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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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Flying through space, powered by sunlight

Yesterday a rocket launched from Cape Canaveral in Florida carrying the LightSail into space!  It's a small spacecraft with a big shiny screen that's pushed by the light of the sun.  

It's just a test - it won't go far.   It will fall to the Earth and burn up.  But next year there will be a more serious test.  And someday, solar-powered space flight may become a force to be reckoned with.

One cool thing is that all this is being paid for private donations, by a Kickstarter campaign!

The LightSail is carried to space in a cute little CubeSat.  It looks like a big toaster, and it weighs just 10 kilograms.   But it holds a sail 32 square meters in area,  made of a shiny plastic called Mylar, just 4.5 microns thick.  This unfolds in a clever way - watch the movie! - to form a big square.

The Sun will push on this with a tiny force. 

Puzzle: How tiny is this force?

Someone named Bill Russell answered this over on Yahoo.  Let me go through his calculation so we can check it.

The momentum of light is given by

p = E/c

where E is the energy of the light, p is the momentum, and c is the speed of light. 

In outer space near earth the sunlight provides 1370 watts per square meter - that's energy per area per time.  We can use the formula above to convert this to momentum per area per time, better known as force per area... or pressure

Russell calculates

(1370 watts / meter²) / c = 9.13 micronewtons / meter²

and concludes the pressure is 9.13 micronewtons per square meter.  His arithmetic checks out, but I think he's neglecting some physics: when the light bounces back off the mirror its momentum completely reverses, so I think we get an extra factor of 2. 

Puzzle 2:  Am I right or am I wrong?

The area of the LightSail is about 32 square meters.  Russell says this gives a total force of

9.13 micronewtons/meter² x 32 meter²

or about 300 micronewtons.   I'd double this and get 600 micronewtons.

Puzzle 3: Once it's out of the box, the LightSail weighs about 4.5 kilograms.  How much will it accelerate due to sunlight?

Here we use Newton's good old

F = ma

and solve for the acceleration a.   But at this point Russell seems to make a serious mistake.  I'll let you see what you think, and fix it if necessary!  Here is his calculation:
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What about the effect of the solar wind ? Is it negligible ?
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Here are some blog posts about the categorical foundations of network theory - a warmup for the workshop we're having in Turin next week.
And now for my next trick... Category theory is a branch of math that puts processes on an equal footing with things - unlike set theory, where… - John Baez - Google+
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The dunes of Mars

This field of dunes lies on the floor of an old crater in Noachis Terra.  That's one of the oldest places on Mars, scarred with many craters, with rocks up to 4 billion years old.  It's in the southern hemisphere, near the giant impact basin called Hellas, which is 2.5 times deeper than the Grand Canyon and 2000 kilometers across!

This is a 'false color' photograph - you'd need to see infrared light to see that the dunes are very different than the rock below.

These are barchans, dunes with a gentle slope on the upwind side and a much steeper slope on the downwind side where horns or a notch can form.  If you know this, you can see the wind is blowing from the southwest.

It's actually a bit of a puzzle where the sand in these dunes came from!   Here's the abstract of a paper by +Lori Fenton on this subject:

No sand transport pathways are visible in a study performed in Noachis Terra, a region in the southern highlands of Mars known for its many intracrater dune fields.Detailed studies were performed of five areas in Noachis Terra, using Mars Orbiter Camera (MOC) wide-angle mosaics, Thermal Emission Imaging System (THEMIS) daytime and nighttime infrared mosaics, MOLA digital elevation and shaded relief maps,and MOC narrow-angle images. The lack of observable sand transport pathways suggests that such pathways are very short, ruling out a distant source of sand. Consistent dune morphology and dune slipface orientations across Noachis Terra suggest formative winds are regional rather than local (e.g., crater slope winds). A sequence of sedimentary units was found in a pit eroded into the floor of Rabe Crater, some of which appear to be shedding dark sand that feeds into the Rabe Crater dune field. The visible and thermal characteristics of these units are similar to other units found across Noachis Terra, leading to the hypothesis that a series of region-wide depositional events occurred at some point in the Martian past and that these deposits are currently exposed by erosion in pits on crater floors and possibly on the intercrater plains. Thus the dunes and sources may be both regional and local: sand may be eroding from a widespread source that only outcrops locally. Sand-bearing layers that extend across part or all of the intercrater plains of Noachis Terra are not likely to be dominated by loess or lacustrine deposits; glacial and/or volcanic origins are considered more plausible.

