Profile

Cover photo
John Baez
Works at Centre for Quantum Technologies
Attended Massachusetts Institute of Technology
Lives in Riverside, California
55,047 followers|37,822,839 views
AboutPostsCollectionsPhotos
People
Have him in circles
55,047 people
Isaac Kuo's profile photo
Emiliano Mordenti's profile photo
‫عبدالسلام المصري‬‎'s profile photo
Jose Ponce's profile photo
Laura Robinson's profile photo
antonio moreno's profile photo
Ronan Lamy's profile photo
Karrington Atkins's profile photo
ryan edge's profile photo
Work
Occupation
I'm a mathematical physicist.
Employment
  • Centre for Quantum Technologies
    Visiting Researcher, 2011 - present
  • U.C. Riverside
    Professor, 1989 - present
Places
Map of the places this user has livedMap of the places this user has livedMap of the places this user has lived
Currently
Riverside, California
Links
YouTube
Contributor to
Story
Tagline
I'm trying to get mathematicians and physicists to help save the planet.
Introduction
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.
Education
  • Massachusetts Institute of Technology
    Mathematics, 1982 - 1986
  • Princeton University
    Mathematics, 1979 - 1982
Basic Information
Gender
Male

Stream

John Baez

Shared publicly  - 
 
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 6 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
82
29
Steve S's profile photoJen Yoeng's profile photoDavid Algie's profile photoSighris Sargon's profile photo
16 comments
 
I won't give away the answer to Puzzle 1, other than to say I first saw the Weaire-Phelan structure recently in a fantastic TED talk titled Math is Forever by Eduardo Sáenz de Cabezón.  You can see the puzzle answer and enjoy the talk here:

https://www.youtube.com/watch?v=pGlZi2SwETc
Add a comment...

John Baez

Shared publicly  - 
 
A 3-dimensional golden star

Here Greg Egan has drawn a dodecahedron with 5 tetrahedra in it.  This picture is 'left-handed': if you look at where the 5 tetrahedra meet, you'll see they swirl counterclockwise as you go out!  If you view this thing in a mirror you'll get a right-handed version. 

Putting them together, you get a dodecahedron with 10 tetrahedra in it.   You can see it here:

http://math.ucr.edu/home/baez/mathematical/dodecahedron_with_10_tetrahedra.gif

The two kinds of tetrahedra are colored yellow and cyan.  Regions belonging to both are colored magenta.  It's pretty - but it's hard to see the tetrahedra, because they overlap a lot!

You can also do something like this starting with a cube.  A cube has 8 corners.  If you take every other corner of the cube, you get the 4 corners of a tetrahedron.  But you can do this in 2 ways.  If you choose both, you get a cube with 2 tetrahedra in it:

http://math.ucr.edu/home/baez/mathematical/cube_with_2_tetrahedra.gif

All this is just the start of a much more elaborate and beautiful story which also involves the golden ratio, the quaternions, and 4-dimensional shapes like the 4-simplex, which has 5 tetrahedral faces, and the 600-cell, which has 600 tetrahedral faces!   You can read it here:

http://blogs.ams.org/visualinsight/2015/05/01/twin-dodecahedra/

I learned some of this story from Adrian Ocneanu at Penn State University.  Greg Egan and I figured out the rest... or most of the rest.  There's an unproven conjecture here, which needs to be true to make the whole story work.  Can you prove it?

Puzzle: If you take a regular 4-simplex whose vertices are unit quaternions, with the first equal to 1, can you prove the other 4 vertices generate a free group on 4 elements?

Hmm, I see that this puzzle has been solved by +Ian Agol and someone else on Mathoverflow:

http://mathoverflow.net/questions/204464/do-unit-quaternions-at-vertices-of-a-regular-4-simplex-one-being-1-generate-a

I don't understand the solution yet, because I don't know what a 'Bass-Serre tree' is... but I'll try to learn about this.  Math is infinite, there's always more to learn.

#geometry #4d  
97
31
Josefina Barrón Hernández's profile photoCharles Filipponi's profile photoBlue Tyson's profile photoAndrew Thall's profile photo
38 comments
 
I am of course not surprised, yet enjoy the discovery that constraining someone else to utter no, has good enough cause to count as the least reprehensible form of constraint.
Add a comment...

