Cover photo
John Baez
Works at Centre for Quantum Technologies
Attended Massachusetts Institute of Technology
Lives in Riverside, California
54,970 followers|37,374,253 views


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" -

"Creep" -

"Blurred Lines" -

"Call Me Maybe" -

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.
swordmorning1's profile photoDinesh Kumar's profile photodestiny ray's profile photoJasmine LeBlanc's profile photo
See also Paul Anka's album "Rock Swings":
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?

Dave Gordon's profile photoBill Reed's profile photoMath Solver's profile photoBlue Tyson's profile photo
+Michael Nelson - cool!  I will have to think about this, but please remember to look at my blog Visual Insight on May 1st when my more mathematical article on twin dodecahedra shows up.  It may give you more ideas! 
Add a comment...

John Baez

Shared publicly  - 

You can now make your own cyborg roach for just $100.   Just buy this kit developed by the company Backyard Brains:

Are you a teacher or parent that wants to teach a student about advanced neurotechnologies? You are in luck! After 3 long years of R&D, the RoboRoach is now ready for its grand release! We are excited to announce the world's first commercially available cyborg! With our RoboRoach you can briefly wirelessly control the left/right movement of a cockroach by microstimulation of the antenna nerves. The RoboRoach is a great way to learn about neural microstimulation, learning, and electronics!

We are recently ran a successfully-funded kickstarter campaign to fund the release of our new RoboRoach! The hardware and firmware development are complete and we are now shipping!

Product Details

The RoboRoach "backpack" weighs 4.4 grams with the battery, and each battery will last over a month! Following a brief surgery you perform on the cockroach to attach the silver electrodes to the antenna, you can attach the backpack to the roach and control its movement for a few minutes before the cockroach adapts. When you return the cockroach to its cage for ~20 minutes, he "forgets" and the stimulation works again. Once you receive your RoboRoach in the mail, follow our online surgery instructions and videos and you will soon be on your way to becoming an expert in neural interfaces. After about 2-7 days, the stimulation stops working altogether, so you can clip the wires and retire the cockroach to your breeder colony to spend the rest of its days making more cockroaches for you and eating your lettuce.

Technical Specs

1x Free iOS or Android 4.3+ application for remote control
1x Bluetooth Roboroach backpack control unit
1x 1632 RoboRoach Battery
3x Electrode Sets (to implant 3 Roaches)

View our RoboRoach Ethics Statement

Backyard Brains has developed ethical guidelines for all our products. You can read more in our statement regarding our use of insect for experiments at:

I feel ethical qualms about taking away the autonomy of an animal this way, and their ethics statement doesn't really address that.  This is the closest they come:

Criticism: Modifying a living creature to make a toy is wrong.

The RoboRoach circuit is not a toy. This new bluetooth version is a powerful low-cost tool for studying neural circuits, allowing for students to make discoveries. High school students in New York, for example, have discovered random stimulation causes much slower adaptation times. We have scientist and high school educator colleagues who are mentoring students in novel behavioral experiments using the RoboRoach circuit. Some highlights will be posted on our website soon.

By focusing on the question of whether the RoboRoach is a "toy", they dodge the harder question of when it's okay to override the nervous system of an animal and make it do what you want.  Perhaps feeling a bit nervous about this, some of the cyborg roach developers say they want to use it as a "rescue robot" that can crawl around and hear people trapped under collapsed buildings.  I think most people would say this is okay, at least if it actually works.

For a critical view on the ethics, see:

For more on how to actually make a RoboRoach, go here:
Scott Ellis's profile photosanti buranapanichkij's profile photoChris Harpner (CSharpner)'s profile photoRachelle M. M. Adams's profile photo
I have zero ethical concerns about this. Having a nervous system isn't sacred. Neither are the emotions that inevitably accompany a nervous system. I kill dozens of bugs a year, and I don't think taking their autonomy is much worse.

I have a rule, no forcing experiments on anything capable of learning calculus. Simple rule, never broke it.
Add a comment...

John Baez

Shared publicly  - 
Endless reflections

If you stood in a spherical room with mirrors for walls, what would you see?  Of course you'd need a flashlight.

This picture gives an idea of what you'd see.  It's small white marble in a mirrored spheroid, as drawn by +Refurio Anachro.  You can see almost endless reflections, forming complex patterns.  These pose many fascinating puzzles!

