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## John Baez

Occupation: I'm a mathematical physicist. (Centre for Quantum Technologies)

Location: Riverside, California

Followers: 57,654

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Views: 53,355,793

Cream of the Crop: 11/05/2011

Added to CircleCount.com: 07/21/2011That's the date, where John Baez has been indexed by CircleCount.com.

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### Most comments: 106

2016-08-27 04:43:12 (106 comments; 26 reshares; 220 +1s; )

**Dark mysteries**

You probably heard the news this week: astronomers found a galaxy that's 98% dark matter.

It's called **Dragonfly 44**. It's extremely faint, so it doesn't have many stars. But we can use redshifts to see how fast those stars are moving - over 40 kilometers per second on average. If you do some calculations, you can see this galaxy would fly apart unless there's a lot of invisible matter providing enough gravity to hold it together. (Or unless something even weirder is happening.)

Something similar is true for most galaxies, including ours. What makes Dragonfly 44 special is that 98 percent of the matter must be invisible. And this is just in the part where we see stars. If we count the outer edges of the galaxy, the **halo**, the percentage could rise to 99% or more!

By comparison, theMilky Way is... more »

### Most reshares: 92

2016-08-08 04:51:46 (0 comments; 92 reshares; 201 +1s; )

**What's wrong with Trump?**

More and more people are wondering. Here's an insightful analysis from a psychiatric social worker. Please reshare. It's not light reading, but we need to understand what we're dealing with here.

--------------------**The Billionaire's Baffling Behavior Explained**

Recently, Mr. Trump's words and actions in various situations have become headline news. Suddenly, many people are alarmed and are questioning temperament, his emotional stability. Dementia? Campaign tactics?

No, it's not an illness that can be treated so he can return to his usual state of health. It's not like when a car's brakes don't work and the mechanic fixes them. It's more like the car came off the assembly line without them.

It's a structural problem.

We're witnessingte... more »

### Most plusones: 339

2016-08-14 02:05:13 (47 comments; 46 reshares; 339 +1s; )

**Not like Earth**

At the end of August, the European Southern Observatory will announce a planet orbiting **Proxima Centauri** - the star closest to our Sun, 4.24 light years away. They're trying to make this planet sound like Earth... and that's cool. But I'll tell you some ways it's not.

Mainly, Proxima Centauri is really different from our Sun!

It's a red dwarf. It puts out just only 0.17% as much energy as our Sun. So any planet with liquid water must be very close to this star.

And because it's cooler than the Sun, Proxima Centauri mainly puts out infrared light - in other words, heat radiation. Its visible luminosity is only 0.005% that of our Sun!

So if you were on a planet as warm as our Earth orbiting Proxima Centauri, it would look very dim - about 3% as bright as our Sun.

Of course,if there... more »

Latest 50 posts

2016-10-24 17:59:12 (10 comments; 15 reshares; 59 +1s; )

**Category theory - an advanced online course!**

Hurrah! My student +Brendan Fong has teamed up with Alexander Campbell and +Emily Riehl to teach an advanced reading course on category theory to 8 students chosen from around the world. We'll all benefit, because these students will write essays on The n-Category Café, a popular math blog. And next summer, they'll give talks at the 2017 International Category Theory Conference at the University of British Columbia.

Emily Riehl has done this before, and it's worked well. So if you're a grad student interested in category theory, you should apply to take this course!

The course is about "functorial semantics" - a great idea going back to Lawvere. The students will read 8 papers on this topic. Here's the ad for the course, written by Emily Riehl

----------

Inearly 2... more »

2016-10-21 15:07:05 (36 comments; 11 reshares; 57 +1s; )

**Activity diagrams****Unified modeling language** or **UML** is a commonly used way to create models of complex systems. Among other things it lets you draw **activity diagrams** with boxes connected by wires, like this one here. To me this is clearly part of "applied category theory". It deserves to be recognized as such!

On Monday, my grad student +Blake Pollard and I went down to San Diego to visit the west coast headquarters of Metron. We're working with this company to develop new mathematics and apply it to Coast Guard search and rescue missions. They have a software environment that resembles UML. We got a tutorial about that, and now I'm going to think about it mathematically. It's an interesting new twist in my attempts to understand complex networks.

For the whole story, go here:

https://... more »

2016-10-19 16:25:09 (15 comments; 14 reshares; 166 +1s; )

**Mini Saturn****Chariklo** orbits the Sun between Saturn and Uranus. Just 250 kilometers across, it has two tiny rings!

Is it an asteroid? Not quite: it's a 'centaur'. In Greek mythology, a centaur was half-human, half-horse. In astronomy, a **centaur** is halfway between an asteroid and a comet. Centaurs live in the outer solar system between Jupiter and Neptune. They don't stay there long - at most a million years. They come from further out, pulled in by the gravity of Neptune, but their orbits are chaotic and they eventually move in toward Jupiter.

Over 300 centaurs have been seen, and scientists believe there are over 40,000 that are bigger than a kilometer across. But Chariklo is the biggest. And it has two rings!