• Lori K. Fenton, Potential sand sources for the dune fields in Noachis Terra, Mars, Journal of Geophysical Research 110 (2005), E11004.  Available at

The image is from a great series of photos taken by the HIRISE satellite, which orbits Mars and takes high resolution images:

• Colorful Dunes,

#mars   #astronomy
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+Steve Esterly - that's interesting.  A cool thing about karst on Earth is that it's deeply connected to biology, since limestone is made of organisms.  But I guess all karst needs is lots of calcium carbonate or similar minerals that can be dissolved underground.  I wouldn't want to conflate karst with mere erosion.
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Fighting global warming: the tide is turning!

Good news!   We, the citizens of the world, may be starting to burn less carbon - not more!

The International Energy Agency claims:

In 2014, global carbon dioxide emissions from energy production stopped growing!

It seems the big difference is China.  They say the Chinese made more electricity from renewable sources, such as hydropower, solar and wind, and burned less coal.  

In fact, a report by Greenpeace says that from April 2014 to April 2015, China's carbon emissions dropped by an amount equal to the entire carbon emissions of the United Kingdom!   

I want to check this, because it would be wonderful - a 5% drop.  They say that if this trend continues, in 2015 China will make the biggest reduction in CO2 emissions every recorded by a single country.

The International Energy Agency also credits Europe's improved attempts to cut carbon emissions for the turnaround.   In the US, carbon emissions has basically been dropping since 2006 - with a big drop in 2009 due to the economic collapse, a partial bounce-back in 2010, but a general downward trend.

In the last 40 years, there were only 3 other times when emissions stood still or fell compared to the previous year, all during economic crises: the early 1980's, 1992, and 2009.  In 2014, however, the global economy expanded by 3%.

So, the tide may be turning!   But please remember: while carbon emissions may start dropping, they're still huge.  The amount of the CO2 in the air shot above 400 parts per million this year.  As Erika Podest of NASA put it:

CO2 concentrations haven't been this high in millions of years. Even more alarming is the rate of increase in the last five decades and the fact that CO2 stays in the atmosphere for hundreds or thousands of years. This milestone is a wake up call that our actions in response to climate change need to match the persistent rise in CO2. Climate change is a threat to life on Earth and we can no longer afford to be spectators.

So let's not slack off now!  The battle has just begun.  We need to cut carbon emissions to almost zero.

Here is the announcement by the International Energy Agency:

"This gives me even more hope that humankind will be able to work together to combat climate change, the most important threat facing us today," said IEA Chief Economist Fatih Birol.

Their full report will come out in June.  Here is the report by Greenpeace EnergyDesk:

I trust them less than the IEA when it comes to using statistics correctly, but someone should be able to verify their claims if true.  The graph here comes from this article:

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+Edward Morbius - Thanks again!  I corrected the copy of your earlier comment on Azimuth and also added a copy of this one.
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Sometimes you see a tiny piece of a story and wonder how it started - and how it will end.
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+JDS Purohit
Perhaps ... or maybe it's just another attempt at 15 minutes of Internet fame. (My suspicion factor was raised when I noted the runner was barefoot. Do Zookeepers work without shoes?)
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An impossible dream

Kepler, the guy who discovered that planets go in ellipses around the Sun, was in love with geometry.  Among other things, he tried to figure out how to tile the plane with regular pentagons (dark blue) and decagons (blue-gray).  They fit nicely at a corner... but he couldn't get it to work.

