually say?  Here's the start of a series that provides clear answers to those questions.  Check it out!

But, this series only covers the first IPCC report, "Working Group 1: the Physical Science Basis".  Do you know clear-headed, reliable people who can write good summaries of the reports from Working Group 2 (impacts and climate adaptation) and WG3 (climate mitigation)?
guest post by Steve Easterbrook In October, I trawled through the final draft of this report, which was released at that time: • Intergovernmental Panel on Climate Change (IPCC), Climate Chang...
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+Andreas Geisler for me, one of the central components of pseudoscience is belief. Without belief we have bullshit. So let's say that Schmidt is a scientist, Singer is a pseudoscientist, and Milloy is a bullshitter.
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Over the weekend the vultures got into the Presidential Palace by pecking through the screens on the balcony windows, and the flapping of their wings stirred up the stagnant time inside, and at dawn on Monday the city awoke out of its lethargy of centuries with the warm, soft breeze of a great man dead and rotting grandeur. Only then did we dare go in without attacking the crumbling walls of reinforced stone, as the more resolute had wished, and without using oxbows to knock the main door off its hinges, as others had proposed, because all that was needed was for someone to give a push and the great armored doors that had resisted the lombards of William Dampier during the building’s heroic days gave way. It was like entering the atmosphere of another age, because the air was thinner in the rubble pits of the vast lair of power, and the silence was more ancient, and things were hard to see in the decrepit light.

Thus begins Autumn of the Patriarch, a mind-bending tale about the long reign of a Central American dictator.  It's a must-read for anyone who loves the power of writing to create new worlds and strange states of mind.

It's by Gabriel García Márquez, who died yesterday after his own long reign, at the age of 87. 

The New Yorker has made a short version available for free. But the full version is what you really need.  As the novel goes on, the sentences grow longer and longer, like vines wrapping around an ancient building and eventually strangling it, like the extravagantly long rule of this eventually senile dictator.  

For example, by page 5 we get this:

In December, when the Caribbean world turned to glass, he would take the closed carriage on a climb along the cornices of crags until he came to the house perched on top of the reefs, and he would spend the afternoon playing dominoes with the former dictators of other nations of the continent, the dethroned fathers of other countries to whom he had granted asylum over the course of many years and who were now growing old in the shadow of his mercy, dreaming in chairs on the terrace about the chimerical vessel of their second chance, talking to themselves, dying dead in the rest home he had built for them on the balcony of the sea after having received all of them as if each were the only one, for they all appeared at dawn in the dress uniform they had put on inside out over their pajamas, with chests of money they had pilfered from the public treasury and suitcases with boxes of decorations, newspaper clippings pasted into old ledgers, and photograph albums they would show him at the first audience, as if they were credentials, saying look, General, that’s me when I was a lieutenant, this was the day I was inaugurated, this was the sixteenth anniversary of my taking power, here, look, General, but he would give them asylum without paying any more attention to them or inspecting credentials, because the only document of identity for an overthrown president should be his death certificate, he would say, and with the same disdain he would listen to the self-deluding little speech of I accept for this short time your noble hospitality while the justice of the people brings the usurper to account, the eternal formula of puerile solemnity which a while later he would hear from the usurper, and then from the usurper’s usurper, as if the God-damned fools didn’t know that in this business of men if you fall, you fall, and he put all of them up for a few months in the Presidential Palace, made them play dominoes until he had fleeced them down to their last cent, and then he took them by the arm over to the window looking out onto the sea, he helped them grieve over this stinking life that only goes in one direction, he consoled them with the illusion that they would go over there, look, he said, over there to that big house that looks like an ocean liner aground on the top of the reefs, where they would have some lodgings with good light and good food, and plenty of time to forget along with other companions of misfortune, and with a terrace overlooking the sea, where he liked to sit on December afternoons not so much for the pleasure of playing dominoes with that bunch of boobs but to enjoy the base good fortune of not being one of them, to look at himself in the instructive mirror of their misery while he wallowed in the great slough of felicity, dreaming alone.

Over time, it bends your reality.  Here it is:

Puzzle: what is the longest sentence ever written by a human?  (Computer-generated sentences don't count here, though I'd be glad to know about those too.)
Remembering Gabriel García Márquez: We've unlocked some of his writing in our archive.
Timely notes from The New Yorker's archive.
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Thanks John.  I have read it and enjoyed it, or maybe more accurately have read a version that is minimally different than it.
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The Wall Street Journal chose David Berlinski to write an editorial against the new IPCC report on climate change.  An interesting choice!

He says that most climate scientists are “intellectual mediocrities and pious charlatans".  But that's not all.   He's part of the intelligent design movement - which, in case you didn't notice, supports a watered-down form of creationism.    But while most of those guys try to act scientific, he comes out explicitly against science.  Or at least against the 'arrogance' of science.  In fact he's appeared in a film called The Arrogance of the Sciences.

In his book The Devil’s Delusion: Atheism and Its Scientific Pretensions, he wrote:

A great many men and women have a dull, hurt, angry sense of being oppressed by the sciences. They are frustrated by endless scientific boasting. They suspect that the scientific community holds them in contempt. They are right to feel this way.