John Baez

​​​Physical  - 
 
 
Mathematical dream worlds

+Jos Leys blends mathematics and art in a delightful way.  You don't need to know math to enjoy this picture.   It's a whimsical and mysterious landscape.  The bright colors make it clownish, but the shadows make it a bit eerie: the sun is setting, and who knows what happens here at night!   You can see more here:

http://www.josleys.com/show_gallery.php?galid=252

On the other hand, if you read the title of this gallery, you'll see there's math here: "the first 3d views of limit sets of Kleinian groups".  And trying to understand this math will lead you on quite a journey.  Let me sketch it here... I apologize for going rather fast.

A Kleinian group is a discrete subgroup of the group called PSL(2,C).  This group shows up in many ways in math and physics. 

Physicists call it the Lorentz group: it's the group generated by rotations and Lorentz transformations, which acts as symmetries in special relativity. 

In math, it's called the group of Möbius transformations or fractional linear transformations.  Those are transformations like this:

z |→ (az + b)/(cz + d)

where z is a complex number and so are a,b,c,d.  These can be seen as transformations of the Riemann sphere - the complex plane together with a point at infinity.  They are, in fact, precisely all the conformal transformations of the Riemann sphere: the transformations that preserve angles. 

But this group PSL(2,C) also acts as conformal transformations of a 3-dimensional ball whose boundary is the Riemann sphere!   And that's important for understanding this picture.

(In physics, this ball is the set of 'mixed states' for a spin-1/2 particle, and the sphere, its boundary, consists of the 'pure states'.  Lorentz transformations act on the mixed states, and they act on the pure states.  But you don't need to know this stuff.)

If you take any point inside the ball and act on it by all the elements in a Kleinian group - a discrete subgroup of PSL(2,C) - you'll get a set S of points in the ball.  The set of points in the Riemann sphere that you can approach by a sequence of points in S is called a limit set of the Kleinian group.  And this set can look really cool! 

In these pictures, Jos Leys has systematically but rather artificially these cool-looking subsets of the Riemann sphere and puffed them up into 3-dimensional spaces: puffing a circle into a sphere, and so on.  This makes the picture nicer, but doesn't have a deep mathematical meaning.

Later, Jos Leys took a deeper approach, using quaternions to make limit sets that are truly 3-dimensional.  You can seem some here:

http://www.josleys.com/show_gallery.php?galid=346

They have a very different look.

For more on the math try these:

https://en.wikipedia.org/wiki/Kleinian_group
https://en.wikipedia.org/wiki/Möbius_transformation

Puzzle: if you put together everything I said, you'll get a physics interpretation of the limit set of a Kleinian group in terms of states of a spin-1/2 particle.  What is it?

#geometry  
36 comments on original post
27
3
Ivaldo Amorim's profile photoKristen Waters's profile photo
Add a comment...

John Baez

Shared publicly  - 
 
We Can't Stop

If you've vaguely heard about that scandalous Miley Cyrus character, but have never brought yourself to actually listen to any of her songs, you might prefer this version of her hit "We Can't Stop", sung in a 1950s doo-wop style by the group Postmodern Jukebox.

Postmodern Jukebox covers lots of modern hits in old-fashioned styles like ragtime, jazz, and bluegrass.  You can find them on YouTube.  The surprising thing is that they're really enjoyable!  First, they just sound nice.  Second, they let you ponder what's left of a modern hit after the glitz has been removed.

The brains behind Postmodern is Scott Bradlee, a musician from New York who fell in love with jazz at the age of 12 after hearing George Gershwin's "Rhapsody in Blue".  He became a jazz musician, but then had the brilliant idea of giving modern songs old-fashioned arrangements.  In 2009  he released "Hello My Ragtime '80s", which combined popular music from the 1980s with ragtime-style piano.  In 2013 he formed Postmodern Jukebox, and they first became famous with this song... probably sung in his living room.  The lead singer is Robyn Adele Anderson.

Some other good ones:

"All About That Bass" - https://www.youtube.com/watch?v=aLnZ1NQm2uk

"Creep" - https://www.youtube.com/watch?v=m3lF2qEA2cw

"Blurred Lines" - https://www.youtube.com/watch?v=Nz-OMn1o22Y

"Call Me Maybe" - https://www.youtube.com/watch?v=5meWI3iX1sE

If you don't know the originals of these songs, you have been living under a rock - which is not necessarily a bad thing.  Now you can catch up without ever entering the modern world!  Go straight to the postmodern world.

