The Golay code

The Golay code is a way to encode 12 bits of data in a 24-bit word in such a way that any error affecting up to 7 bits can be detected.

This animated gif by Greg Egan, based on one by +Gerard Westendorp, shows how the Golay code works.

First, take the 12 bits to be coded and imagine them sitting on the faces of a great dodecahedron. This is a shape with the same vertices as an icosahedron, but with 12 large pentagons as faces. These pentagons intersect each other.

Then we need to describe 12 parity bits: additional bits, computed from the bits we want to code, whose function is to help us detect that errors have occurred, or perhaps even correct them.

We put the 12 parity bits at the 12 vertices of the icosahedron. And here's the rule: the parity bit at any vertex is the sum, modulo 2, of the bits on faces that do not contain that vertex.

So, as we turn on bits at various faces of the great dodecahedron—that is, change them from 0 to 1, while keeping the other face bits equal to 0—bits at various vertices will ‘light up’, as shown in this picture!

The Golay code is a wonderful entry point to a world of beautiful mathematics, all connected to the magic properties of the number 24. So, it's nice to be able to visualize the Golay code.

To dig a bit deeper, read my post on Visual Insight:

http://blogs.ams.org/visualinsight/2015/12/01/golay-code/

Among other things you'll meet a Mathieu group, which is the symmetry group of the Golay code! This Mathieu group, called M24, is a stepping-stone to an amazing group called the Monster.



Welcome to the US - here's your complimentary bullet-proof vest

The latest mass shooting in the US struck a city near where I live: San Bernardino, California. I'm so sick and tired and disgusted by this kind of news. It makes me want to move to another country. This one doesn't even have the will to do anything.

So far, in 209 out of 336 days this year, at least one shooting left 4 or more people injured or dead in the United States. Some days there are more mass shootings - one day there were 6. A total of 462 people have died and 1,314 have been wounded in such attacks this year. Of course, many more people are shot in the US: so far this year, 12,223 died and 24,722 were wounded. But mass shootings have a specially damaging effect to society, beyond the mere body count.

Data from here:

http://mobile.nytimes.com/2015/12/03/us/how-often-do-mass-shootings-occur-on-average-every-day-records-show.html

http://www.gunviolencearchive.org/


The joy of tilings

Ever since I was a kid, I've loved the ways you could tile the plane with regular polygons. Some are used for floor tiles - pondering these is a great way to stay entertained while sitting in public restrooms. But unfortunately, a lot of the fancier ones have not come into wide use.

There are 3 regular tilings: you can use equal-sized regular triangles, squares or hexagons to tile the plane. If you let yourself use several kinds of regular polygons in the same tiling but demand that every vertex look alike, you get 8 more choices: the uniform tilings.

Only recently did I learn about the n-uniform tilings, where you relax a bit and let there be n different kinds of vertices.

The picture here supposedly shows a 4-uniform tiling. But I must be sleepy or something because I'm only seeing 3 kinds of vertices. I see one kind where:

a blue dodecagon, a green hexagon and a red square meet

one where:

a red square, a green hexagon, a red square and a yellow triangle meet

and another where:

a red square, a green hexagon, a red square and a yellow triangle meet.

The last two sound the same! But they're different in this way: no symmetry of the whole tiling can carry the first to the second. You see, one is closer to a blue dodecagon than the other!

I just see these 3 kinds. I'm allowing reflections as symmetries. Otherwise I could get one more kind... but I'm pretty sure reflections are allowed in this game!

Puzzle 1: what's the 4th kind of vertex?

According to the experts, there are 20 2-uniform tilings. There are 61 3-uniform tilings. There are 151 4-uniform tilings. There are 332 5-uniform tilings. There are 673 6-uniform tilings. And I guess the list stops there only because people got tired!

Of course I'd be delighted if I had spotted an error in this list. But I probably just need more coffee! That's how it often works in math.

This picture was drawn by +Tom Ruen. You can find it, along with lots more, here:

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

I think some of these should be deployed as bathroom tiles in public restrooms. We supposedly have this great, high-tech civilization, yet we're not taking full advantage of math in the decorative arts!

Puzzle 2: what uniform tiling is this 4-uniform tiling based on, and how?

#geometry




+rasha kamel of the +Azimuth Project pointed us to a report in Science Daily which says:

Planet Earth experienced a global climate shift in the late 1980s on an unprecedented scale, fueled by anthropogenic warming and a volcanic eruption, according to new research. Scientists say that a major step change, or ‘regime shift,’ in Earth’s biophysical systems, from the upper atmosphere to the depths of the ocean and from the Arctic to Antarctica, was centered around 1987, and was sparked by the El Chichón volcanic eruption in Mexico five years earlier.

That got me curious, so I read the actual paper this rather sensationalized report is based on. For more, visit my blog!