A while ago I told you about a 'super Saturn' - an object in another solar system with ringsalmost a... more »

2016-10-18 15:31:45 (0 comments; 5 reshares; 38 +1s; )

**Trump's Russian connections**

This page created by the Financial Times is a good overview of Donald Trump's connections to Russia over the last 30 years:

https://ig.ft.com/sites/trumps-russian-connections/

2016-10-16 17:24:21 (30 comments; 8 reshares; 63 +1s; )

**Triamond**

The structure of a diamond crystal is fascinating. But there’s an equally fascinating form of carbon, called the **triamond**, that’s theoretically possible but never yet seen in nature.

In the triamond, each carbon atom is bonded to three others at 120° angles, with one double bond and two single bonds. Its bonds lie in a plane, so we get a plane for each atom.

But here’s the tricky part: for any two neighboring atoms, these planes are different. In fact, if we draw the bond planes for all the atoms in the triamond, they come in four kinds, parallel to the faces of a regular tetrahedron!

If we discount the difference between single and double bonds, the triamond is highly symmetrical. There’s a symmetry carrying any atom and any of its bonds to any other atom and any of its bonds. However, the triamond has an inherent handedness, orchirality... more »

2016-10-15 16:35:08 (27 comments; 37 reshares; 121 +1s; )

**Alien machinery**

That's what it looks like to me. But it's an image created by Greg Egan, the science fiction author. And there's a story behind it.

Egan and I figured out a bunch of stuff about the **McGee graph**, a highly symmetrical graph with 24 vertices and 36 edges. I wrote an article about it on Visual Insight, my blog for beautiful math pictures.

Later I got an email from Ed Pegg, Jr saying he'd worked out a **unit-distance embedding** of the McGee graph: a way of drawing it in the plane so that any two vertices connected by an edge are distance 1 apart. He wanted to know if this was **rigid** or **flexible**. In other words, he wanted to know whether you can change its shape slightly while it remains a unit-distance embedding.

Egan thought about it a lot and did a lot of computations and discovered thatthis un... more »

2016-10-14 01:28:44 (16 comments; 16 reshares; 169 +1s; )

**Super Saturn**

About 400 light years away, there's something with rings like Saturn — but much, much bigger!

It's called **J1407b**. It could be a huge planet. Or it could be a star so small that it never lit up: a **brown dwarf**.

One of Saturn's largest visible rings, the **F ring**, is about 140 thousand kilometers in radius. But J1407b's rings are almost a thousand times bigger. It has rings 90 million kilometers in radius!

That's 2/3 as big as the Earth's orbit around the Sun. That's insane! It's so huge that scientists don't know why the ring doesn’t get ripped apart by the gravity of the star it orbits.

One theory is that the rings are spinning in a **retrograde** way — in other words, backwards. If you have a planet moving clockwise around a star, andits rings are turning ... more »

2016-10-12 14:39:40 (13 comments; 16 reshares; 137 +1s; )

**Gas, Solid, Liquid, Darkness**

Here Canadian photographer David Burdeny captured an iceberg rising straight out of the ocean. It seems to divide the world into four parts.

He took this photo in 2007 in the **Weddell Sea**, one of the two big dents in Antarctica separated by the huge peninsula called **West Antarctica**. Scientists have found that the Weddell Sea has the clearest water of any sea. But it's a dangerous place, according to historian Thomas R. Henry's book White Continent:

The Weddell Sea is, according to the testimony of all who have sailed through its berg-filled waters, the most treacherous and dismal region on earth. The Ross Sea is relatively peaceful, predictable, and safe.

The **Ross Sea** is the other big dent in Antarctica - look at the map here:

https://en.wikipedia.org/wiki/Weddell_Sea

Da... more »

2016-10-11 18:03:22 (9 comments; 26 reshares; 123 +1s; )

**The golden ratio**

When I was in junior high, my uncle the physicist Albert Baez gave me a calculator. This was back in 1971, when electronic calculators were pretty rare. I immediately got turned on to experimental math, at a very basic level.

One thing I did was hit "1 + 1/(1 + 1/(1 + ..." and watch the numbers get closer and closer to the **golden ratio**:

Φ = (1 + sqrt(5))/2 = 1.6180339...

I tried other options, like this:

2 + 1/(2 + 1/(2 + ...

and this:

3 + 5/(3 + 5/(3 + ...

Eventually I figured out a nice formula for all expressions like these. I was very proud of it.**Puzzle 1:** what's the formula?

Only much later did I learn that people know how to find a formula for such infinite fractions, called **continued fractions**, whenever they repeat. Forexamp... more »

2016-10-10 17:09:45 (13 comments; 37 reshares; 137 +1s; )

**McGee graph**

On MathOverflow, someone named მამუკა ჯიბლაძე made this cool animation of the McGee graph, which has 24 dots and 36 edges.

This movie illustrates a **symmetry** of the McGee graph. In other words, if you let the picture make a quarter turn, it looks just the same, even though the dots have moved.

In fact, even if you let the graph make a full turn, the dots have moved from their original position! Why? Because the red edges have flipped upside down. So, you need to let the graph make 2 full turns before everything returns to its original position.