What's interesting, of course, is how well these songs do with old-fashioned arrangements.  At a certain basic level, like the chord progressions, American popular music is remarkably slow to change.
351
88
Sheila Crotty's profile photoSiyavash Nekuruh's profile photoPaul Harris's profile photoSimon Morris's profile photo
78 comments
 
Of course, as you probably know, I'm not actually taking about music, but the whole package: the looks, the moves, the fashion. With even the simplest software (as I've demonstrated) it's easy for anyone to run circles around it. One of the central themes of Tofflers' Third Wave and sequel Revolutionary Wealth was laying bare Mass Culture, Mass Consensus and Mass Synchronization as the girders of the largely by-gone Industrial Age. See, for instance: http://www.skypoint.com/members/mfinley/toffler.htm. For a broader discussion, do a search on "Mass Culture" + "Toffler" on Google.

Music: we've been part of a large and growing Underground for a long time in the college radio (like WMSE http://www.wmse.org) & club scene. Among us Millennials and Gen Xers, it has eclipsed the Mass Culture venue of commercial radio, whose very mention evokes the same reaction that Windows does to mobile users, Google & Apple fans, etc. (which now dominate the OS scene, with Windows under 25%.)

Digital production has taken off in the last few years, giving everyone access to production (with the rise of such sites as http://www.dance-music.org/). It's because of this surge (and my creation and mastery of an ever-growing part of the technology) that the question has loomed so greatly in my mind lately. If it is truly this easy even at this early stage in evolution to run circles around A&E Royalty -- the full package -- then how long will it be before their entire fan base is decimated all of them fully embracing the Tofflerian creed of the Prosumer? I think we're on the verge of a Twilight of the Stars.
Add a comment...

John Baez

Shared publicly  - 
 
The insanity of infinite reflections

This picture by +John Valentine shows a ball inside a mirrored spheroid... together with all its reflections! The real ball is at lower right.  The rest are reflections.  They form crazy patterns - the kind of thing mathematicians think about when they can't sleep at night.

This is like a picture I showed you earlier, made by +Refurio Anachro. But now the ball is lit from three directions with soft red, green, and blue lights, so we can see things more clearly.  The view simulates an ultra-wide-angle camera. 

This is just a low-resolution closeup of a much bigger and more detailed picture!  You can get other views here, along with a good discussion:

https://plus.google.com/u/0/114187364719055671781/posts/RzdjJwARTu6

You can get a really big version here:

https://sites.google.com/site/csjohnv/Ball001d-16k-low.jpg?attredirects=0&d=1

This is 16384 × 16384 pixels and about 16 megabytes.  If you know a nice way to display such a big image online, which makes it easy to zoom in on pieces, please try it!

Puzzle 1: what creates the black 'zone of invisibility', and the fractal hexagonal patterns near the zone of invisibility?

I don't really know the answer in detail - this could be a great math project.

I've watched a number of movies where the climactic final scene involves people fighting inside a hall of mirrors, where it's hard to tell who is real and who is a reflection.  Orson Welles' 1947 classic Lady from Shanghai may be the first - if you haven't seen that, you should definitely watch it.  Another that stands out is Bruce Lee's Enter the Dragon

Puzzle 2: what other movies or stories do you know involving this theme?
80
24
Nicholas Meyler's profile photomust mj's profile photoJAYAPRAKASH TP's profile photoMath Solver's profile photo
39 comments
 
I'm looking forward to it!
Add a comment...

John Baez

Shared publicly  - 
 
This is an official photo of the Canadian Supreme Court!  They dress like Santa Claus because of their curious role in the Canadian legal system.  I hadn't known about this until +Allen Knutson posted about it.

If you feel a verdict from a lower court has been unfair, on Christmas Eve you put a note in a sock explaining your case, and hang it on your fireplace.  Then, one of the Supreme Court members will come down your chimney and either grant your wish or leave you coals.  They know who's naughty and who's nice, thanks to an extensive system of legal clerks who dress as elves.

If you don't believe this is a real picture of the Canadian Supreme Court, do a Google image search!   Here, I'll make it easy:

https://www.google.com/search?q=canadian+supreme+court&tbm=isch

There is a long history of goofy outfits for judges.  British judges, even ones who aren't bald, are required to wear a wig of horse hair!  This was the origin of the Wig Party, who used to be the main opposition to the Tories.  And the Australians have a kangaroo court, who jump to decisions.

