CircleCount helps you better understand Google+

2013-05-25 22:25:21 (12 comments, 2 reshares, 10 +1s)

Puzzle: How many lines through the origin can you find that are all at equal angles to each other?

You can certainly find at least 4, as shown here.  Can you do better?  (If you give up and look up the answer,  please don't post it here.)

Of course my question is about lines in 3-dimensional space.  But you could also ask the question in other dimensions.  In the plane you can find 3 lines that are all at equal angles from each other.  And in 7 dimensions, you can find 28!

That's actually not hard to see.  Just take this vector in 8 dimensional space:

(3, 3, -1,- 1, -1, -1, -1, -1)

and permute its coordinates in all possible ways.  You get

8 × 7 / 2 = 28

different vectors, all with the same length.  The dot product of two different vectors in this collection is 4 if they both have a 3 in the same place, and-4 otherw... more »

2013-05-24 22:30:52 (32 comments, 46 reshares, 105 +1s)

In The Hitchhiker's Guide to the Galaxy by Douglas Adams, the number 42 is the "Answer to the Ultimate Question of Life, the Universe, and Everything".  But he didn't say what the question was!  Let me reveal that now.

If you try to get several regular polygons to meet snugly at a point in the plane, what's the most sides any of the polygons can have?  42. 

The picture shows an equilateral triangle, a regular heptagon and a regular 42-gon meeting snugly at a point.  The reason this works is that

(1/2 - 1/3) + (1/2 - 1/7) + (1/2 - 1/42) = 1

There are 17 solutions of

(1/2 - 1/p) + (1/2 - 1/q) + (1/2 - 1/r) = 1

with p ≤ q ≤ r, but this one features the biggest number of all!

But why is this so important?  Well, you can take

(1/2 - 1/3) + (1/2 - 1/7) + (1/2 - 1/42) = 1
and rew... more »

2013-05-24 16:04:22 (8 comments, 1 reshares, 31 +1s)

The Chen–Gackstatter surfaces are a family of minimal surfaces that generalize the Enneper surfaces by adding handles!  Earlier in this #minimalsurface thread I showed you a whole series of Enneper surfaces.  For each one, we can add any number of handles and get a Chen–Gackstatter surface.  In the pictures here, we are taking the first Enneper surface, the one I showed you the most pictures of, and adding handles.

Like the Enneper surfaces, the Chen–Gackstatter surfaces intersect themselves... but in these pictures created by Anders Sandberg, they've been cut off before they get a chance to do that.

These surfaces aren't arbitrary things: for example, the first surface here is the only genus one orientable complete minimal surface of total curvature -8π.  I can explain those buzzwords if you like... or click on the links here:
http://e... more »

2013-05-24 01:09:00 (19 comments, 8 reshares, 42 +1s)

2013-05-23 18:37:00 (36 comments, 49 reshares, 108 +1s)

At last - a textbook on category theory for scientists!  And it's free!

• David Spivak, Category Theory for Scientists, http://arxiv.org/abs/1302.6946.

"This course is an attempt to extol the virtues of a new branch of mathematics, called category theory, which was invented for powerful communication of ideas between different fields and subfields within mathematics. By powerful communication of ideas I actually mean something precise. Different branches of mathematics can be formalized into categories. These categories can then be connected together by functors. And the sense in which these functors provide powerful communication of ideas is that facts and theorems proven in one category can be transferred through a connecting functor to yield proofs of an analogous theorem in another category. A functor is like a conductor of mathematical truth."
... more »

2013-05-22 19:19:13 (67 comments, 16 reshares, 100 +1s)

I'm helping the American Mathematical Society start a new "mathematical images" feature on their website.  I need your help. 

First: what should we call it?  "Mathematical Images" would be good except that they already have a "Mathematical Imagery" page:

http://www.ams.org/mathimagery/

That page is about art based on math, while mine will be about explaining math using pictures and movies.  

Second: if you have a good image that helps explain some serious math, please let me know.  It's best if you are the creator of the image, because the creator will need to fill out a permission form to let the AMS use the image.

What's this image here?  It's by Greg Egan.  You can take a 3-holed torus and tile its surface with 56 triangles, where 7 triangles meet at each vertex... and this is amovie o... more »

2013-05-21 21:06:58 (8 comments, 2 reshares, 30 +1s)

I think this finally puts to rest alternative explanations of the aftermath of the disputed 2009 election: it was the final phase of a rolling coup by the clerical government, which is now exercising near-total control over the parallel and now-powerless civilian government. 