First, just to be clear: spheroid is a sphere that has been stretched or squashed along one axis.  This is a prolate spheroid, meaning it's been stretched: it's about 10% taller than it is wide.   The reflections in here are more complicated than in a sphere.

Refurio writes:

The pattern is made of reflections of the little white marble you can see on the right hand side. To the slightly bluish mirror it appears pure white, but I have shaded the unreflected marble afterwards to make it easier to identify.

More precisely, the marble is a sphere, with radius a tenth that of the equatorial circle of the spheroid, and touching it there from the inside. I’ve placed it 90° away from one of the two ‘straight’ positions to make the image less symmetric and more interesting.

The idea behind the marble was that we could pick a point and highlight all rays coming close to it. But the presence of the marble changes things: since it extends into the spheroid, it will catch high flying rays that might not have gotten reflected in the vicinity of our chosen point. Coloring a patch of the spheroid’s surface, or punching a hole in it, would not have produced some rather beautiful artifacts you see here.

That large, wavy, most bright reflection to the left, and all the similar ones, would resolve to a number of separate elongated images of our spot. And the smaller blots further inside, the biggest one looking like two intersecting elliptic discs, would look more like a single one. And the marble-thick, brighter appearing region all around the rim.

Aside from that, the marble works like a flashlight. Think of the pattern as a fixed, static thing, produced by all possible rays bouncing within the ellipsoid. Moving the flashlight will illuminate different parts of it. Some points will be especially hard to illuminate. Two of them are the foci of the prolate spheroid: they’re the the dark points that appear to attract reflections that can never reach them, just above and below the center.

To dig deeper into the math, visit my blog:

xiaoyi jing's profile photoMichelle Beissel's profile photoStefano Muccinelli's profile photoMath Solver's profile photo
Yes, there should be some vaguely number-theoretic characterization of the points that take a long time to get hit by reflections... a bit like the results about numbers that are hard to approximate well by rationals with small denominators.
Add a comment...

John Baez

Shared publicly  - 
Life is a game of information and entropy.  We're having a workshop on this from Wednesday April 8th to Friday April 10th!  I hope you can  join us!  To watch live streaming videos of the workshop, go to the link here.  Go down to where it says Investigative Workshop: Information and Entropy in Biological Systems.  Then click where it says live link. There’s nothing there now - but there should be when the show starts.

You should also be able to watch videos of talks after the conference.   And you can see some talk slides now!

Here's the schedule of talks. The hours are in Eastern Daylight Time: add 4 hours to get Greenwich Mean Time. The talks start at 10 am EDT, which is 2 pm GMT. 

Wednesday April 8

• 9:45-10:00 — the usual introductory fussing around.

• 10:00-10:30 — John Baez, Information and entropy in biological systems.  Slides here:

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Chris Lee, Empirical information, potential information and disinformation. Slides here:

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — John Harte, Maximum entropy as a foundation for theory building in ecology.  Slides here:

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Annette Ostling, The neutral theory of biodiversity and other competitors to the principle of maximum entropy.

• 2:45-3:00 — questions, coffee.

• 3:00-5:30 — break up into groups for discussions.

• 5:30 — reception.

Thursday April 9

• 10:00-10:30 — David Wolpert, The Landauer limit and thermodynamics of biological organisms.

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Susanne Still, Efficient computation and data modeling.

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — Matina Donaldson-Matasci, The fitness value of information in an uncertain environment.  Paper here:

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Roderick Dewar, Maximum entropy and maximum entropy production in biological systems: survival of the likeliest?

• 2:45-3:00 — questions, coffee.

• 3:00-6:00 — break up into groups for discussions.

Friday April 10

• 10:00-10:30 — Marc Harper, Information transport and evolutionary dynamics.  Slides here:

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Tobias Fritz, Characterizations of Shannon and Rényi entropy.  Slides here:

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — Christina Cobbold, Biodiversity measures and the role of species similarity.

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Tom Leinster, Maximizing biological diversity.

• 2:45-3:00 — questions, coffee.