So, this movie illustrates 2 × 4 = 8 symmetries of the McGee graph. But the McGee graph actually has a total of 32 symmetries. These symmetries are precisely the transformations of the "affine line over Z/8". For details, try this:

http://blogs.ams.org/visualinsight... more »

2016-10-08 20:46:35 (23 comments; 36 reshares; 154 +1s; )

**Vortex versus antivortex**

No, I'm not trying to hypnotize you! These animations by Greg Egan show a **vortex** at left and an **antivortex** at right - two patterns that frequently occur in a 2-dimensional magnet if the spins are forced to lie in a plane. Kosterlitz and Thouless just won the Nobel prize for their work on such magnets.

The pictures are changing with time, with each little vector rotating at a constant rate - but that's just to show that there are many different possible vortex configurations, and also many different antivortex configurations.

For a better explanation, read my article:

https://johncarlosbaez.wordpress.com/2016/10/07/kosterlitz-thouless-transition/

I just wanted to show you these cool animations, which Egan added to the comments. Also check out +Simon Willerton's animations and SimonBurt... more »

2016-10-07 20:36:03 (18 comments; 37 reshares; 119 +1s; )

**Nobel Prize in Physics Goes to Another Weird Thing Nobody Understands**

That was the headline this week in Wired. Kosterlitz, Thouless and Haldane won the Nobel for their work on topological phase transitions. It's beautiful, truly fundamental work - but journalists were unable to explain it.

Indeed, most of them could barely pronounce 'topological'. In case you're wondering, the stress is on the third syllable.

So what's a 'topological phase transition'? It's not so complicated. Check out my blog article, which is graced by wonderful illustrations created by Brian Skinner, a physics postdoc at MIT.

And for more details, read his blog:

http://www.ribbonfarm.com/2015/09/24/samuel-becketts-guide-to-particles-and-antiparticles/

He doesn't explain topological phase transitions, but hedescribes... more »

2016-10-07 00:39:12 (31 comments; 14 reshares; 78 +1s; )

**Diamonds are forever?**

This shows the pattern of carbon atoms in a diamond. Each atom is connected to 4 neighbors. Its neighbors are the corners of a regular tetrahedron!

The mathematics of this pattern is beautiful, and I explain it here:

http://blogs.ams.org/visualinsight/2016/10/01/diamond-cubic/

I also explain **hyperdiamonds** in 4 or more dimensions. The hyperdiamond in 8 dimensions is especially awesome: it's called the **E8 lattice**, and it's connected to string theory, the octonions and more.

In 1888, Cecil Rhodes started a company called De Beers to sell the diamonds dug up by slaves in Botswana, Namibia, and South Africa. De Beers got a total monopoly on diamonds. To keep the price up, they wanted a slogan to make diamonds into the jewel of choice for weddings:

After unsuccessfuly trying tocreate a... more »

2016-10-05 16:23:05 (31 comments; 8 reshares; 93 +1s; )

**Systems of systems**

In January of this year, I was contacted by a company called Metron Scientific Solutions. They asked if I’d like to join them in a project to use category theory to design and evaluate complex, adaptive systems of systems.

What’s a **system of systems**?

It’s a system made of many disparate parts, each of which is a complex system in its own right. The biosphere is a system of systems. But so far, people usually use this buzzword for large human-engineered systems where the different components are made by different organizations, perhaps over a long period of time, with changing and/or incompatible standards. This makes it impossible to fine-tune everything in a top-down way and have everything fit together seamlessly.

So, systems of systems are inherently messy. And yet we need them.

Metron was applying for a grantfrom ... more »

2016-10-04 17:53:10 (15 comments; 14 reshares; 62 +1s; )

**Mathematical Enchantments**

If you like math, and you haven't tried my friend's +James Propp's blog, give a try! He's an expert on combinatorics who enjoys explaining things. The article here is about Ramanujan and his mysterious formulas.

Propp also looking for someone who is good at making animated gifs of mathematics! If that person could be you, leave a comment here, or email me.

Here's an example of what he wants:

------------------------

In the Spring I'll be posting a Mathematical Enchantments essay on signed area, and thought I'd ask now for help with supporting animations.

One thing I'd like to include is a GIF loop whose first half animates

ac+ad+bd+bc=(a+b)(c+d)

with a small rectangle becoming big and whose second half animates

ac-ad... more »

2016-10-02 00:21:59 (19 comments; 9 reshares; 94 +1s; )

**The icosahedron that got away**

Iron pyrite can form crystals shaped like icosahedra. They aren't regular solids, with equilateral triangles as faces - that would violate the laws of math! Iron pyrite is a cubical crystal, and you can't make a regular icosahedron using little cubes.

These crystals are called **pseudoicosahedra**. They take advantage of how the golden ratio can be approximated using Fibonacci numbers:

1/1 = 1

3/2 = 1.5

5/3 = 1.6666...

8/5 = 1.6125

and so on, getting closer to

Φ = 1.6180339....

They call iron pyrite **fool's gold** - and it can fool you into thinking its proportions attain the golden ratio.

Recently the curator of the Museum of Evolution, Palaeontology and Mineralogy in Uppsala, Sweden, emailed me and told me that the handsome pseudoicosahedron shown herewas... more »

2016-09-30 19:52:28 (56 comments; 13 reshares; 98 +1s; )

**An infinite corridor of universes**

Einstein's equations for gravity have some amazing solutions. Some describe things we see: the Big Bang and black holes. Others don't - like white holes, wormholes, and the infinite corridor of universes shown here.