Puzzle 1: why does the Canadian supreme court really dress like this?

Puzzle 2: why do British judges wear wigs?

Puzzle 3: what's the origin of the phrase 'kangaroo court'?

If you look up the answers using Google, you get special extra credit: it means you know how to use the internet.
58
2
John Baez's profile photoDan Piponi's profile photoRicardo Buring's profile photoAdrian Colley's profile photo
22 comments
 
Lucky for you, you broke the pattern by finding something not funny. Otherwise I would have had to invent comedic modal logic and construct a fixed point of the modal operator within it.
Add a comment...

John Baez

Shared publicly  - 
 
The flood after the impact

On Mars, an asteroid impact can cause a flood!

This is a place called Hephaestus Fossae, on the northern hemisphere of Mars.  The image has been colored to show the elevation: green and yellow shades represent shallow ground, while blue and purple stand for deep depressions, as much as 4 kilometers deep.

You can see a few dozen impact craters, some small and some big, up to 20 kilometers across.   But I'm sure you instantly noticed the cool part: the long and intricate canyons and riverbeds.  These were created by the same impact that made the largest crater!

When a comet or an asteroid crashes at high speed into a planet, the collision dramatically heats up the surface at the impact site.  In the case of the large crater seen in this image, the heat melted the soil – a mixture of rock, dust and also, hidden deep down, water ice – resulting in a massive flood.  And before drying up, this hot mud carved a complex pattern of channels while flowing across the planet’s surface!

The melted rock-ice mixture also made the debris blankets surrounding the largest crater.  Since there aren't similar structures near the small craters in this image, scientists believe that only the most powerful impacts were able to dig deep enough to release part of the frozen reservoir of water lying beneath the surface.

Why is it called 'Hephaestus Fossae'?  Hephaestus was the Greek god of fire.  Fossae are channels or canyons.  So it's a good name.

Puzzle: about when did this large impact occur? 

I don't know!

This picture was taken by the high-resolution stereo camera on ESA’s Mars Express orbiter on 28 December 2007, and my post is paraphrased from this article:

http://www.esa.int/spaceinimages/Images/2009/06/The_flood_after_the_impact

#mars #astronomy  
147
22
Philipp Birken's profile photoKevin Maria's profile photoPeter Nonnenmann's profile photomary lemire's profile photo
18 comments
 
+Benoit Hudson - yes, it's more pretty, though perhaps only when it's big.  I love the deep canyons and gulches rather far from the crater... which show up as dark blue cracks in the original post.
Add a comment...

John Baez

Shared publicly  - 
 
In 1901, you could pay 50 cents to ride an airship to the Moon

This article by Ron Miller is so cool I'm just going to quote some:

The passengers wait eagerly in the ornate lobby of the enormous spaceport. Soon, a signal indicates that their spaceship is ready for boarding. As they wait, special displays instruct them about how their spaceship functions and what to expect once they leave Earth's atmosphere. Aboard the giant spacecraft — as luxuriously appointed as any yacht — they are soon on their way to a vacation on the Moon.

No, this isn't a vision of the future of space tourism. It's what happened in 1901, when people could pay a princely half dollar for a ticket to ride into space.

[...]

Thompson spared no expense in creating the illusion of a trip to the Moon. To house his show, he erected an eighty-foot-high, 40,000-square-foot building that for sheer opulence put European opera houses to shame. It cost a staggering $84,000 to construct... at a time when a comfortable home could be built for $2000.

For fifty cents — twice the price of any other attraction on the midway, such as the ever-popular "Upside-Down House" — customers of "Thompson's Aerial Navigation Company" took a trip to the moon on a thirty-seat spaceship named "Luna". The spaceship resembled a cross between a dirigible and an excursion steamer, with the addition of enormous red canvas wings that flapped like a bird's. The wings were worked by a system of pulleys and the sensation of wind was created by hidden fans. A series of moving canvas backdrops provided the effect of clouds passing by and the earth dropping into the distance. Lighting and sound effects added to the illusion.

[...]