Then he discovered he could do better if he also used 5-pointed stars!

Can you tile the whole plane with these three shapes?  No!  The picture here is very tempting... but if you continue you quickly run into trouble.  It's an impossible dream.

However, Kepler figured out that he could go on forever if he also used overlapping decagons, which he called 'monsters'.  Look at this picture he drew:

If he had worked even harder, he might have found the Penrose tilings, or similar things discovered by Islamic tiling artists.  Read the whole story here:

• Craig Kaplan, The trouble with five,

How did Kepler fall in love with geometry?  He actually started as a theologian.   Let me quote the story as told in the wonderful blog The Renaissance Mathematicus:

Kepler was born into a family that had known better times, his mother was an innkeeper and his father was a mercenary. Under normal circumstances he probably would not have expected to receive much in the way of education but the local feudal ruler was quite advanced in his way and believed in providing financial support for deserving scholars. Kepler whose intelligence was obvious from an early age won scholarships to school and to the University of Tübingen where he had the luck to study under Michael Mästlin one of the very few convinced Copernican in the later part of the 16th century. Having completed his BA Kepler went on to do a master degree in theology as he was a very devote believer and wished to become a theologian. Recognising his mathematical talents and realising that his religious views were dangerously heterodox, they would cause him much trouble later in life, his teacher, Mästlin, decided it would be wiser to send him off to work as a school maths teacher in the Austrian province.

Although obeying his superiors and heading off to Graz to teach Protestant school boys the joys of Euclid, Kepler was far from happy as he saw his purpose in life in serving his God and not Urania (the Greek muse of astronomy). After having made the discovery that I will shortly describe Kepler found a compromise between his desire to serve God and his activities in astronomy. In a letter to Mästlin in 1595 he wrote:

I am in a hurry to publish, dearest teacher, but not for my benefit… I am devoting my effort so that these things can be published as quickly as possible for the glory of God, who wants to be recognised from the Book of Nature… Just as I pledged myself to God, so my intention remains. I wanted to be a theologian, and for a while I was anguished. But, now see how God is also glorified in astronomy, through my efforts.

So what was the process of thought that led to this conversion from a God glorifying theologian to a God glorifying astronomer and what was the discovery that he was so eager to publish? Kepler’s God was a geometer who had created a rational, mathematical universe who wanted his believers to discover the geometrical rules of construction of that universe and reveal them to his glory. Nothing is the universe was pure chance or without meaning everything that God had created had a purpose and a reason and the function of the scientist was to uncover those reasons. In another letter to Mästlin Kepler asked whether:

you have ever heard or read there to be anything, which devised an explanation for the arrangement of the planets? The Creator undertook nothing without reason. Therefore, there will be reason why Saturn should be nearly twice as high as Jupiter, Mars a little more than the Earth, [the Earth a little more] than Venus and Jupiter, moreover, more than three times as high as Mars.

The discovery that Kepler made and which started him on his road to the complete reform of astronomy was the answer to both the question as to the distance between the planets and also why there were exactly six of them: as stated above, everything created by God was done for a purpose.

On the 19th July 1595 Kepler was explaining to his students the regular cycle of the conjunctions of Saturn and Jupiter, planetary conjunctions played a central role in astrology. These conjunctions rotating around the ecliptic, the apparent path of the sun around the Earth, created a series of rotating equilateral triangles. Suddenly Kepler realised that the inscribed and circumscribed circles generated by his triangles were in approximately the same ratio as Saturn’s orbit to Jupiter’s. Thinking that he had found a solution to the problem of the distances between the planets he tried out various two-dimensional models without success. On the next day a flash of intuition provided him with the required three-dimensional solution, as he wrote to Mästlin:

I give you the proposition in words just as it came to me and at that very moment: “The Earth is the circle which is the measure of all. Construct a dodecahedron round it. The circle surrounding that will be Mars. Round Mars construct a tetrahedron. The circle surrounding that will be Jupiter. Round Jupiter construct a cube. The circle surrounding it will be Saturn. Now construct an icosahedron inside the Earth. The circle inscribed within that will be Venus. Inside Venus inscribe an octahedron. The circle inscribed inside that will be Mercury.”