As far as we know, all real-world black holes were formed at some moment in time by collapsing matter. But it's easier to find solutions of Einstein's equations that describe an eternal black hole whose shape doesn't change with time.

A rotating eternal black hole is called a **Kerr black hole**, because this solution of Einstein's equation was first found by Roy Kerr in 1963. However, he just found part of the solution - not the whole picture here!

You see, when you solve Einstein's equations, you get a world obeying the rules of general relativity. Butsometimes, ... more »

2016-09-24 16:14:25 (28 comments; 27 reshares; 212 +1s; )

**Solar wind**

This is the **solar wind**, the stream of particles coming from the Sun. It was photographed by **STEREO**. That's the **Solar Terrestrial Relations Observatory**, a pair of satellites we put into orbit around the Sun at the same distance as the Earth, back in 2006. One is ahead of the Earth, one is behind. Together, they can make stereo movies of the Sun!

One interesting thing is that there's no sharp boundary between the 'outer atmosphere' of the Sun, called the **corona**, and the solar wind. It's all just hot gas, after all! STEREO has been studying how this gas leaves the corona and forms the solar wind. This picture is a computer-enhanced movie of that process, taken near the Sun's edge.

What's the solar wind made of? When you take hydrogen and helium and heat them up so much that theelectrons... more »

2016-09-23 19:07:29 (26 comments; 11 reshares; 73 +1s; )

**Life on the Infinite Farm**

This is a great book about infinity - for kids. For example, there's a cow named Gracie with infinitely many legs. She likes new shoes, but she wants to keep wearing all her old shoes. What does she do?

Life on the Infinite Farm is by Richard Evan Schwartz, and it's free here:

https://www.math.brown.edu/~res/farm.pdf

Later it will be published on paper by the American Mathematical Society. I really like turning the pages when I'm reading a book to a child. Is that old-fashioned? What do modern parents think?

Gracie's tale is just a retelling of the first **Hilbert Hotel** story. There's a hotel with infinitely many rooms. Unfortunately they're all full. A guest walks in. What do you do?

You move the guest in room 1 to room 2, the guest in room 2 to room 3, andso on. Now... more »

2016-09-22 06:29:24 (18 comments; 25 reshares; 115 +1s; )

**Poncelet's Porism**

If you can fit a triangle snugly between two circles, you can always slide the triangle around. The triangle may have to change shape, but it stays snug! All 3 corners keep touching the outside circle, and all 3 sides keep touching the inside circle.

That's really cool. But even better, it also works for polygons with more than 3 sides!

This amazing fact is called **Poncelet's Porism**.

A **porism** is like a theorem, but much cooler. **Poncelet** was a French engineer and mathematician who wrote a famous book on 'projective geometry' in 1822.

What's a porism, really?

Well, Euclid is famous for his Elements, but he also wrote a more advanced book called Porisms. Unfortunately that book is lost. I hear that someone checked it out from the library of Alexandria andnever ret... more »

2016-09-21 15:17:59 (40 comments; 12 reshares; 60 +1s; )

**Black hole versus white hole**

Last time I showed you a Schwarzschild black hole... but not the whole hole.

Besides the **horizon**, which is the imaginary surface that light can only go in, that picture had a mysterious "antihorizon", where light can only come out. When you look at this black hole, what you actually see is the antihorizon. The simplest thing is to assume no light is coming out of the antihorizon. Then the black hole will look black.

But I didn't say what was behind the antihorizon!

In a real-world black hole there's no antihorizon, so all this is just for fun. And even in the Schwarzschild black hole, you can never actually cross the antihorizon - unless you can go faster than light. So there's no real need to say what's behind the antihorizon. And we can just decree that no light comesout of it.

2016-09-20 18:21:39 (90 comments; 26 reshares; 75 +1s; )

**Understanding black holes**

This is a diagram of a **Schwarzschild black hole** - a non-rotating, uncharged black hole that has been around forever.

Real-world black holes are different. They aren't eternal - they were formed by collapsing matter. They're also rotating. But the Schwarzschild black hole is simple: you can write down a formula for it. So this is the one to start with, when you're studying black holes.

This is a **Penrose diagram**. It shows time as going up, and just one dimension of space going across. The key to Penrose diagrams is that light moves along diagonal lines. In these diagrams the speed of light is 1. So it moves one inch across for each inch it moves up - that is, forwards in time.

The whole universe outside the black hole is squashed to a diamond. The **singularity** is the wiggly line attop. T... more »

2016-09-18 18:48:33 (58 comments; 19 reshares; 85 +1s; )

**The mystical hexagram theorem**

The picture explains this amazing result, which was discovered by Pascal in 1639, when he was only sixteen.

Take six points on an ellipse, called A,B,C,D,E,F. Connect each point to the next by a line.

The red lines intersect in a point G.

The yellow lines intersect in a point H.

The blue lines intersect in a point K.

And then the cool part:

The points G, H and K lie on a line!