Every half hour, at the sound of a gong and the rattle of anchor chain, the "Luna" — "a fine steel airship of the latest pattern", according to one newspaper — rocked from side to side and then rose into the sky under the power of its beating wings. The passengers, sitting on steamer chairs, see clouds floating by, then a model of Buffalo far below, complete with the exposition itself and its hundreds of blinking lights. The city soon falls into the distance as the entire planet earth comes into view. Soon, the ship is surrounded the twinkling stars of outer space. After surviving a terrific — and spectacular — electrical storm the "Luna" and its passengers sets down in a lunar crater.

Read the whole thing here, and look at pictures:

http://io9.com/5914655/in-1901-you-could-pay-50-cents-to-ride-an-airship-to-the-moon

Thanks to +Matt McIrvin for pointing it out!
146
44
pankaj chauhan's profile photoTURBO BURN DIGITAL RADIO's profile photoNathy Taaj's profile photoLord Stanley Wraithton's profile photo
35 comments
 
+Alexander Fretheim
It is not that we kill, all animals do that as you say. I was referring to the way we've turned violence into mass entertainment in ways that are not a requirement of our survival ... which I'm sure no other species here on Earth has. And, this entertainment is sought out by many who, if asked, would speak out against it. This is what I find so troubling about the electrocution of one elephant being shown over and over, all in the interest of entertainment.
Add a comment...

John Baez

Shared publicly  - 
 
The battle has begun

We're starting to fight global warming.  It's going to be a difficult war, and it's not clear we'll win.  But it's the most exciting, suspenseful story that we're all part of.

As the engineer Saul Griffith said:

"It's not like the Manhattan Project, it's like the whole of World War II, only with all the antagonists on the same side this time. It's damn near impossible, but it is necessary. And the world has to decide to do it."

A few promising signs:

1) On Wednesday morning, the governor of California set a goal of cutting carbon emissions by 40% below 1990 levels by 2030.  This matches the target set by the EU in October.   Both California and the EU are aiming to cut emissions 80% by 2050.  The governor said:

“We must demonstrate that reducing carbon is compatible with an abundant economy and human well-being.  Taking significant amounts of carbon out of our economy without harming its vibrancy is exactly the sort of challenge at which California excels. This is exciting, it is bold and it is absolutely necessary if we are to have any chance of stopping potentially catastrophic changes to our climate system."

At the national level, the US is dragging its heels.  But the states don't need to wait!  California, Oregon, Washington and British Columbia have signed a regional agreement to reduce carbon emissions, and the governor has signed separate accords with leaders in Mexico, China, Japan, Israel and Peru.  

2) Copenhagen has an ambitious plan to go carbon-neutral by 2025:

http://www.theguardian.com/environment/2013/apr/12/copenhagen-push-carbon-neutral-2025

As with California, the goal is not to save the world single-handedly, but to figure out how things can be done, so others can copy.

3) Pope Francis is getting serious about global warming.  He's already said he believes it's mostly manmade and that a Christian who does not protect God’s creation “is a Christian who does not care about the work of God”.  Now he's written an encyclical about it - the most significant sort of papal document.  This will come out in June.

I'm not a big fan of the Catholic church.  But it's important that everyone claiming to be a moral leader use their influence to get people to take this issue seriously, so I applaud this move.

Cardinal Peter Turkson, who is taking the lead on this, said increasing use of fossil fuels is disrupting Earth on an “almost unfathomable scale”, and says we need a “full conversion” of hearts and minds on this issue. 

http://www.theguardian.com/environment/2015/apr/28/vatican-climate-change-summit-to-highlight-moral-duty-for-action
Gov. Jerry Brown accelerated California's effort to slash greenhouse gas emissions Wednesday, burnishing the state's reputation as a pacesetter in the battle against climate change.
57
6
Sourena Yadegari's profile photoScott GrantSmith's profile photoIan Petersen's profile photoAlex Bikbaev's profile photo
20 comments
 
+John Baez: "However, it will take some sort of words to convince governments to stop subsidizing fossil fuel consumption: they do it for political reasons that are deeply entrenched" -- what's even sadder is that many countries that are doing this can't even remotely afford the cost of the subsidies. India has steep subsidies for fuel, but part of the price of that is the fact that millions continue to starve in the country every year. :(
Add a comment...