This is the last phase of a trend which began in the 1980s. If we put Montazeri, Rafsanjani, and Khatami on a continuum, the trend is relatively clear:

By 1989, Montazeri -- who was a genuine moderate, and Khomeini original successor -- was outside the Guardian Council's acceptable range of political competition. He was singled out for public ridicule and lived out the rest of his life under house arrest.

By 2005, it was clear that neither Khatami nor any of his political affiliates would be permitted to stand for office. By the standards of the late 1980s, Khatami wasn't particularly moderate... more »

2013-05-21 23:27:16 (4 comments, 1 reshares, 6 +1s)

It can be pretty surprising when you calculate something real valued and you get almost but not quite an integer. One of the best examples is Ramanujan's constant, exp(pi*sqrt(163)) [1].

I came across another one today.

It's well know that the harmonic series 1+1/2+1/3+1/4+1/5+... doesn't converge [2].

You can show that the alternating series 1-1/2+1/3-1/4+1/5+... does converge.

What happens if we pick the sign before each term at random with a 0.5 chance of picking plus or minus?

Then the series converges with probability 1 and there is a probability distribution function (PDF) for the sum. At values 2 and -2 the PDF takes a value that is almost but not quite 1/8: 0.124999999999999999999999999999999999999999764…

This is closely related to the Borwein integrals [3] which give another nice example of something almost but notqu... more »

2013-05-21 20:44:55 (23 comments, 0 reshares, 13 +1s)

Puzzle: what's the significance of the number

22.92067661926415034816....  ?

(It's easy to look this up online if you get stuck.)

2013-05-21 15:18:36 (10 comments, 0 reshares, 27 +1s)

The Taklamakan Desert, surrounded by mountains to the north and south, is really quite amazingly deep.   You can see it nicely in this image made by the CALIPSO satellite.  There's almost a 4-kilometer drop when you descend from the Kunlun Shan mountains into this desert... and quite a rise to the north, too, when you hit the Tian Shan mountains.

CALIPSO stands for Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations.  It's a French/American project to study clouds, dust and the like by shining a laser beam down to the Earth  and measuring the amount that gets scattered back. 

This technique is called lidar.  The word 'lidar' was originally created by combining the words 'light' and 'radar'.  However, everyone tends to assume this word is an acronym.  So, people often spell it LIDAR - and then they had to come up withsome wor... more »

2013-05-21 04:45:05 (31 comments, 9 reshares, 35 +1s)

As time goes on we're getting more data on global warming.  The Earth has been warming more slowly than most of the models used in the 2007 Intergovernment Panel on Climate Change predicted.  But how much more slowly, and why?  The details matter, and they're tricky. 

Suppose the concentration of CO2 in the air goes up linearly until it doubles in 70 years, and then it stops changing.   The transient climate response (TCR) is how much the Earth's average temperature will increase by the time the CO2 doubles.  But the temperature will continue to rise after that!  The amount it goes up in the long term is the equilibrium climate sensitivity (ECS). 

A new paper estimates using data from the last decade that the ECS has a 90% chance of being 1.2–3.9 °C, with a best estimate of about 2 °C.   This is shown in dark red at left. 

This wouldbe good news ... more »

2013-05-20 14:01:34 (90 comments, 595 reshares, 943 +1s)

On April 17, a paper arrived in the inbox of the Annals of Mathematics. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.  And it had!  Rumors swept through the mathematics community that a great advance had been made by a researcher no one seemed to know — someone whose talents had been so overlooked after he earned his doctorate in 1992 that he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.

For the whole story read this:

https://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/

Thanks to +Ian Agol forpointing ... more »

2013-05-20 05:23:53 (10 comments, 16 reshares, 95 +1s)

I want one of these.  Yes, it's just a computer-generated image made by Tom Beddard.  But I want a real one! 

You can see more of these, and bigger, here:

http://butdoesitfloat.com/And-it-occurred-to-me-that-these-must-be-holographic-viral-projections

The title of this series is from Terence McKenna, the flipped-out ethnobotanist:

And it occurred to me that these must be holographic viral projections from an autonomous continuum that was somehow intersecting my own.

Yow!  You can see more of Tom Beddard's work here:

http://www.subblue.com/

both pictures and videos.  It's worth a look! 

Thanks to +Cliff Harvey for pointing this out.

#fractals  

2013-05-19 22:56:49 (8 comments, 16 reshares, 58 +1s)

Another movie of the Enneper surface - this one made by +Ron Avitzur.  The two holes are very clear in this closeup, and so is the symmetry of this elegant shape. 

