Archive for the ‘Mathematics’ Category

25 Mar

How to save the world

Posted by Max Goldstein in Best of Dethorning STEM, Education, FIRST, Internet, Tau, TED.

The end of World War I was a bad time to be an optimist. It wasn’t that millions of young men had died or that western Europe had been transfigured into a hellish bombed-out landscape, although that was certainly true. It was the inescapable philosophical consideration that civilization had done this to itself. The “progress” of the industrial revolution and German unification led inexorably to total war. Civilization itself was fundamentally flawed and unsustainable; the only alternative was to admit Rousseau was right and go back to the trees.

Of course, that’s not what happened, and twenty years later they were at it again. The technology changed dramatically, but it didn’t change the fact that people were still killing each other, only how they did it. The changes that mattered were the social institutions built afterwards. Instead of the outrageous reparations in the Treaty of Versailles, there was the conciliatory Marshall Plan. Instead of the League of Nations, there was the United Nations. It wasn’t technological improvements that saved lives and improved the quality of living after the war. It was the people, with their resiliency, their forgiveness, and their intent not to make the same mistake twice.

We now find ourselves, once again, on the brink of destruction. It is not destruction by military means, but rather, economic and environmental means. Natural resources are being depleted faster than they can be renewed, if they can be renewed at all. Industrialization has spread concrete, steel, and chemicals across previously untouched land. The established political institutions are being challenged by forces as diverse as the Arab Spring and the Occupy movement. The economy is still largely in shambles. And then there’s the small matter of climate change. And so on. We’ve heard it all before. At TED 2012, this grim view was presented by Paul Gilding (talk, follow-up blog post). He’s pretty blunt about it: the earth is full.

Around a third of the world lives on less the two dollars a day. They have dramatically different cultures, education, living conditions, access to technology than the typical American or European. You honestly think that they’re the ones that are going to fix the problems? The people who are illiterate, innumerate, and don’t know where their next meal is coming from are going to fix climate change?

Depending on your answer, I have two different responses. I’ll give both of them, but you might want to think about it first. Continue reading »

14 Mar

A new place of activists: math

Posted by Max Goldstein in Mathematics, Tau, Tufts.

This article originally appeared in the Tufts Daily on March 14, 2012.

Remember the unit circle? Of course you don’t. It’s a bunch of numbers lost in the fog of high school geometry. But it’s not your fault. It’s pi’s fault. Pi is wrong, and I want you to help make it right.

I don’t mean that pi is factually wrong; the ratio of a circle’s circumference to its diameter hasn’t changed. I mean that it’s the wrong choice of the circle constant because it leads to weird and unnatural situations. Let me explain.

Mathematicians don’t like to measure circles in degrees. They prefer radians, which are just a way of making every circle look like the unit circle, regardless of size. Because the unit circle has a radius of one, its diameter is two and its circumference is two−pi. Therefore, every circle has a circumference of two−pi radians. Pi radians is only half a circle. That’s all the math you need. I promise.

So, in classic textbook tradition, let’s apply math to a real−world situation where you would never actually need it. Say you’re cutting up your favorite circular fruit−filled pastry and your friend wants a mathematically precise amount. Where do you cut? The problem is that one pie isn’t one−pi — it’s two−pi. If you want an eighth of a pie, it’s a quarter pi, measured along the crust. It’s also really confusing, measured from anywhere.

Continue reading »

5 Sep

Deranged

Posted by Max Goldstein in Epistemology, Internet, Mathematics.

No, I’m not checking into a psych ward. A derangement is the mathematical term for a question, such as this, that appeared in my twitter feed:

How many ways can I rearrange the letters ABCDEFGH so that no letter is in its original position? What a deceptively simple question!

We eventually found the answer (14833), but the more interesting point is how we found the answer. R. Wright had an analytical solution, while Andy Rundquist used brute force. I, however, saw this as a member of a set of problems, namely, how can I rearrange the first n letters of the alphabet so each one gets moved? I then proceeded to try small values of n.

For n=1, you’re stuck with A, so f(1)=0. For n=2, AB, becomes BA, so f(2)=1. For n=3, ABC becomes CAB and BCA, so f(3)=2. But {0,1,2…} isn’t distinct enough. I needed another term.

I started writing an organized list: ABCD, ABDC, ADBC, ADCB… but I got bored so I googled and found someone had already typed up all 24 permutations. So I copied them into a text document and eliminated the bad ones, the same process I was going to do on paper. I was left with nine.

Then I surfed over to oeis.org, the Online Encyclopedia or Integer Sequences, and entered 1,2,9. The first item to come up was sequence 000166. There was, as usual for the OEIS, a lot of math jargon, and I didn’t know the term derangement yet. I could tell the sequence had something to do with factorials and combinatorics and had a recursive definition, but I wasn’t sure, so I checked the next sequence. I knew the correct sequence would have to increase O(n!), never dropping to 0 nor exceeding n! as n got big. The second sequence listed was not bounded above by n!, which meant it said there were more permutations meeting the no repeats criteria than there were permutations total, so that couldn’t be it. The next sequences had the numbers in no particular order, so they were out too. I tweeted that I “suspected” the first sequence. If I had thought to google “subfactorial” or “derangements”, I could have found less technical information telling me this was indeed the case.

We live in the 21st century. There has been a lot discovered already, and in the last 10 years a lot of it has been put online for all to see. We do not need to reinvent the wheel, as long as we know when to ignore the used wheel salesman. (“Ugg not sure about square. That work?”) Notice how my thought process includes sanity checks, broadening or narrowing as appropriate, and using existing resources? I call this the exploration of the known. (Known to humanity, but not to you.)

The truth is out there. You just have to know where and how to look.

1 Aug

Tau and Pi

Posted by Max Goldstein in Mathematics, Tau.

Interlude

You’re at a fancy restaurant and have just finished a sumptuous feast. You don’t think you could eat another bite when the waitress brings out the desert cart, and lo and behold there’s a steaming hot fresh apple pie. The waitress looks at you and asks, “how much pie would you like?”

Let’s say you want an eighth of a pie. You – who we’ll call the pie-eater – of course meant an eighth of the area of the pie. But the waitress, who we’ll call the pie-cutter, can’t cut the pie like a rectangular cake, in straight parallel and perpendicular lines. She has to cut it like a circle, using what we’ll call the pie-cutter’s algorithm:

  1. Pick a starting point on the edge of the circle.
  2. Go some fraction of the total distance around the circle, around the circle.
  3. Cut from that new point to the center.
  4. Cut from the center to the starting point.

The question of course is how far around the edge of the pie (circumference of the circle) does the pie-cutter go? Continue reading »

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