I'm teaching a course on 'algebraic groups' starting on Thursday, so I need to review a bit of the history of projective geometry. This result of Pascal, called the **Hexagrammum Mysticum Theorem**, was the first exciting theorem about projective geometry after the old work of Pappus. So I'll mention it in my course! But I don't really understand why it's true. Do you know a niceexplana... more »

2016-09-17 02:15:36 (92 comments; 22 reshares; 110 +1s; )

**Exploring black holes - with cats!**

There should be a series of videos exploring black holes with cats.

So far all we have is this gif made by +Dragana Biocanin. A cat can orbit just above the photon sphere of a non-rotating black hole, moving at almost the speed of light. It's impossible for a cat to orbit below the photon sphere. As long as it's outside the event horizon it can accelerate upwards and escape the black hole's gravitational pull. But if it crosses the event horizon, it's doomed!

The **event horizon** is an imaginary surface in spacetime that's defined by this property: once a cat crosses this surface, it can't come back without going faster than light! This property involves events in the future, so there's no guaranteed way for the cat to tell when it's crossing an event horizon.

Forexample, if... more »

2016-09-16 01:13:54 (35 comments; 21 reshares; 90 +1s; )

**Light moves around a rotating black hole**

This gif by +Leo Stein shows a photon orbiting a black hole. Since the black hole is rotating, the photon traces out a complicated path. You can play around with the options here:

https://duetosymmetry.com/tool/kerr-circular-photon-orbits/

If a black hole is not rotating, light can only orbit it on circles that lie on a special sphere: the **photon sphere**.

But if the black hole is rotating, photon orbits are more complicated! They always lie on some sphere or other — but now there's a range of spheres of different radii on which photons can move!

The cool part is how a rotating massive object — a black hole, the Sun or even the Earth — warps spacetime in a way that tends to drag objects along with its rotation. This is called **frame-dragging**.

Frame-dragging... more »

2016-09-15 03:41:43 (15 comments; 20 reshares; 165 +1s; )

**Just because someone's on crutches doesn't mean they're handicapped**

Nomads kick ass. James Dator explains:

The World Nomad Games concluded on Friday in what can only be described as the greatest week-long sporting event on the planet. The games, intended to showcase ethnic sports of Central Asia, featured things you have never heard of, athletes you’ll never learn about and sports that sound absolutely terrifying.

There were 16 sports with medals up for grabs. These are the ones that are the absolute wildest.**Cirit**

This Turkish equestrian sport involves teams of riders chasing each other and throwing javelins at each other while on horseback. Yes, seriously.**Er Enish**

It’s wrestling, except you’re on a horse. You win by pulling your opponent off their horse.**Kok-boru**

more »

2016-09-13 00:41:47 (75 comments; 19 reshares; 104 +1s; )

**An even bigger particle accelerator?**

This is Chen-Ning Yang. He helped create **Yang-Mills theory** - the wonderful theory that describes all the forces in nature except gravity. He helped find the **Yang-Baxter equations**, which describe what particles do when they move around on a thin sheet of matter, tracing out braids.

He's one of China's top particle physicists... and he's come out against building a new, bigger particle accelerator! This is a big deal, because only China has the will to pay for the next machine.

In 2012, two months after the Large Hadron Collider (near Geneva) found the Higgs boson, a Chinese institute called for a bigger machine: the **Circular Electron Positron Collider** or **CEPC**.

This machine would be a ring 80 kilometers around. It would collide electrons and positrons at an energy of250 GeV... more »

2016-09-11 06:38:13 (25 comments; 23 reshares; 121 +1s; )

**Just above the photon sphere**

This gif shows what it's like to orbit a non-rotating black hole just above its **photon sphere**.

That's the imaginary sphere where you'd need to move at the speed of light to maintain a circular orbit. At the photon sphere, the horizon of the black hole looks like a perfectly straight line!

But since you can't move at the speed of light, this gif shows you orbiting slightly above the photon sphere, a bit slower than light.

We cannot go to such a place - not yet, anyway. The gravity would rip us to shreds if we tried. But thanks to physics, we can figure out what it would be like to be there! And that is a wonderful thing.

The red stuff drawn on the black hole is just to help you imagine your motion. You would not really see that stuff.

The light above the blackhole is st... more »

2016-09-09 06:43:28 (50 comments; 5 reshares; 50 +1s; )

**The Ultimate Question, and its Answer**

+David Madore has a lot of great stuff on his website - videos of black holes, a discussion of infinities, and more. He has an interesting story that claims to tell you the Ultimate Question, and its Answer.

(No, it's not 42.)

I like it, but I can't tell how much sense it makes.

Here's the key part:**What is the Ultimate Question, and what is its Answer? The answer to that is, of course: “The Ultimate Question is ‘What is the Ultimate Question, and what is its Answer?’ and its answer is what has just been given.”. This is completely obvious: there is no difference between the question “What color was Alexander's white horse?” and the question “What is the answer to the question ‘What color was Alexander's white horse?’?”. Consequently, the Ultimate Questionis “What is the Answer t... more »**

**
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2016-09-08 08:39:04 (36 comments; 14 reshares; 76 +1s; )

**Cosmic censorship**

Einstein's theory of gravity predicts **singularities**: places where you can fall off the edge of spacetime. In reality, these may be places where Einstein's theory breaks down! We don't really know.