This model, while approximately true, is now considered completely silly!   We no longer think there should be a simple geometrical explanation of why planets in our Solar System have the orbits they do.

So: a genius can have a beautiful idea in a flash of inspiration and it can still be wrong.

But Kepler didn't stop there!  He kept working on planetary orbits until he noticed that Mars didn't move in a circle around the Sun.  He noticed that it moved in an ellipse!  Starting there, he found the correct laws governing planetary motion... which later helped Newton invent classical mechanics.

So it pays to be persistent - but also not get stuck believing your first good idea.

Read The Renaissance Mathematicus here:

Puzzle: can you tile the plane with shapes, each of which has at least the symmetry group of a regular pentagon? 

So, regular pentagons and decagons are allowed, and so are regular 5-pointed stars, and many other things... but not Kepler's monsters.  The tiling itself does not need to repeat in a periodic way.

#geometry #astronomy  
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+John Wehrle There's no chance that Chomsky is right at this point, though.
His paradigm is alive only through heavy propaganda... Europe and the US east coast have all but abandoned it... but as mentioned, that's irrelevant here.
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Here are some blog posts about the categorical foundations of network theory - a warmup for the workshop we're having in Turin next week.
And now for my next trick... Category theory is a branch of math that puts processes on an equal footing with things - unlike set theory, where… - John Baez - Google+
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Only in North America! 
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And now for my next trick...

Category theory is a branch of math that puts processes on an equal footing with things - unlike set theory, where everything is fundamentally a thing.   Can we use category theory to help understand the complex processes that underlie biology and ecology? 

I believe so, and I'm hoping this is a good way for fancy-schmancy mathematicians like me to help the world.  But it will take a while.  I think we should start by seeing what category theory has to say about some related subjects that are better understood: chemistry, electrical engineering, classical mechanics, and the like.

We're having a workshop about this next week - and to organize our thoughts we've been writing some blog articles.  Check 'em out!

• John Baez, Categorical foundations of network theory - an introduction to the workshop and what it's about.

• David Spivak, A networked world.

• Eugene Lerman, Networks of dynamical systems.

• Tobias Fritz, Resource convertibility - an introduction to the mathematics of 'resources'.

• John Baez, Categories in control - about my paper with Jason Erbele on using categories to study signal flow diagrams in control theory.

• John Baez, A compositional framework for passive linear networks - about my paper with Brendan Fong on using categories to study electrical circuit diagrams.

• John Baez, Decorated cospans - about Brendan Fong's paper providing mathematical infrastructure for the study of networks.

• John Baez and Brendan Fong, Cospans, wiring diagrams, and the behavioral approach - an attempt to reflect on how our work connects to that of David Spivak.

• Brendan Fong, Resource theories - about Brendan's new paper with Hugo Nava-Kopp on resource theories.

• John Baez, PROPs for linear systems - about Simon Wadsley and Nick Woods' generalization of a result in my paper with Jason Erbele, describing categories where the morphisms are linear maps.

The picture, by the way, was drawn by Federica Ferraris and appears in this book:

• John Baez and Jacob Biamonte, Quantum techniques for stochastic physics,

It's about Petri nets and reaction networks - two kinds of networks that appear in chemistry and population biology.
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Nevermind that last question. It's self-evident. (Especially after the same relation was written with correct names later down the paper)
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Why is this true?

The spooky-smart mathematician Srinivasa Ramanujan came up with this formula around 1913.  Why is it true?

I don't know, let's see...

In 1735, a young fellow named Euler stunned the world by cracking a famous puzzle that had been unsolved for almost a century: the Basel problem.  The problem was to sum the reciprocals of perfect squares:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... = ???