In case you haven't been been following my #minimalsurface thread, this is one of the simplest minimal surfaces: you can write down a pretty simple equation for it, which I explained earlier.  Soon I want to talk about some more complicated ones!

2013-05-18 15:35:48 (7 comments, 8 reshares, 72 +1s)

When you get bored, sometimes it's nice to take a little walk.

This is yet another photo of Eyjafjallajökull, taken by the daredevil photographer Skarphedinn Thrainsson... or Skarphéðinn Þráinsson if you can handle those runic letters "edh" (ð) and "thorn" (Þ).  You can see more of his amazing photos here:

http://photos.skarpi.is/

#volcanoes  

2013-05-18 02:46:29 (0 comments, 0 reshares, 1 +1s)

Another amazing shot of the Icelandic volcano Eyjafjallajökull, taken in March 2010 by Bjarni T.  I would love to be one of those hikers!

For more pictures of this volcano, try:

http://www.fotopedia.com/wiki/2010_eruptions_of_Eyjafjallaj%C3%B6kull/#!/items/flickr-4480596041

#volcano  

2013-05-18 02:46:10 (0 comments, 1 reshares, 3 +1s)

THE POWER OF NATURE.  Remember that volcano in Iceland that erupted in the spring of 2010 and shut down air travel in parts of Europe?  It's called Eyjafjallajökull, and this photo of it was taken by Marco Fulle.

As ash rises through the air from a volcano, static electricity builds up and causes lightning!  But nobody is sure how it works.  "What is mostly agreed upon," says geologist Brentwood Higman, "is that the process starts when particles separate, either after a collision or when a larger particle breaks in two. Then some difference in the aerodynamics of these particles causes the positively charged particles to be systematically separated from the negatively charged particles."

#volcano

2013-05-17 13:59:29 (32 comments, 0 reshares, 29 +1s)

A brief housekeeping note:

Without a fix or workaround for the Read More issue, G+ no longer does what I need it to. Accordingly, I am on hiatus until I either get it working or find another platform which works for what I'm doing. Comment threads in particular are impossible to read -- reading my last post and its comments took 31 individual clicks.

Don't take this as a personal dig at the G+ staff; it's either a problem which will be fixed or a strategic decision which excludes my use case entirely. Neither is likely catastrophic or idiotic.

EDIT: I've found that Replies and More is a reasonable workaround for the text-hostile changes to the interface. Once it autoexpands comments, I suspect it'll work great. Unfortunately, however, requiring a workaround to make G+ work well for text is probably going to make some people drop off my stream... more »

2013-05-16 05:31:31 (99 comments, 4 reshares, 105 +1s)

I hate the new G+ layout.  I'm going to quit posting here until I get so lonely and bored I come crawling back with my tail between my legs...

2013-05-18 03:06:26 (10 comments, 25 reshares, 89 +1s)

Another amazing shot of the Icelandic volcano Eyjafjallajökull, taken in March 2010 by Bjarni T.  I would love to be one of those hikers!

For more pictures of this volcano, try:

http://www.fotopedia.com/wiki/2010_eruptions_of_Eyjafjallaj%C3%B6kull/#!/items/flickr-4480596041

#volcanoes  

2013-05-18 03:06:06 (7 comments, 24 reshares, 78 +1s)

THE POWER OF NATURE.  Remember that volcano in Iceland that erupted in the spring of 2010 and shut down air travel in parts of Europe?  It's called Eyjafjallajökull, and this photo of it was taken by Marco Fulle.

As ash rises through the air from a volcano, static electricity builds up and causes lightning!  But nobody is sure how it works.  "What is mostly agreed upon," says geologist Brentwood Higman, "is that the process starts when particles separate, either after a collision or when a larger particle breaks in two. Then some difference in the aerodynamics of these particles causes the positively charged particles to be systematically separated from the negatively charged particles."

#volcanoes

2013-05-14 14:28:13 (16 comments, 18 reshares, 54 +1s)

Check out these crazy ways planets could move!   This one looks like a gravitational atom!  You can run the animations on your browser:

http://www.maths.manchester.ac.uk/~jm/Choreographies/

These orbits are unlikely to ever occur, and probably most of them are unstable.   So finding them is mainly a challenge in pure math... but it's a game that uses some pretty interesting ideas, not just computer simulations.  These animations were created with the help of a full classification of highly symmetrical solutions of the gravitational n-body problem that lie on a plane:

• James Montaldi and Katrina Steckles, Classification of symmetry groups for planar n-body choreographies, http://arxiv.org/abs/1305.0470.

Abstract:  Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of then-body... more »