• 3:00-6:00 — break up into groups for discussions.
Sankua Chao's profile photoRanjith V's profile photoDave Gordon's profile photoPol Nasam's profile photo
Thanks for saving the day!  People can see his slides here:

The second half is the dramatic part.
Add a comment...
I can't get this tune out of my head - and I don't want to!  Luzmila Carpio is a Bolivian singer who sings in Quechua, a native American language that was once banned in Peru.   She sings in a deeply traditional style - but here she's been remixed by El Remolón, a minimalist techno producer from Buenos Aires.   The result is striking: a sweet, delicate lark in the chilly modern world.

This tune is part of an album Luzmila Carpio Meets ZZK.  You can hear the whole thing here:

Following the philosophy of ZZK Records, all remixes were made in collaboration with Luzmila Carpio, who had the final say over what was done.   But alas, only this one track pleases me.

As a child, Luzmila Carpio learned the songs of the Quechua and Aymara indigenous peoples who inhabit the mountains and valleys of Northern Potosí in Bolivia.  As a teenager, she moved to the mid-sized city of Oruro.  She soon gained fame for her voice, and her song "Siway Azucena" was the first truly indigenous tune to have widespread popular success in Bolivia.

I don't understand her career, but she later went to Paris, and in 2006 she became Bolivia's ambassador to France!  This lasted until 2010, and the next year she was made a Grand Officer of the Order of Merit of the French Republic.

El Remolón - 'the lazy one' - is really Andrés Schteingart.

There are over 10 million speakers of various related Quechan languages in Peru, Argentina, Bolivia and other countries.  The Inca were just one of the peoples who spoke these languages.  By now Quecha and Spanish have blended.  So you actually know some Quechan words: coca, condor, guano, jerky, llama, puma, quinine, quinoa, vicuña and possibly gaucho!

Apparently there are a bunch of people who speak Quechan in Queens, New York and Paterson, New Jersey.  I'm always fascinated by how people change and adapt, and this song is a metaphor for that.

For some more traditional music by Luzmila Carpio, go here:

For more to read:
Irina Tcherednichenko's profile photoJulien Baboud's profile photostephen robson's profile photoAmparo Cabal's profile photo
+Nick DeWaal that sounds super interesting! Is the rest of their language equally modular?
Add a comment...
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,

The  animated gif is from here:

#spnetwork #bdelloid #rotifer doi:10.1371/journal.pgen.1003035
Sibel Aktas's profile photoMichelle Beissel's profile photoDave Gordon's profile photoKurt Hoell's profile photo
+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  - 
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:

You can get a really big version here:

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?
Nicholas Meyler's profile photomust mj's profile photoJAYAPRAKASH TP's profile photoMath Solver's profile photo
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:

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.
John Baez's profile photoDan Piponi's profile photoRicardo Buring's profile photoAdrian Colley's profile photo
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  - 
Talks from the 8th dimension

This week I'm visiting Penn State University.  If you're nearby, you can hear 2 talks about fun geometry stuff:

Split octonions and the rolling ball, 2:30 – 3:20 p.m, Tuesday April 14th, 106 McAllister Building.

Learn what happens when you roll a ball on another ball exactly 3 times as big!   The geometry of objects rolling without slipping or twisting is always fun - but in this particular case the problem gets extra symmetries, which are best understood using an 8-dimensional number system called the split octonions.  What's so great about exactly three times as big?  I'll explain!

The exceptional Jordan algebra and the Leech lattice, 12:05 – 1:20 pm, Wednesday April 15th, 114 McAllister Building.

There's a specially beautiful way to pack balls in 24 dimensions, called the Leech lattice.  When physicists classified the algebras that could describe observables in quantum mechanics, they found a weird possibility: a 27-dimensional one called the exceptional Jordan algebra.   It turns out that the Leech lattice fits into the exceptional Jordan algebra in a nice way... which comes from the octonions.  So all this stuff fits together!  This talk is part of the "Geometry Luncheon Seminar", where mathematicians eat lunch and talk about mind-blowing geometry.

The first talk is about work I did with +John Huerta and James Dolan, and it will feature some fun animations made by Geoffrey Dixon.  The second is about work with Greg Egan.

The actual reason I'm at Penn State is to give a guest lecture at John Roe's undergrad course on "Mathematics for Sustainability".  I want to teach a course on math and environmental issues.  It'll be good to hear how he's been doing this.  But I thought it would be fun to talk about some other things too.