Why not just go look? Unfortunately, most singularities are hidden. You can't see them and come back and tell us what you saw.

One singularity that's not hidden, according to Einstein's theory, is the Big Bang.

Look into the sky in any direction and you are really looking back in time, because it takes time for light to travel. If you look far enough - with the right kind of equipment - you can see a faint glow of microwaves left over from the Big Bang. Actually this glow is from hot gas that cooled down enough to become transparent 380,000 years after the Big Bang. It's harder to seebeyond that... more »

2016-09-06 02:19:14 (35 comments; 9 reshares; 80 +1s; )

**Singularities - when spacetime goes bad**

Physicists don't like it when things become infinite, but for Einstein's theory of gravity, infinities are deeply connected to its most dramatic successful predictions: black holes and the Big Bang!

In this theory, the density of the Universe approaches infinity as we go back in time to the Big Bang. The density of a star approaches infinity when it collapses to form a black hole. The curvature of spacetime approaches infinity, too! These situations where spacetime goes bad are called **singularities**.

Einstein didn't like singularities. They could mean that his theory - general relativity - breaks down under extreme conditions. But so far, they fit what we see very well.

In the 1960's, two guys named Roger Penrose and Stephen Hawking became famous by proving **singularitytheorems... more »**

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**2016-09-05 04:37:57 (31 comments; 11 reshares; 66 +1s; )

**The photon sphere**

A nonrotating black hole is surrounded by an imaginary sphere called the **event horizon**. If you cross this sphere, you are doomed to fall in.

If you carry a flashlight and try to shine light straight out, light emitted at the instant you cross the event horizon will basically stay there! Why? Because to stay on the horizon you must move outwards at the speed of light. As the Red Queen said in Alice in Wonderland:

"Now, here, you see, it takes all the running you can do, to keep in the same place."

But there's another imaginary sphere outside the event horizon, called the **photon sphere**. This is where light can go in circles around the black hole!

This picture by +David Madore shows the view from the photon sphere. The black hole occupies exactly half the sky! As he says:

This ... more »

2016-09-04 03:52:21 (4 comments; 6 reshares; 54 +1s; )

**Kummer surface**

This picture by +Abdelaziz Nait Merzouk almost completes my series on surfaces with the maximum number of **nodes** - those places where the tips of two cones meet. This surface has 16 nodes, the most possible for a surface described by a **quartic** equation: a polynomial equation of degree 4.

Unlike other surfaces in this series, it's connected to some pretty deep math. Take a complex curve of genus 2, form its Jacobian variety, mod out by the Kummer involution, and you get a surface like this! Whee! Where's my "mad scientist" emoji? 😃

For an explanation of all that, go here:

http://blogs.ams.org/visualinsight/2016/09/01/kummers-quartic-surface/

As usual, mathematicians seem to have a perverse desire to make things complicated. For example, a "complex curve of genus 2" isbasically j... more »

2016-09-02 00:37:26 (93 comments; 18 reshares; 199 +1s; )

**Black Saturns**

Imagine a black hole with a black ring. Physicists call such a thing a **black Saturn**.

Nobody has ever seen one. But we can still study them.

You see, we know the equation that describes black holes. It's called **Einstein's equation**, the basic formula in Einstein's theory of gravity.

We know this equation has solutions with a round **event horizon** - a surface that you can't escape if you fall through it. These are black holes. And we've seen plenty of black holes - or at least the hot gas falling into black holes.

Could there be a **black ring** - an event horizon shaped like a ring? It would need to spin so it wouldn't collapse.

Nobody has ever seen a black ring... and there's a reason why! They're mathematically impossible. There's nosolution of... more »

2016-09-01 06:06:53 (47 comments; 29 reshares; 85 +1s; )

**Waiting for the motivation fairy**

Go ahead! Procrastinate! If you wait long enough, a fairy princess may suddenly appear and cast a spell motivating you to get work done.

Here's what Hugh Kearns and Maria Gardiner have to say about this approach. It contains good advice:**If you were trying to set up ideal conditions for procrastination, conducting a research project would provide them. Such projects tend to be large and time-consuming: completing a doctoral research project, for example, often takes three years or more. Deadlines and endpoints are often fuzzy and ill-defined. Then there’s the reward structure: you can put in a lot of effort with little to no positive feedback along the way, and the rewards, if there are any, take a long time to come.****Add to this the fact that scientists are often perfectionists with demanding, ifnot id... more »**

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**2016-08-31 05:17:00 (40 comments; 6 reshares; 73 +1s; )

**Cayley's nodal cubic surface****Arthur Cayley**, born in 1821, was one of the first great pure mathematicians in Britain. A guy named Newton had done some stuff connected to physics... and ultimately Newton's line of work led to Stephen Hawking, who occupies the same position at Cambridge, called the Lucasian Professorship. But Cayley's work was more abstract.

He liked math a lot, and went to Cambridge at the age of 17, but when he got a degree and needed a job at the age of 25, he became a lawyer. He worked as a lawyer for 14 years. But then a new position opened up: the Sadleirian Professorship. This professor was supposed to

explain and teach the principles of pure mathematics, and to apply himself to the advancement of that science

and that's what Cayley did!