Flow

I've been staying at home for the last two days writing a paper about information and entropy in biological systems.  My wife is away, and I'm trying to keep distractions to a bare minimum, trying to get into that state where I'm completely absorbed, there's always something to do, and it's lots of fun.  That's what I love about writing.  At first I feel stuck, frustrated.  But gradually the ideas start falling into place - and once they do, I don't want to be anywhere else!  

This state is called flow, and it's great.  But life can't be all flow, it seems.

I like this chart.  I like any chart that takes psychology and maps it down to a few axes in a reasonably plausible way.  I don't have to 'believe in it' to enjoy a neat picture that pretends to tame the wild mess of the soul. 

Apparently this chart goes back to Mihaly Csikszentmihaly's theory of 'flow'.  According to Wikipedia:

In his seminal work, Flow: The Psychology of Optimal Experience, Csíkszentmihályi outlines his theory that people are happiest when they are in a state of flow— a state of concentration or complete absorption with the activity at hand and the situation. It is a state in which people are so involved in an activity that nothing else seems to matter.  The idea of flow is identical to the feeling of being in the zone or in the groove. The flow state is an optimal state of intrinsic motivation, where the person is fully immersed in what he is doing. This is a feeling everyone has at times, characterized by a feeling of great absorption, engagement, fulfillment, and skill—and during which temporal concerns (time, food, ego-self, etc.) are typically ignored.

In an interview with Wired magazine, Csíkszentmihályi described flow as "being completely involved in an activity for its own sake. The ego falls away. Time flies. Every action, movement, and thought follows inevitably from the previous one, like playing jazz. Your whole being is involved, and you're using your skills to the utmost."

Csikszentmihályi characterized nine component states of achieving flow including “challenge-skill balance, merging of action and awareness, clarity of goals, immediate and unambiguous feedback, concentration on the task at hand, paradox of control, transformation of time, loss of self-consciousness, and autotelic experience.”

What does autotelic mean?  It seems to mean 'internally driven', as opposed to seeking external rewards.  Csíkszentmihályi says "An autotelic person needs few material possessions and little entertainment, comfort, power, or fame because so much of what he or she does is already rewarding."  Anyway, back to the Wikipedia article:

To achieve a flow state, a balance must be struck between the challenge of the task and the skill of the performer. If the task is too easy or too difficult, flow cannot occur. Both skill level and challenge level must be matched and high; if skill and challenge are low and matched, then apathy results.

But in this chart, 'apathy' is just one of 8 options, the one diametrically opposite to 'flow'.  I like the idea of how 'relaxation' is somewhere between flow and boredom, but I'm not sure it feels next to 'control'. 

It's all very thought-provoking.  We have these different modes, or moods, and we bounce between them without very much thought about what they're for and what's the overall structure of the space of these moods.

Moods seem like the opposite of mathematics and logic, but there's probably a science of moods which we haven't fully understood yet - in part because when we're in a mood, it dominates us and prevents us from thinking about it analytically.

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

Bullshit

If you think these quotes are profound, you may be stupid.

That's the conclusion of a new paper, "On the reception and detection of pseudo-profound bullshit".

The authors are actually a bit more precise:

Those more receptive to bullshit are less reflective, lower in cognitive ability (i.e., verbal and fluid intelligence, numeracy), are more prone to ontological confusions and conspiratorial ideation, are more likely to hold religious and paranormal beliefs, and are more likely to endorse complementary and alternative medicine.

They took a sample of 200 people and measured their "pseudo-profound bullshit sensitivity" by comparing their profundity ratings for pseudo-profound bullshit and legitimately meaningful motivational quotations. Then they correlated this "pseudo-profound bullshit sensitivity" with other traits.

The results are not very surprising to me. It would be great if we could figure out how to make people less susceptible to pseudo-profound bullshit. This might be a first step.

The other big question is: where did they get their pseudo-profound bullshit? What's a reliable source of this valuable commodity?

The first two quotes here are from Deepak Chopra's Twitter feed. The rest are from a website that randomly generates pseudo-profound new age quotes - it's here:

http://sebpearce.com/bullshit/

You can get the paper for free here:

Gordon Pennycook, James Allan Cheyne, Nathaniel Barr, Derek J. Koehler, Jonathan A. Fugelsang, On the reception and detection of pseudo-profound bullshit, Judgment and Decision Making 10 (2015), 549-563. Available at
http://journal.sjdm.org/15/15923a/jdm15923a.html.






The Ultimate Question

This Saturday, I'm giving a public lecture in Montreal.

In The Hitchhiker's Guide to the Galaxy, the number 42 was revealed to be the Answer to the Ultimate Question of Life, the Universe, and Everything. But we never learned what the question was!

That's what I will explain.

