Stories tagged math

All together now

by Anonymous on May. 15th, 2013

What happens if you start 32 metronomes at different times on a stable surface? Not much. They'll tick-tock out of sync until the cows come home. But what happens when you start the same 32 metronomes on an unfixed surface? You get to witness a nifty (and mesmerizing) example of coupled oscillations. Watch and learn.

Pick a number: The PIN you punch in at an ATM can have a higher or lower level of security based on the order of the numbers you pick.
Pick a number: The PIN you punch in at an ATM can have a higher or lower level of security based on the order of the numbers you pick.Courtesy Victor van Werkhooven
What are the most popular PIN numbers people use? What are the least? How can you best ensure that your PIN is something crooks would have a lesser chance of figuring out? This is a pretty neat report that talks about what works, and what doesn't, when selecting a personal identification number to use with some financial or electronic devices. You'd be surprised how often people commonly make mistakes that lead to easy uncovering of their PINs.

Jan
07
2011

It's a big drawer: But I'm a grown man. This could be close...
It's a big drawer: But I'm a grown man. This could be close...Courtesy JGordon
The human body is more or less the density of water: 1000 kg per cubic meter. (I tend to sink, so I'm probably a little denser, but I think we're close enough here.)

I weigh about 150 pounds, or 68 kg (2.2 pounds per kilogram).
1000kg / 1m^3 = 68kg / ?m^3
? = .068
So the volume of my body is about 0.068 cubic meters.

Each cubic meter equals about 35.3146667 cubic feet.
.068 x 35.315 = 2.401
So the volume of my body is about 2.401 cubic feet

My file drawer is 1 foot wide by 9 inches high (or .75 feet) by 2 feet and 2 inches deep (or 2.167 feet)
1 x .75 x 2.167 = 1.62525
So the volume of my desk drawer is 1.62525 cubic feet

1.62525 < 2.401
So the answer is NO. If you were to blend me up in a giant food processor, and then pour me into my file drawer, I would literally be overflowing.

That's sort of a relief.

Then again, it depends on how much denser I am than water. Maybe I would fit.

101010: 101010 = 42.
101010: 101010 = 42.Courtesy jimmowatt
"101010 (base two (binary)) equals 42 (base ten). Oddly enough, this is evenly divisible by the number of days in a week (7 (lucky)); and equally oddly, is also evenly divisible by the number 6 (which is generally designated as being unlucky). Both a Ying and Yang situation seem to be incorporated into this date." HubPages.com

10 (base ten) = 1010 (base two)

(base ten): 10 x 10 = 100

(base two): 10 x 10 = 100

42 is the answer to everything

In Hitchhiker's Guide to the Galaxy, the
"Answer to Life, the Universe, and Everything" was 42.
Fortytwoday.com
.

May
07
2010

One man, one vote.: But how those votes are counted can lead to some surprisingly complex mathematics.
One man, one vote.: But how those votes are counted can lead to some surprisingly complex mathematics.Courtesy Theresa Thompson

Winston Churchill once quipped, "democracy is the worst form of government, except for all the others." Though said tongue-in-cheek, a recent article in New Scientist shows that, mathematically at least, Winnie was on to something.

Every election has winners and losers. Different countries have different systems for determining the winners, and dealing with the losers. And, it turns out, each of those systems has mathematical quirks which prevent the results from perfectly matching the will of the people.

  • The "winner-take-all" system used in America is certainly very simple and straight-forward. The problem is, the thousands--or even millions--of people who voted for the losing candidate end up with no elected official representing their views. (In the recent British elections, the Liberal Democratic party won 23% of all individual votes cast, but ended up with less than 9% of the seats in Parliament.) And in a race with three or more candidates, you can get a winner who carries less than 50% of the vote.
  • Some countries get around this by using "proportional representation:" they count votes cast for each political party, rather than for individual candidates, and divvy up the legislature that way. The voters' voice is fairly represented. But if one party controls more than half the seats, it can effectively shut the minor parties out. And if no party has a majority, they end up sharing power in ways that do not reflect their numbers. (Again, the British elections are a good example. The leading Conservative Party won 37% of the vote and 47% of the seats--not quite enough for a majority. They may form an alliance with the Liberal Democrats. The two parties would share power 50%-50%--quite a boon for the LibDems, who control only 9% of the seats!)
  • A few countries have tried "ordered voting," in which voters rank all candidates in order of preference, and then conducting run-offs until someone gets 50% of the vote. But this can lead to a strange situation where nobody wins!
  • And dividing the electorate into districts can shift power in unexpected ways. (We had a Buzz exhibit last year explaining how the Electoral College redistributes power.)