He helped invent group theory - he was the firstto give ... more »

2016-08-27 04:43:12 (106 comments; 26 reshares; 220 +1s; )

**Dark mysteries**

You probably heard the news this week: astronomers found a galaxy that's 98% dark matter.

It's called **Dragonfly 44**. It's extremely faint, so it doesn't have many stars. But we can use redshifts to see how fast those stars are moving - over 40 kilometers per second on average. If you do some calculations, you can see this galaxy would fly apart unless there's a lot of invisible matter providing enough gravity to hold it together. (Or unless something even weirder is happening.)

Something similar is true for most galaxies, including ours. What makes Dragonfly 44 special is that 98 percent of the matter must be invisible. And this is just in the part where we see stars. If we count the outer edges of the galaxy, the **halo**, the percentage could rise to 99% or more!

By comparison, theMilky Way is... more »

2016-08-26 05:35:36 (30 comments; 23 reshares; 104 +1s; )

**The driverless taxi is here**

Singapore now has the world's first driverless taxi!

Yes, just one so far. Only 10 people are allowed to use it, and it will stay in the most futuristic part of town, near the research centers Biopolis and Fusionopolis. But the company **nuTonomy** hopes to make this service commercially available by 2018, with a fleet of 75 cabs. And it wants to boost the number to thousands by 2019.

Singapore just barely beat Pittsburgh: Uber plans to offer driverless rides there in a few weeks.

Here's a story from May 2016:**During this test drive, there were people; there was construction; there was even a fairly busy intersection.****Being able to understand traffic lights, navigate to a destination and not just detect obstacles but figure out when and how to pass them is no small feat for anaut... more »**

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**2016-08-25 10:03:24 (40 comments; 3 reshares; 47 +1s; )

**Transitions**

I'm ashamed to say I've never been to a "rave" and danced the night away. But if I ever do, this is what I want to hear.

Because I somehow missed this sort of scene - spending my youth more quietly - I'm only now getting into various kinds of electronic dance music that I should have known about a long time ago. I started with Photek's Modus Operandi, a masterpiece of icy cold, sometimes jazzy, vaguely sinister drum-and-bass. Then I picked up Richie Hawtin's Consumed, just because I liked the spooky look of this CD, and discovered he too has the mysterious, "chilly" esthetic I often enjoy... though not the virtuosity Photek can muster.

(I must sound strange. Though I've become cheerful and romantic in the second half of my life, I'm still extremely fussy about music that acts that way. Why? I ... more »

2016-08-24 11:38:23 (31 comments; 21 reshares; 138 +1s; )

**Quantum cryptography in space**

Last week China launched **Micius**, the first of 20 satellites that will use quantum entanglement to create almost unbreakable codes.

This satellite will broadcast pairs of photons to two ground stations. These photons will be **entangled** - correlated in a way that's only possible through quantum mechanics. If you share an entangled pair of photons with a friend, you can use them as a key to decode the messages you send each other. And if someone tries to intercept this key, you can detect it! No third party can access entangled information without affecting it.

This idea has already been tested over long distances - it's not just a crazy dream. What's new is sending entangled photons from satellites orbiting the Earth. China's new system is called **QUESS**: Quantum Experiments at SpaceScale.<... more »

2016-08-23 14:32:26 (52 comments; 12 reshares; 101 +1s; )

**Jump for joy**

Dolphins do this. Why? Maybe just for fun. If you've ever seen the amazing games they play with air bubbles, you'll know what I mean. If you haven't, check this out:

https://plus.google.com/117663015413546257905/posts/W8AAhgY1tCz

It was one of my most popular posts!

But people actually debate this question. Here's what they say at Dolphins-World :**Why do dolphins jump out of the water?****There is an ongoing debate about why dolphins jump out of the water. Scientists think about different reasons for this behavior.****Among them, some think that dolphins jump while traveling to save energy as going through the air consume less energy than going through the water.****Some other think that jumping is to get a better view of distant things, mainly food. So, inthis w... more »**

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**2016-08-21 02:02:52 (45 comments; 12 reshares; 117 +1s; )

**Yu Jianchun - self-taught math whiz**

Henan is one of the poorer provinces of China. But there are beautiful mountains in the county of Xinxian. That's where Yu Jianchun grew up. Until recently he was a package delivery worker. He says he barely knows calculus. But he's been working on number theory. It took him 8 years to get anyone to pay attention to his discoveries. But recently he was invited to give a talk at Zhejiang University!

Yu is modest:**"I'm slow-witted. I need to spend far more time studying math problems than others. Although I am sensitive to numbers, I barely have any knowledge about calculus or geometry."**

But he's made some discoveries about **Carmichael numbers**. I won't define those, but they're **pseudoprimes**: they pass a test for being prime that Fermat invented, butthey'... more »

2016-08-18 07:02:26 (62 comments; 52 reshares; 104 +1s; )

**The Equation Group**

We live in a world of shadowy struggles. A team of hackers called the **Equation Group** has remarkable powers:

• They can reprogram your hard drive firmware. This lets them put software on your machine that will survive even if you reformat your hard drive and reinstall your operating system. They can create an invisible, persistent area in your hard drive, store data there, and collect it later.