My talk is part of the Canadian Mathematical Society winter meeting. But it's open to the public, and I'll keep the math simple and fun. So, come along and bring your kids!

It's from 6 to 7 pm on December 5th at the Hyatt Regency Montreal, at 1255 Jeanne-Mance. It's in "Rooms Soprano A & B", but I won't be singing - they must have confused me with my cousin Joan.

If you can't come to the talk, you can still see the slides here:

http://math.ucr.edu/home/baez/42/

Earlier that day I'll be at Prakash Panangaden's session on Logic, Category Theory and Computation, and I'll give a talk there at 10 am.


Superforecasting

+Stewart Brand describes a talk by Philip Tetlock, author of Superforecasting: The Art and Science of Prediction:

Will Syria’s President Assad still be in power at the end of next year? Will Russia and China hold joint naval exercises in the Mediterranean in the next six months? Will the Oil Volatility Index fall below 25 in 2016? Will the Arctic sea ice mass be lower next summer than it was last summer?

Five hundred such questions of geopolitical import were posed in tournament mode to thousands of amateur forecasters by IARPA—the Intelligence Advanced Research Projects Activity--between 2011 and 2015. (Tetlock mentioned that senior US intelligence officials opposed the project, but younger-generation staff were able to push it through.) Extremely careful score was kept, and before long the most adept amateur “superforecasters” were doing 30 percent better than professional intelligence officers with access to classified information. They were also better than prediction markets and drastically better than famous pundits and politicians, who Tetlock described as engaging in deliberately vague “ideological kabuki dance."

What made the amateurs so powerful was Tetlock’s insistence that they score geopolitical predictions the way meteorologists score weather predictions and then learn how to improve their scores accordingly. Meteorologists predict in percentages—“there is a 70 percent chance of rain on Thursday.” It takes time and statistics to find out how good a particular meteorologist is. If 7 out of 10 such times it in fact rained, the meteorologist gets a high score for calibration (the right percentage) and for resolution (it mostly did rain). Superforecasters, remarkably, assigned probability estimates of 72-76 percent to things that happened and 24-28 percent to things that didn’t.

How did they do that? They learned, Tetlock said, to avoid falling for the “gambler’s fallacy”—detecting nonexistent patterns. They learned objectivity—the aggressive open-mindedness it takes to set aside personal theories of public events. They learned to not overcompensate for previous mistakes—the way American intelligence professionals overcompensated for the false negative of 9/11 with the false positive of mass weapons in Saddam’s Iraq. They learned to analyze from the outside in—Assad is a dictator; most dictators stay in office a very long time; consider any current news out of Syria in that light. And they learned to balance between over-adjustment to new evidence (“This changes everything”) and under-adjustment (“This is just a blip”), and between over-confidence ("100 percent!”) and over-timidity (“Um, 50 percent”). “You only win a forecasting tournament,” Tetlock said, “by being decisive—justifiably decisive."

Much of the best forecasting came from teams that learned to collaborate adroitly. Diversity on the teams helped. One important trick was to give extra weight to the best individual forecasters. Another was to “extremize” to compensate for the conservatism of aggregate forecasts—if everyone says the chances are around 66 percent, then the real chances are probably higher.

In the Q & A following his talk Tetlock was asked if the US intelligence community would incorporate the lessons of its forecasting tournament. He said he is cautiously optimistic. Pressed for a number, he declared, “Ten years from now I would offer the probability of .7 that there will be ten times more numerical probability estimates in national intelligence estimates than there were in 2005.”

Asked about long-term forecasting, he replied, “Here’s my long-term prediction for Long Now. When the Long Now audience of 2515 looks back on the audience of 2015, their level of contempt for how we go about judging political debate will be roughly comparable to the level of contempt we have for the 1692 Salem witch trials."

You can hear Tetlock's whole talk at the link, at least for a while.



What does ISIS want?

Why did ISIS make these seemingly absurd attacks on Paris and a Russan plane? How can it possibly help them to have both Europe and Russia motivated to annihilate them?

To find out, we can start by reading what they say. They could be lying, but still....

They say they want to destroy the "gray zone" of tolerance where Muslims and "infidels" can peacefully coexist. They want to bring about a war against Muslims worldwide, to force Muslims into their arms. And they want conditions in predominantly Muslim nations to deteriorate, to create lawless zones that they can occupy.

Shortly after the Charlie Hebdo attacks in Paris, ISIS put out this statement:

“The world today is divided into two camps. Bush spoke the truth when he said, ‘either you are with us or you are with the terrorists.’ Meaning either you are with the crusade or you are with Islam.