Euler showed that the answer was π²/6:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... = π²/6

He also showed you could rewrite this sum as a product over primes:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... =

(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ...

That's actually the easy part: it's a cute trick called the Euler product formula.

So we know

(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ... = π²/6

If you think about it, Ramanujan's formula is saying that

(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ...

is 2/5 as big.  So, proving it is the same as showing

(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ... = π²/15

Maybe the next step is to use the same idea as the Euler product formula.  I think this gives

(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ... =

1/1² - 1/2² - 1/3² + 1/4²  - 1/5² + 1/6² - 1/7² + ...

where the signs at right follow a fancy pattern: we get 1/n² whenever n is the product of an even number of primes, and -1/n² when n is the product of an odd number of primes.  For example, 4 = 2 x 2 is the product of an even number of primes, so we get 1/4².

So I'm left wanting to know why this strange sum

1/1² - 1/2² - 1/3² + 1/4² - 1/5² + 1/6² - 1/7² + ...

equals π²/15.  Ramanujan, dead since 1920, is still messing with my mind! 

The formula is supposed to be in here:

• Srinivasa Ramanujan, Modular equations and approximations to π, Quart. J. Pure. Appl. Math. 45 (1913-1914), 350-372.  Also available at ://

But I don't see it!

Here you can see how Euler solved the Basel problem:

It's a great example of his brilliant tactics, many of which were far from rigorous by today's standards... but can be made rigorous.

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+Rita the dog
Looking at your extended list and noticing how quickly the occurrence of primes peters out, I have begum tp think that we are looking at another application of Richard Guy's Strong Law of Small Numbers. (This also applies to my conjecture that you busted.) "There aren't enough small numbers to meet the many demands made of them," says Richard Guy and from this it follows that we should expect numerical coincidences that at first seem to be real patterns.


Note to +John Baez: Have you done any posts on coincidental patterns that masquerade as real? If not that would be fun!
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A galaxy - falling

This galaxy is suffering!  It's falling into a large cluster of galaxies, pulled by their gravity.   You can see this in 3 ways:

1.  The reddish disk of dust and gas looks bent.  There aren't many atoms between galaxies, but there are still some. So the galaxy is moving through the wind of integalactic space!   And it's having trouble holding onto the loosely bound dust and gas near its edge.  They're getting blown away.

2.  The blue disk of stars is not bent.  It extends beyond the disk of dust and gas, which is where stars are formed.  This suggests that the dust and gas is being stripped from the galaxy after these stars were formed!

3.  Streamers of dust and gas can be seen trailing behind the motion of the galaxy - near the top.  On the other hand, the blue stars near the leading edge of the galaxy have no dust and gas left to hide them.

This phenomenon is called ram pressure stripping, and it can kill a galaxy, shutting down the production of new stars.   Here we are seeing it damage the galaxy NGC 4402, which is currently falling into the Virgo cluster - a cluster of galaxies about 65 million light years away.

Apparently there's about 1 atom per cubic centimeter in our galaxy - on average, though some regions are vastly more dense than others.   But in the space between galaxies in clusters it's more like 1/1000 of that.  Not much!  But enough to kill off the formation of new star systems, life, civilizations...

I got most of my information from here:

and I got the picture from here:

The photo was taken at the WIYN 3.5-meter telescope on Kitt Peak, which is fitted with some 'adaptive optics' to compensate for the jittery motion of the image due to variable atmospheric conditions and telescope vibrations.

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+JDS Purohit - I don't know.  I bet you can look up its velocity and distance to the center of the Virgo Cluster and do an estimate.
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Making yourself into a superhero

I enjoyed this true story by Kelly McEvers:

We met in a bar in Flagstaff, Arizona. I'd just moved back from Cambodia and I was going out for one of my first beers back in the States. Not long into the first one, I notice this Amazon of a woman with huge blond and red-streaked hair and frosty lips, wearing a short red tank dress and at least 50 bracelets. She's six feet tall and showing a lot of leg. People at the bar swivel their heads to watch her every move.