John Baez

Shared publicly  - 
 
Mathematical dream worlds

+Jos Leys blends mathematics and art in a delightful way.  You don't need to know math to enjoy this picture.   It's a whimsical and mysterious landscape.  The bright colors make it clownish, but the shadows make it a bit eerie: the sun is setting, and who knows what happens here at night!   You can see more here:

http://www.josleys.com/show_gallery.php?galid=252

On the other hand, if you read the title of this gallery, you'll see there's math here: "the first 3d views of limit sets of Kleinian groups".  And trying to understand this math will lead you on quite a journey.  Let me sketch it here... I apologize for going rather fast.

A Kleinian group is a discrete subgroup of the group called PSL(2,C).  This group shows up in many ways in math and physics. 

Physicists call it the Lorentz group: it's the group generated by rotations and Lorentz transformations, which acts as symmetries in special relativity. 

In math, it's called the group of Möbius transformations or fractional linear transformations.  Those are transformations like this:

z |→ (az + b)/(cz + d)

where z is a complex number and so are a,b,c,d.  These can be seen as transformations of the Riemann sphere - the complex plane together with a point at infinity.  They are, in fact, precisely all the conformal transformations of the Riemann sphere: the transformations that preserve angles. 

But this group PSL(2,C) also acts as conformal transformations of a 3-dimensional ball whose boundary is the Riemann sphere!   And that's important for understanding this picture.

(In physics, this ball is the set of 'mixed states' for a spin-1/2 particle, and the sphere, its boundary, consists of the 'pure states'.  Lorentz transformations act on the mixed states, and they act on the pure states.  But you don't need to know this stuff.)

If you take any point inside the ball and act on it by all the elements in a Kleinian group - a discrete subgroup of PSL(2,C) - you'll get a set S of points in the ball.  The set of points in the Riemann sphere that you can approach by a sequence of points in S is called a limit set of the Kleinian group.  And this set can look really cool! 

In these pictures, Jos Leys has systematically but rather artificially taken these cool-looking subsets of the Riemann sphere and puffed them up into 3-dimensional spaces: puffing a circle into a sphere, and so on.  This makes the picture nicer, but doesn't have a deep mathematical meaning.

Later, Jos Leys took a deeper approach, using quaternions to make limit sets that are truly 3-dimensional.  You can seem some here:

http://www.josleys.com/show_gallery.php?galid=346

They have a very different look.

For more on the math try these:

https://en.wikipedia.org/wiki/Kleinian_group
https://en.wikipedia.org/wiki/Möbius_transformation

Puzzle: if you put together everything I said, you'll get a physics interpretation of the limit set of a Kleinian group in terms of states of a spin-1/2 particle.  What is it?

#geometry  
101
24
Ivaldo Amorim's profile photoaltivo fonizon's profile photoKristen Waters's profile photoDave Gordon's profile photo
36 comments
Ray Lee
 
+John Baez That does help. A Lorentz transform can make momentum sharper, so the entropy must be going down. So I believe the trend line at least.

Thank you for your patience in walking me through this!
Add a comment...

John Baez

Shared publicly  - 
 
The toughest animal on the planet

A rotifer is an animal that lives in water and sweeps food into its mouth with small hairs.  There are many kinds, most less than a millimeter in length.  They can eat anything smaller than their head.

The toughest are the bdelloid rotifers.  These can survive being completely dried out for up to 9 years!  When they dry out, they sometimes crack.  Even their DNA can crack... but when they get wet, they come back to life!

Thanks to this strange lifestyle, their DNA gets mixed with other DNA.   Up to 10% of their active genes come from bacteria, fungi and algae!!! 

Scientists have found DNA from 500 different species in the genes of a rotifer from Australia.  "It's a genetic mosaic. It takes pieces of DNA from all over the place," said one of the study's authors. "Its biochemistry is a mosaic in the same way. It's a real mishmash of activities."

Perhaps because of this, bdelloid rotifers don't bother to have sex. 

Their ability to survive dry conditions makes them great at living in desert lakes and mud puddles that dry up.  But they also use this ability to beat some parasites.  When they dry out, the parasites die... but the rotifers survive!

On top of all this, bdelloid rotifers can survive high doses of radiation.  My guess is that this is just a side-effect of having really good genetic repair mechanisms.

Puzzle 1: what does 'bdelloid' mean?

Puzzle 2: what other words begin with 'bd'... and why?