He also appeared in the film Expelled: No Intelligence Allowed, which argues that the mainstream science establishment suppresses academics who find evidence of intelligent design in nature... and said:

It'd be nice to see the scientific establishment lose some of its prestige and power.

In short: he's trying to use resentment against science to fight against the IPCC, evolutionary biology... and even particle physics!  Yes, he's written an article saying the Higgs boson isn't such a big deal:

• David Berlinski, The ineffable Higgs, Evolution News and Views

For more about his Wall Street Journal article, read:

• Steven T. Corneliussen, Wall Street Journal opinion video: “The Arrogance of the Sciences”, PhysicsToday,
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Exactly -- I knew that GHGs and aerosols were the big forcings, but it's been interesting to see what else there was.  Until now I didn't realize that low-altitude ozone pollution caused warming.  And I didn't realize that contrails mattered at all.  And, I thought that aerosols had fallen a lot since the 1970s, due to pollution control; seems they haven't.  Maybe they have in the rich world but that improvement has been offset by industrialization in the developing world?

So much to learn...
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Le chat du Café Rostand

I've spent several summers in Paris working on math with Paul-André Melliès, often at the Café Rostand right across from the Jardin du Luxembourg.  I'm back here again for a week, so we had coffee at this café... and we saw the cat that owns the place! 

The huge and peaceful Luxembourg Gardens, created by the order of Queen Marie de Medici in 1612, are on the west edge of the Latin Quarter.  People stroll around, sail toy boats in a pool, eat roast chestnuts, and generally chill out.

Since the Middle Ages, the Latin Quarter has been the place for scholars in Paris.  There are lots of old colleges here.  The Sorbonne goes back to 1254!  The Collège de France is a comparative newcomer, established only in 1530.   And there are more. 

I'm staying in the Hôtel des 3 Collèges right near the Luxembourg Gardens... so I wanted to know which colleges it's named after!  Turns out to be the Collège de Sorbonne, the College de Cholets -  which used to be located right down the street - and the Collège de Cluny, which used to be right here!   

It's gone now, but the air is thick with history... so I don't feel bad for paying 2 euros for an espresso.  You're not just paying for the coffee, you're paying for the whole experience - including the cat!  At least that's the right way to think about it to keep from getting grumpy. 

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It gets much worse...
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If a ball can roll on it, it must be real

(But if only a picture of a ball can roll on it...)

I can't find who created this image - can you?  I can find lots of copies using Google Image Search, but I haven't found the original.

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If you're an iPad user, check out the Monument game...
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Event Announcements and Discussion  - 
John Baez originally shared:
Some of you might be interested in this.   Some funding may be available to support travel and accommodation for students and young researchers. See the webpage for more details!


  * 19--22 May 2014, University of Oxford, UK

This school will present a range of lectures on quantum structures in
physics and computer science, with a focus on abstract algebraic
techniques, including category theory. It is an ideal event for PhD
students, as well as more established researchers, who would like to
learn more about these exciting topics close to the research frontier.
The courses will be accessible to anyone who has taken a first course in quantum information. Everyone is welcome to participate, and registration information can be found on the school web page.

The school will feature the following courses:

  * Samson Abramsky, "Contextual semantics"
  * Jon Barrett, "Correlations and contexts: the quantum no-go theorems"
  * Bart Jacobs, "Kadison duality"
  * Prakash Panangaden, "Stone, Gelfand and Tannaka dualities"
  * Peter Selinger, "Number-theoretic methods in quantum information theory"
  * Aleks Kissinger and Bob Coecke, "Quantum picturalism"
  * Chris Heunen and Jamie Vicary, "Categorical quantum computing"

This event is twinned with a conference in honour of Prakash
Panangaden on the occasion of his 60th birthday, to be held on the
dates 23--25 May 2014 at the University of Oxford:

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Have him in circles
52,339 people
I'm a mathematical physicist.
  • Centre for Quantum Technologies
    Visiting Researcher, 2011 - present
  • U.C. Riverside
    Professor, 1989 - present
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Riverside, California
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I'm trying to get mathematicians and physicists to help save the planet.
I teach at U. C. Riverside and work on mathematical physics — which I interpret broadly as ‘math that could be of interest in physics, and physics that could be of interest in math’. I’ve spent a lot of time on quantum gravity and n-categories, but now I want to work on more practical things, too.

Why? I keep realizing more and more that our little planet is in deep trouble! The deep secrets of math and physics are endlessly engrossing — but they can wait, and other things can’t.

So, I’ve cooked up a plan to get scientists and engineers interested in saving the planet: it's called the Azimuth Project.  It includes a wiki, a blog, and a discussion forum.  I also have an Azimuth page here on Google+, where you can keep track of news related to energy, the environment and sustainability.

Check them out, and join the team!  Or drop me a line here.
  • Massachusetts Institute of Technology
    Mathematics, 1982 - 1986
  • Princeton University
    Mathematics, 1979 - 1982
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