She stands next to me to order a drink, and in this throaty voice says, "What are those?" pointing to my cigarettes. I tell her they're Cambodian. Her eyes light up and she shoots out a long, tan arm, and points at a table in the corner. She orders me there. Before I can say no, I'm following her to my seat.

She tells me she's an international private investigator, a bounty hunter, and a bail bonds enforcer, and that her name is Zora. I sit there for hours listening to her. Within a week, she takes me to Las Vegas. We drive there in her red Mustang. As always, there's a Colt .380 under the driver's seat and a .45 Megastar in the trunk.

In Vegas, we skip the casinos and head straight for the male strip clubs, where Zora drops at least $200 on lap dances from buff guys with names like Roman. Her getup is the same as before – teeth, hair, jewelry, and the ubiquitous tank dress, which, I realize, is the best way to show off her tattoos.

One is this big circle with blue and white swirls in it, kind of like a bowling ball, on her left shoulder. Every guy she meets asks her about it, and when they hear her answer, they sometimes propose marriage. Turns out the tattoo is a magic globe she holds in her dreams. And in these dreams, it gives her superpowers.

Zora: Ever since I remember, I've had the dreams. And they're very vivid. But it varies. It usually involves fighting, sometimes with guns, sometimes with superhero powers. Lightning from my fists and all that. And I usually have super strength, and I can fly, and I have all those things.

And it's my most common set of dreams. And it varies. Sometimes it's medieval, sometimes it's futuristic, sometimes it's present day, sometimes it's like a guerrilla war in Latin America.

Kelly: Can you describe that Zora to me, the Zora in dreams?

Zora: Very powerful athletically, but beyond the rules of nature that this world allows.  Six foot five and long, like almost impossibly long silver hair. This sort of otherworldly quality to her, where her voice did not sound normal. It sounded, like, almost musical.

And it became something that I aspired to be. I aspired to be this sort of superhero, this sort of person who would fight for a cause. That was my motivation in life. Ever since I was 10 or 11, I decided that that was my goal.

Zora took the dreams seriously. So seriously that at the age of 12, she sat down and composed a list of some 30 skills she needed to learn if she wanted to become as close to a superhero as any mortal could be. She even gave herself a deadline – to master these skills by the time she was 23.

Zora: I don't know what's in these.

Zora pulls out the old spiral notebook that was her diary at the age of 13 and turns to the inside back cover.

Zora: There's the list.

Kelly: Wow. Why don't you go ahead and read it.

Zora: OK. The list included martial arts, electronics, chemistry, metaphysics, hang gliding, helicopter and airplane flying, parachuting, mountain climbing, survival....

Throughout her teens and 20s, each time she started a new diary, she would update the list and write it in the back of the book, each one with the same format, each one titled "The List."

Zora: Weaponry, rafting, scuba diving, herbology – yes, I, studied that -- CPR, first aid and mountain emergency kind of medicine....

The list also includes bodybuilding, archery, demolitions, and explosives. She wanted to learn how to hunt animals and track men.

Zora: Major physical conditioning....

And the most incredible thing about all of this is that Zora accomplished nearly every item on the list.

Zora: Throwing stars and compound bows and throwing knives and -- yes, it was a very interesting pastime.

To keep up with the goals set by the list, she sped through school. Starting in the seventh grade, she began completing entire school years during the summer term and finished high school by the time she was 15. She got her BA at 18, a master's at 20, and completed the coursework for a PhD in Geopolitics by the time she was 21. She wanted to live like Indiana Jones, spending half her time in the classroom and half her time saving the world in the jungles of Peru.

Zora: Item number four – camel, elephant riding. Evasive driving and stunts....

When you're a kid, you have these romantic visions of what you'll be when you grow up. But how many people are so diligent they commit their dreams to paper and make it their life's work to achieve them? How many keep a list, amending it, adding to it, ticking things off as they go along, well into their adult lives?