Here's the paper that found 10% of active genes and 40% of all enzyme activity in bdelloid rotifers involve foreign DNA:

• Chiara Boschetti, Adrian Carr, Alastair Crisp, Isobel Eyres, Yuan Wang-Koh, Esther Lubzens, Timothy G. Barraclough, Gos Micklem and Alan Tunnacliffe, Biochemical diversification through foreign gene expression in bdelloid rotifers, PLOS Genetics, 15 November 2012, http://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1003035.

The  animated gif is from here:

http://merismo.tumblr.com/post/43868329996/gif-rotifer-with-cilia-on-corona-present-mastax

#spnetwork #bdelloid #rotifer doi:10.1371/journal.pgen.1003035
179
69
Dave Gordon's profile photoKurt Hoell's profile photoAndrew Thall's profile phototej prakash's profile photo
33 comments
 
+John Baez _Some_ sense it sure makes, but isn't the structure/valuation of the Drake equation behind the status of paradox of Fermi's paradox, arbitrary enough to justify comparing it for perspective, to the case of modeling the Earth as a finite horizontal surface -- whose boundary it then makes some sense to not approach, for fear of falling off?

So, "shut up and hope..." ok but also "maintain process enough to not risk groundlessly persisting forever".

BTW, I've had time to regret not protesting earlier that I find the like of bdelloid rotifers more evocative of checkers than of chess.
Add a comment...

John Baez

Shared publicly  - 
 
Twin dodecahedra

Here Greg Egan has drawn two regular dodecahedra, in red and blue.  They share some corners - and these are the corners of a cube, shown in green! 

I learned some cool facts about this from Adrian Ocneanu when I was at Penn State.  First some easy stuff.  You can take some corners of a regular dodecahedron and make them into the corners of a cube.  But not every symmetry of the cube is a symmetry of the dodecahedron!  If you give the cube a 90° rotation around any face, you get a new dodecahedron.  Check it out: doing this rotation switches the red and green dodecahedra.  These are called twin dodecahedra.

But there are actually 5 different ways to take a regular dodecahedron and make them into the corners of a cube.  And each one gives your dodecahedron a different twin!  So, a dodecahedron actually has 5 twins.

But here's the cool part.  Suppose you take one of these twins.  It, too, will have 5 twins.  One of these will be the dodecahedron you started with.  But the other 4 will be new dodecahedra: that is, dodecahedra rotated in new ways.

How many different dodecahedra can you get by continuing to take twins?  Infinitely many!

In fact, we can draw a graph - a thing with dots and edges - that explains what's going on.  Start with a dot for our original dodecahedron.  Draw dots for all the dodecahedra you can get by repeatedly taking twins.  Connect two dots with an edge if and only if they are twins of each other.

The resulting graph is a tree: in other words, it has no loops in it!  If you start at your original dodecahedron, and keep walking along edges of this graph by taking twins, you'll never get back to where you started except by undoing all your steps.

Ocneanu's proof of this is very nice, using some 4-dimensional geometry and group theory.  I will have to outline it somewhere, because Ocneanu is famous for not publishing most of his work.  But I like how you can state the end result without these more sophisticated concepts.

Here are some puzzles.

You can also choose some corners of a cube and make them into the corners of a regular tetrahedron.  You can fit 2 tetrahedra in the cube this way.  These are a bit like the 5 cubes in the dodecahedron, but there's a big difference.

Here's the difference.  In the first case, every symmetry of the tetrahedron is a symmetry of the cube it's in.  But in the second case not every symmetry of the cube is a symmetry of the dodecahedron.  That's why we get 'twin dodecahedra' but not 'twin cubes'.

Puzzle 1: If you inscribe a tetrahedron in a cube and then inscribe the cube in a dodecahedron, is every symmetry of the tetrahedron a symmetry of the dodecahedron?

Puzzle 2: How many ways are there to inscribe a tetrahedron in a dodecahedron?  More precisely: how many ways are there to choose some corners of a regular dodecahedron and have them be the corners of a regular tetrahedron?

And a harder one:

Puzzle 3: Trees are related to free groups.  What free group is responsible for Ocneanu's result?

#geometry  
108
28
Bill Reed's profile photoMath Solver's profile photoBlue Tyson's profile photoAlexander Yu. Vlasov's profile photo
84 comments
 
For much more on twin dodecahedra, read this:

http://blogs.ams.org/visualinsight/2015/05/01/twin-dodecahedra/
Add a comment...
John's Collections