In their English-language magazine Dabiq, they went further and predicted the "extinction" of the "gray zone":

The Muslims in the West will quickly find themselves between one of two choices, they either apostatize and adopt the kufrī [infidel] religion propagated by Bush, Obama, Blair, Cameron, Sarkozy, and Hollande in the name of Islam so as to live amongst the kuffār [infidels] without hardship, or they perform hijrah [emigrate] to the Islamic State and thereby escape persecution from the crusader governments and citizens... Muslims in the crusader countries will find themselves driven to abandon their homes for a place to live in the Khilāfah [caliphate], as the crusaders increase persecution against Muslims living in Western lands so as to force them into a tolerable sect of apostasy in the name of 'Islam' before forcing them into blatant Christianity and democracy.

In 2004, an Al Qaeda strategist named Abu Bakr Naji put out a book called Management of Savagery. I haven't read this book - it may not be translated into English - but here's what Wikipedia says about it:

Management of Savagery discusses the need to create and manage nationalist and religious resentment and violence in order to create long-term propaganda opportunities for jihadist groups. Notably, Naji discusses the value of provoking military responses from superpowers in order to recruit and train guerilla fighters and to create martyrs. Naji suggests that a long-lasting strategy of attrition will reveal fundamental weaknesses in the ability of superpowers to defeat committed jihadists.

Management of Savagery argues that carrying out a campaign of constant violent attacks in Muslim states will eventually exhaust their ability and will to enforce their authority, and that as the writ of the state withers away, chaos—or "savagery"—will ensue. Jihadists can take advantage of this savagery to win popular support, or at least acquiescence, by implementing security, providing social services, and imposing Sharia. As these territories increase, they can become the nucleus of a new caliphate. Naji nominated Jordan, Saudi Arabia, Yemen, North Africa, Nigeria and Pakistan as potential targets, due to their geography, weak military presence in remote areas, existing jihadist presence, and easy accessibility of weapons.

Naji professes to have been inspired by Ibn Taymiyya, the influential 14th century Islamic scholar and theologian.

A number of media outlets have compared the attempts by the Islamic State of Iraq and the Levant to establish territorial control in Iraq and Syria with the strategy outlined in Management of Savagery. The premier issue of the Islamic State's online magazine, Dabiq, contained discussion of guerrilla warfare and tactics that closely resembled the writings and terminology used in Management of Savagery, although the book was not mentioned directly. Journalist Hassan Hassan, writing in The Guardian, reported an ISIL-affiliated cleric as saying that Management of Savagery is widely read among the group’s commanders and some of its rank-and-file fighters. It was also mentioned by another member of ISIL in a list of books and ideologues that influence the group.

You can read issues of Dabiq at the Clarion Project website:

http://www.clarionproject.org/news/islamic-state-isis-isil-propaganda-magazine-dabiq

It pays to know your enemy.

The Wikipedia article on Management of Savagery:

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

I found the quotes from Dabiq here:

https://www.opendemocracy.net/nafeez-ahmed/isis-wants-destroy-greyzone-how-we-defend

Elliptic functions

You probably know about the sine and cosine.  These are the most basic functions that are periodic:

sin(x + 2π) = sin(x)

Elliptic functions are functions of two variables, x and y, that are periodic in two directions:

f(x + 2π,y) = f(x,y)

and

f(x,y + 2π) = f(x,y)

This movie is a way of illustrating an elliptic function.

What makes elliptic functions so special is that you can think of them as functions of a single complex variable:

z = x + iy

and then they have a derivative in the special sense you learn about in a course on complex functions!

It's a lot harder for a complex function to have a derivative than an ordinary real function.  A function like

f(x,y) = sin(x) sin(y)

is periodic in two directions, but it doesn't have a derivative df/dz.  Mysterious as this may sound, this is the reason elliptic functions are so special.

In the late 1800s, all the best mathematicians thought about elliptic functions, so there are 'Jacobi elliptic functions' and 'Weierstrass elliptic functions' and many more.  Now they're less popular, but they're still incredibly important.  You need to think about them if you want to deeply understand how long the perimeter of an ellipse is.  They're also important in physics, and fundamental to the proof of Fermat's Last Theorem.

+Gerard Westendorp has been making mathematical illustrations for a long time, so if you like such things, circle him!

An elliptic function actually has a derivative everywhere except at certain points where the function 'blows up' - that is, becomes infinite.  These points are called poles.   You can prove an elliptic function has to have poles unless it is constant (and thus too boring to talk about). 

Because an elliptic function is periodic in two directions, its poles make a repeating pattern in the plane, which you can see in this movie.  The poles are the points from which checkerboard pattern keeps expanding outward.  The zeros of the elliptic function - the points where it's zero - are the points where the checkerboard keeps shrinking inward.

For more on elliptic functions, you could try this:

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

It's a bit scary: for example, instead of saying a function has a derivative except at points where it blows up, they say it's meromorphic.    It means the same thing, but it's shorter and it makes you sound more educated.