After finishing the course work for her PhD, Zora decided to quit school, disappointed at the lack of cliff-hanging adventure in her doctoral program. And since superheroes who live in the real world need jobs, she decided to seek employment at the only place that would allow her to put all the skills from the list to use. Zora wanted to become an agent in the CIA.

But then the story takes some surprising twists!  Listen to it here:

The picture here is, of course, not Zora.  It's Charlize Theron playing  'Aeon Flux' - a kind of superhero invented by a high school friend of mine, the animator Peter Chung.
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Hmm, okay, sounds interesting.
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The architecture of water

Water is fascinating, for many reasons.   It takes more energy to heat than most substances.  It's one of the few substances that expands when it freezes.  It forms complicated patterns in its liquid state, which are just beginning to be understood.  There are at least 18 kinds of ice, which exist at different temperatures and pressures.  Snowflakes are endlessly subtle.  

And ice can form cages that trap other molecules!  Here you see the 3 main kinds.

They're called clathrate hydrates.  There's a lot under the sea beds near the north and south pole - they contain huge amounts of methane.   At some moments in the Earth's history they may have erupted explosively, causing rapid global warming.  

But let's focus on the fun part: the geometry!  Each of the 3 types of clathrate hydrates is an architectural masterpiece.

Type I consists of water molecules arranged in two types of cages: small and large.  The small cage, shown in green, is dodecahedron.  It's not a regular dodecahedron, but it still has 12 pentagonal sides.  The large cage, shown in red, has 12 pentagons and 2 hexagons.   The two kinds of cage fit together into a repeating pattern where each unit cell - each block in the pattern - has 46 water molecules.

Puzzle 1: This pattern is called the Weaire-Phelan structure.  Why is it famous, and what does it have to do with the 2008 Olympics?

You can see little balls in the cages.  These stand for molecules that can get trapped in the cages.   They're politely called guests.   The type I clathrate often holds carbon dioxide or methane as a guest.

Type II is again made of two types of cages – small and large.  The small cage is again a dodecahedron.  The large cage, shown in blue, has 12 pentagons and 4 hexagons.  These fit together to form a unit cell with 136 water molecules.

The type II clathrate tends to hold oxygen or nitrogen as a guest.

Type H is the rarest and most complicated kind of clathrate hydrate.  It's built from three types of cages: small, medium and huge.  The small cage is again a dodecahedron, shown in green.  The medium cage - shown in yellow - has 3 squares, 6 pentagons and 3 hexagons as faces.  The huge cage - shown in orange - has 12 pentagons and 8 hexagons.  The cages fit together to form a unit cell with 34 water molecules.

The type H clathrate is only possible when there are two different guest gas molecules - one small and one very large, like butane - to make it stable.   People think there are lots of type H clathrates in the Gulf of Mexico, where there are lots of heavy hydrocarbons in the sea bottom.

Puzzle 2: how many cages of each kind are there in the type I clathrate hydrate?

Puzzle 3: how many cages of each kind are there in the type II?

Puzzle 4: how many cages of each kind are there in the type H?

These last puzzles are easier than they sound.  But here's one that's a bit different:

Puzzle 5: the medium cage in the type H clathrate - shown in yellow - has 3 squares, 6 pentagons and 3 hexagons as faces.   Which of these numbers are adjustable?  For example: could we have a convex polyhedron with a different number of squares, but the same number of pentagons and hexagons?

The picture is from here:

• Timothy A. Strobel, Keith C. Hester, Carolyn A. Koh, Amadeu K. Sum, E. Dendy Sloan Jr., Properties of the clathrates of hydrogen and developments in their applicability for hydrogen storage, Chemical Physics Letters 478 (27 August 2009), 97–109.

#geometry #water
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Yes, I meant 4 hexagons.  I'll fix that - thanks!
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I'm a mathematical physicist.
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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.

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  • Massachusetts Institute of Technology
    Mathematics, 1982 - 1986
  • Princeton University
    Mathematics, 1979 - 1982
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