In 1963, American economist Kenneth Arrow considered all these quirks and tried to describe the perfect voting system. He then proved that it was mathematically impossible. (Of course, this assumes the system he described really is perfect--I'm not so sure.)

It seems to me, though, that the problem isn't with democracy, but rather with representative democracy. The people of Minnesota elect only one governor, only one senator (at a time). And there's no way one person is going to perfectly reflect public opinion--be 53% in favor of issue A and 61% opposed to issue B. And even if they were, they still have to make a series of yes-or-no decisions, and be either 100% for 100% against any given bill.

The only way to have a perfect democracy is to have everybody vote on every issue, a system that would be far too cumbersome to work. Churchill was right: democracy is messy, but it's the best thing we've got.

Two girls tackle their math together: Are they inadvertently taught to fear it?
Two girls tackle their math together: Are they inadvertently taught to fear it?Courtesy gr8matt
Researchers at the University fo Chicago have published a new report in PNAS that shows math anxiety in elementary school teachers (which are predominantly female) is passed on to the young girls in their classes. The research is reported on the Smithsonian's Surprising Science blog site.

http://blogs.smithsonianmag.com/science/2010/01/26/elementary-school-tea...

New record for pi?

by Anonymous on Jan. 06th, 2010

Pi are squared; cake are round: Photo courtesy LeJyBy at Flickr Creative Commons
Pi are squared; cake are round: Photo courtesy LeJyBy at Flickr Creative Commons
Using a desktop computer, a scientist says he's calculated pi to almost 2.7 trillion digits! That's enough information to fill more than a thousand gigabytes (one terrabyte) of hard drive space, and would take more than 49,000 years of around-the-clock counting to count at one number per second. Could this mean more slices for everyone? Let's hope so.

SOURCE
BBC report

Dec
26
2009

GOCE Satellite: The Gravity field and steady-state Ocean Circulation Explorer
GOCE Satellite: The Gravity field and steady-state Ocean Circulation ExplorerCourtesy ESA
Can it be true? Yes, for a mere $5,544 dollars round-trip airfare to Greenland! In March 2009, the European Space Agency launched the Gravity field and steady-state Ocean Circulation Explorer (GOCE) into orbit around our planet, which is now transmitting detailed data about the Earth’s gravity. The GOCE satellite uses a gradiometer to map tiny variations in the Earth’s gravity caused by the planet’s rotation, mountains, ocean trenches, and interior density. New maps illustrating gravity gradients on the Earth are being produced from the information beamed back from GOCE. Preliminary data suggests that there is a negative shift in gravity in the northeastern region of Greenland where the Earth’s tug is a little less, which means you might weigh a fraction of a pound lighter there (a very small fraction, so it may not be worth the plane fare)!

In America, NASA and Stanford University are also working on the gravity issue. Gravity Probe B (GP-B) is a satellite orbiting 642 km (400 miles) above the Earth and uses four gyroscopes and a telescope to measure two physical effects of Einstein’s Theory of General Relativity on the Earth: the Geodetic Effect, which is the amount the earth warps its spacetime, and the Frame-Dragging Effect, the amount of spacetime the earth drags with it as it rotates. (Spacetime is the combination of the three dimensions of space with the one dimension of time into a mathematical model.)