I'll also visit one of my old haunts, the Institute for Gravitation and the Cosmos, where Abhay Ashtekar, Eugenio Bianchi and others are working on loop quantum gravity.  And I'll talk to +Jason Morton about network theory.  It should be a busy, fun week.

But first I have to work on my talks...

This image here, made by Jason Hise, shows a 24-cell, a regular polytope in 4 dimensions.  There's a sculpture of this shape in the math department at Penn State!  It was designed by the mathematician Adrian Ocneanu.  I haven't been here since it was built so it will be fun to see:
nilblau punktde's profile photoEric Green's profile photoRicky Moore-Daniels's profile photoNauka Znanost's profile photo
Oh, that's where the Mathematical Biology Institute is.  For some reason I just heard of it recently, when at the National Institute for Mathematical and Biological Synthesis.
Add a comment...

John Baez

Shared publicly  - 
American hero

On Monday night, artists built this monument to Edward Snowden in Brooklyn.  The next day, it was taken down.   Will there be a permanent one someday?

Martin Luther King was put in jail 29 times, and now there's a monument to him in Washington DC.  But it was built only in 2011, forty-three years after King was killed.

If Snowden ever gets a monument, here are some quotes of his they can carve on it:

There can be no faith in government if our highest offices are excused from scrutiny - they should be setting the example of transparency.

I would rather be without a state than without a voice.

I don't see myself as a hero because what I'm doing is self-interested: I don't want to live in a world where there's no privacy and therefore no room for intellectual exploration and creativity.

After the statue was removed by park officers, a group of artists who call themselves "The Illuminator" — not related to those who built the original sculpture — used laptops and projection equipment to cast an image of Snowden in a haze of smoke at the spot where the sculpture had been.
Alberto M's profile photoMichelle Hood's profile photoЭдвард Джо́зеф Сноуден's profile photoAndréa Negra Guarani's profile photo
I'm already well past the point where I think the United States, along with both its former and current leadership, needs to be brought before the International Criminal Court for its actions and am calling on this very action to be taken. Blunt use of force is neither a necessary nor sufficient condition for gross violations of the edicts covered by the Universal Declaration of Human Rights to pass into the jurisdiction of this body.
Add a comment...

John Baez

Shared publicly  - 
The harvest

I live on the edge of the desert in southern California.  We tore up our lawn and planted beautiful plants that use less water.  Drip irrigation instead of sprayers! 

But we do indulge in some citrus trees.  Here's the harvest!

Satsumas in front - they're like mandarins, but different.  Meyer lemons at rear left - they're sweeter than ordinary lemons. Grapefruits at rear right - they're not very big, perhaps because our tree is still young and struggling.

What's a 'mandarin'?   It's the mandarin orange, Citrus reticulata, often marketed as a 'tangerine'.  According to DNA studies, the mandarin is one of the 4 ancestors of all other citrus species, which arose through hybridization and breeding.   The other 3 are the the citron, the pomelo, and something called a papeda. 

Among these 4 citrus ancestors, mandarins are the only really sweet ones, so they were used to create many of the fruits people like now.

For example, a Meyer lemon is probably a cross between a true lemon and a mandarin or an orange.  A grapefruit is a cross between an orange and a pomelo - a huge fruit that looks like a grapefruit on steroids.  And an orange is itself probably a cross between a pomelo and a mandarin!

It's all very complicated:

Luckily you don't need to know this stuff to enjoy growing and eating citrus!

John Baez's profile photoJürgen Christoffel's profile photoAmina Shrestha's profile photoAmiram Mizne's profile photo
Good luck on those trees - having fruit trees has really improved my life, and it's not just the fruit.
Add a comment...
Have him in circles
54,970 people
Robert Bevins's profile photo
Terry Floyd's profile photo
Kayvion Walton's profile photo
Beverly vel's profile photo
Isaac Kuo's profile photo
Anemon Villas's profile photo
Cory Mahnen's profile photo
Stig Norland's profile photo
Fei Wang's profile photo
I'm a mathematical physicist.
  • Centre for Quantum Technologies
    Visiting Researcher, 2011 - present
  • U.C. Riverside
    Professor, 1989 - present
Map of the places this user has livedMap of the places this user has livedMap of the places this user has lived
Riverside, California
Contributor to
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
Basic Information