• They can retrieve data from networks not connected to the internet. They can use an infected USB stick with a hidden storage area to collect information from a computer. When this USB stick is later plugged into a computer they've subverted that does have an internet connection, they can retrieve this information.

• Since 2001, the Equation Group has infected thousands of computers in over 30 countries, focusing ongovernment ... more »

2016-08-16 05:31:43 (16 comments; 15 reshares; 71 +1s; )

**Points at infinity**

+Abdelaziz Nait Merzouk made this amazing movie. Be patient! It may take a while to load.

This is a surface living in **projective space**. Projective space is like ordinary 3-dimensional space except that it has some extra points called **points at infinity**. In ordinary space, parallel lines never meet. In projective space they do! They meet at one of these points at infinity!

It's not as weird as you think.

Imagine two parallel train tracks. They never meet... but they look like they meet at some point on the horizon. That's the idea of a point at infinity. In projective space, points on the horizon are actual points!

The geometry of projective space is important for understanding perspective, so mathematicians started working on it in the Renaissance and got really good at it bythe 1800s. Th... more »

2016-08-14 02:05:13 (47 comments; 46 reshares; 339 +1s; )

**Not like Earth**

At the end of August, the European Southern Observatory will announce a planet orbiting **Proxima Centauri** - the star closest to our Sun, 4.24 light years away. They're trying to make this planet sound like Earth... and that's cool. But I'll tell you some ways it's not.

Mainly, Proxima Centauri is really different from our Sun!

It's a red dwarf. It puts out just only 0.17% as much energy as our Sun. So any planet with liquid water must be very close to this star.

And because it's cooler than the Sun, Proxima Centauri mainly puts out infrared light - in other words, heat radiation. Its visible luminosity is only 0.005% that of our Sun!

So if you were on a planet as warm as our Earth orbiting Proxima Centauri, it would look very dim - about 3% as bright as our Sun.

Of course,if there... more »

2016-08-13 06:32:47 (27 comments; 31 reshares; 158 +1s; )

**And even in defeat... victory!**

This shows great presence of mind, and a sense of humor.

2016-08-12 05:18:30 (93 comments; 41 reshares; 253 +1s; )

**How big is a proton?**

We thought we knew. New measurements say we were 4% off. That may not seem like much - but it's enough to be a serious problem!

We can measure the proton radius by bouncing electrons off it, or by carefully studying the energy levels of a hydrogen atom. People have measured it many times, and the different measurements agree pretty well. Here's the answer:

0.8775 ± 0.0051 femtometers

A **femtometer** is 10 to the minus 15th meters, or a quadrilionth of a meter.

But you can make a version of hydrogen with a muon replacing the electron. The muon is the electron's big brother. It's almost the same, but 207 times heavier. So, **muonic hydrogen** is about 1/207 times as big across. And that makes the effects of the proton radius easier to detect!

So, in principle, weshould be abl... more »

2016-08-09 12:26:55 (16 comments; 10 reshares; 63 +1s; )

**WWW: the Wood Wide Web**

The world wide web was not the first powerful communication network! Long before came the **wood wide web**, underground in every forest.

This picture was produced by a program called **Mycelium**, which takes a picture and evolves it using the rules by which fungi send out tiny threads... sort of like roots... that absorb nutrients.

A **mycelium** is the name for this network of threads formed by a fungus - or a bunch of fungi. A mycelium can be huge! In his book Mycelium Running, Paul Stamets writes:

"Is this the largest organism in the world? This 2,400-acre [970-hectare] site in eastern Oregon had a contiguous growth of mycelium before logging roads cut through it. Estimated at 1,665 football fields in size and 2,200 years old, this one fungus has killed the forest above it several times over, and in sodoing ... more »

2016-08-08 04:51:46 (0 comments; 92 reshares; 201 +1s; )

**What's wrong with Trump?**

More and more people are wondering. Here's an insightful analysis from a psychiatric social worker. Please reshare. It's not light reading, but we need to understand what we're dealing with here.

--------------------**The Billionaire's Baffling Behavior Explained**

Recently, Mr. Trump's words and actions in various situations have become headline news. Suddenly, many people are alarmed and are questioning temperament, his emotional stability. Dementia? Campaign tactics?

No, it's not an illness that can be treated so he can return to his usual state of health. It's not like when a car's brakes don't work and the mechanic fixes them. It's more like the car came off the assembly line without them.

It's a structural problem.

We're witnessingte... more »

2016-08-05 15:24:03 (30 comments; 12 reshares; 88 +1s; )

**The quest for beauty**

This is a nice interview of Miranda Cheng, an assistant professor at the University of Amsterdam working on the math of string theory. She's on a quest to understand the connection between strings and some mysterious functions in Ramanujan's lost notebook.

It's sort of spooky how much of Ramanujan's work makes more sense with the help of string theory. But perhaps it shouldn't be surprising. String theory hasn't really done anything to predict the results of experiments - it's mainly attractive because of its mathematical beauty. Ramanujan, too, was motivated by the quest for beauty. He got there sooner... but he only saw the tip of the vast iceberg we're exploring now.

What's "mathematical beauty"? Here's what Cheng says:**It’s kind of hard to say why it isbeautiful. I... more »**

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