Quick overview time. The Theory of General Relativity is simply defined as: matter telling spacetime how to curve, and curved spacetime telling matter how to move. Imagine that the Earth (matter) is a bowling ball and spacetime is a trampoline. If you place the bowling ball in the center of the trampoline it stretches the trampoline down. Matter (the bowling ball) curves or distorts the spacetime (trampoline). Now toss a smaller ball, like a marble, onto the trampoline. Naturally, it will roll towards the bowling ball, but the bowling ball isn’t ‘attracting’ the marble, the path or movement of the marble towards the center is affected by the deformed shape of the trampoline. The spacetime (trampoline) is telling the matter (marble) how to move. This is different than Newton’s theory of gravity, which implies that the earth is attracting or pulling objects towards it in a straight line. Of course, this is just a simplified explanation; the real physics can be more complicated because of other factors like acceleration.

Albert Einstein
Albert EinsteinCourtesy none
So what is the point of all this high-tech gravity testing? First of all, our current understanding of the structure of the universe and the motion of matter is based on Albert Einstein’s Theory of General Relativity; elaborate concepts and mathematical equations conceived by a genius long before we had the technology to directly test them for accuracy. The Theory of General Relativity is the cornerstone of modern physics, used to describe the universe and everything in it, and yet it is the least tested of Einstein’s amazing theories. Testing the Frame-Dragging Effect is particularly exciting for physicists because they can use the data about the Earth’s influence on spacetime to measure the properties of black holes and quasars.

Second, the data from the GOCE satellite will help accurately measure the real acceleration due to gravity on the earth, which can vary from 9.78 to 9.83 meters per second squared around the planet. This will help scientists analyze ocean circulation and sea level changes, which are influenced by our climate and climate change. The information that the GOCE beams back will also assist researchers studying geological processes such as earthquakes and volcanoes.

So, as I gobble down another mouthful of leftover turkey and mashed potatoes, I can feel confident that my holiday weight gain and the structure of the universe are of grave importance to the physicists of the world!

Nov
21
2009

Robots that "think for themselves"

Fly&amp;#039;s eyes: Can the nerves, eyes, and brain function of a fly be modeled within a computer circuit?
Fly's eyes: Can the nerves, eyes, and brain function of a fly be modeled within a computer circuit?Courtesy NeilsPhotography
Engineers are trying to design machines that can "think for themselves" when on surveillance or search and rescue missions. Somehow the machines has to look at its environment and decide what to do.
Have you ever tried to catch a fly? They are pretty good at seeing your hand and knowing just how to escape your grasp.

If we can figure out how a fly can do it ...

Can we figure out how a fly is able see, and find food, and escape from our fly swatters? With today's super microscopes, I am sure that we can visualize and model every nerve connection, muscle fiber, and eye facet.

Computational biology

David O’Carroll, a computational neuroscientist who studies insect vision at Australia’s University of Adelaide has been studying the optical flight circuits of flies, measuring their cell-by-cell activity. In a paper published in Public Library of Science Computational Biology, O’Carroll and fellow University of Adelaide biologist Russell Brinkworth describe an

algorithm composed of a series of five equations through which data from cameras can be run. Each equation represents tricks used by fly circuits to handle changing levels of brightness, contrast and motion, and their parameters constantly shift in response to input.

The "fly brain" circuits are small and use only a fraction of a milliwatt

“It’s amazing work,” said Sean Humbert, who builds miniaturized, autonomous flying robots,

“For traditional navigational sensing, you need lots of payload to do the computation. But the payload on these robots is very small — a gram, a couple of Tic Tacs. You’re not going to stuff dual-core processors into a couple Tic Tacs.

Learn more - mathematical modeling of insect biology

Secret Math of Fly Eyes Could Overhaul Robot Vision Wired Science
Robust Models for Optic Flow Coding in Natural Scenes Inspired by Insect Biology Computational Biology

We have constructed a full model for motion processing in the insect visual pathway incorporating known or suspected elements in as much detail as possible. We have found that it is only once all elements are present that the system performs robustly, with reduction or removal of elements dramatically limiting performance. The implementation of this new algorithm could provide a very useful and robust velocity estimator for artificial navigation systems.