Stories tagged Math

Oct
10
2013

Inspiration for the Science Museum of Minnesota's spiral logo comes from the Fibonacci sequence of numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...

The sequence begins with zero (implied) and the number one. Each subsequent number is the sum of the previous two numbers.

0 + 1 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
13 + 21 = 34
21 + 34 = 55
34 + 55 = 89
55 + 89 = 144
89 + 144 = 233
144 + 233 = 377
And so on...

The Fibonacci sequence often appears when measuring patterns in nature, including the spiral shape of many shells. The sequence is named after an accomplished Italian mathematician known as Leonardo Fibonacci (c. 1170 – c. 1250).

The Science Museum of Minnesota began using the spiral logo around 1999. Because Fibonacci numbers can be difficult to explain, and the proportions of the SMM logo do not exactly match the Fibonacci sequence, the museum often describes the SMM logo in the following way:

The Science Museum of Minnesota symbol is based on a spiral. This symbol represents eternity, infinity and the spiral of life. The design can be found in all forms of life from the earth to the sea to the galaxy. The shape suggests movement, and represents the characteristics of science in an ever-changing and expanding universe.

Jan
11
2013

Nano at home!: Forget those super-sterile clean rooms.  The DIY Nano app lets you explore nanoscale science in the comfort of your own home!
Nano at home!: Forget those super-sterile clean rooms. The DIY Nano app lets you explore nanoscale science in the comfort of your own home!Courtesy NISE Network
When things get really really small (nanoscale small), they behave completely differently! For example, gold at the nanoscale can look purple, orange, or red; static electricity has a greater effect on nanoparticles than gravity; and aluminum (the stuff your benign soda cans are made of) is explosive at the nanoscale!

If you want to experience some of these nanoscale phenomena first-hand, check out whatisnano.org, or download the DIY Nano app. The website and the app were both created by the Nanoscale Informal Science Education Network (NISE Net for short), and have videos and activity guides, complete with instructions and material lists, so you can do some nano experiments at home! The app was a Parents' Choice award winner for 2012, and was featured in Wired Magazine's review of apps. Definitely worth a look!

Have fun exploring nanoscale properties!

Oct
27
2011

Sometime on or around October 31st, the world's population will hit seven billion people.

Population map: A map from the Worldmapper World Population Atlas: www.worldpopulationatlas.org (c) Sasi Research Group, University of Sheffield
Population map: A map from the Worldmapper World Population Atlas: www.worldpopulationatlas.org (c) Sasi Research Group, University of SheffieldCourtesy Worldmapper.org / CC BY-NC-ND 2.0

(There are a lot of challenges to supporting seven billion people. Want to know more about that? Check out the University of Minnesota's Institute on the Environment, where folks are working to find solutions to some of those problems.)

That's all fascinating and all, but...what about me? Luckily, the BBC has come to the rescue with a lovely little interactive that's, well, all about me. Or you. Whatever.

For example, according to the BBC calculator,

  • At the time of my birth, I was the 3,840,942,641st person to live on Earth.
  • And the 78,068,048,685th person to live since history began.
  • This morning, the United States has a population of 311,284,287 people,
  • In the US, there are 484 births, 288 deaths, and a gain of 113 immigrants every hour, for a total average yearly population growth of 0.9%. (The country with the fastest growing population is Qatar, experiencing an increase of 514 people every day. The country with the fastest shrinking population is Moldova, experiencing a decrease of 106 people every day.)
  • As a woman in the United States, I can look forward to an average life expectancy of 80.5 years. Men in the US, however, enjoy an average life expectancy of only 75.4 years, dragging the US total average down to 78 years. Thanks, men. (The Japanese have the longest average life expectancy in the world, at 82.7 years, while folks in the Central African Republic have an average life expectancy of only 45.9 years.)

Not too shabby!

To give you a sense of just how fast our population is growing, here's a crazy little fact: by mid-century, the world's urban population will equal the size of the world's global population in 2004. Wow. Cities are efficient, and concentrate us so that we can use land for other purposes, but they're also ecological hotspots. Curious about how your household measures up? Try the household flux calculator, or check out the Q&A with Scientist on the Spot Daniel Nidzgorski.

Not enough for you? Check out "Seven billion in seven stories" and "Will people numbers keep rising?"

Oh, and let us know: #whatsyournumber ?

May
28
2010

A few weeks ago, I assumed that some of our readers were bored with the same ol’ climate change arguments. I know you know what I’m talking about: the Cuddly-Animals-are-Dying and the Catastrophic-Disasters-Will-End-the-Human-Race arguments come to mind first. Now, I’m not saying there isn’t some merit to these frames, but c’mon! Can’t we get a little variety?

Stephen Polasky: This UofM professor and IonE fellow has some BIG ideas about $, the earth, and climate change.
Stephen Polasky: This UofM professor and IonE fellow has some BIG ideas about $, the earth, and climate change.Courtesy University of Minnesota

Lucky for you, University of Minnesota professor and Institute on the Environment fellow Stephen Polasky thinks creatively. In April, he gave a presentation on how adopting inclusive wealth could ultimately reduce climate change and its effects. And since virtually everybody likes money, I’m going to go out on a limb and bet you want to know more about the ca-ching!$

Here’s the skinny:

Economists say that just about everything has a monetary value, and how much something is worth plays largely into the decisions politicians make. Scientists like Polasky are increasingly saying that these traditional accounting methods do a poor job assessing value to natural resources, and these mistakes are leading us to make irrational choices. As an alternative, Polasky suggests adopting inclusive wealth theory.

This is not going to cut it.: Inclusive wealth is a really complicated theory for both scientists and economists.
This is not going to cut it.: Inclusive wealth is a really complicated theory for both scientists and economists.Courtesy happyeclaire (Flickr)

Ready for the good stuff??

Economists and scientists both agree that the environment has worth, called natural capital, but they disagree on how much. In fact, not only do economists and scientists disagree with each other, but they disagree amongst themselves! To be fair, determining something’s worth can be extremely difficult. Because there are already economic markets for some natural resources like trees (i.e. lumber) and metals (i.e. gold), it’s easier to assess their value. Most ecosystem services, however, like the flood control provided by wetlands, are more difficult to put a dollar value on.

Inclusive wealth theory says that our decisions should be made on economic assessments that include true representations of the value of natural resources (difficult as that may be).

Politicians make important decisions regarding environmental policies, including actions that affect climate change. When politicians are choosing between multiple policy options, they are conducting policy analysis. One criterion that politicians pretty much always use is a cost-benefit ratio, or cost efficiency. In order to do that, politicians must determine the value of each policy option and weight the outcome against the rest. (It might sound complicated, but you do this same process informally everyday when you make decisions regarding what to eat for breakfast and whether to walk or ride your bike to school/work.)

Timber!: Land use changes, like logging, release greenhouse gases like carbon dioxide into the atmosphere, ultimately contributing to climate change.
Timber!: Land use changes, like logging, release greenhouse gases like carbon dioxide into the atmosphere, ultimately contributing to climate change.Courtesy Ben Cody

Polasky and other like-minded individuals argue that under traditional accounting methods, politicians’ cost-benefit ratios are distorted – they are not accurately representing the true worth of the environment. Furthermore, as a result, we’re making some pretty big, bad decisions. According to Polasky, the solution is simple in theory, but difficult in practice: adopt inclusive wealth theory to more accurately measure environmental worth. If we increase the value of the environment in our analysis, the cost-benefit ratios will change and perhaps favor decisions that are more environmentally friendly. That is, under inclusive wealth, we might finally see how important it is to take climate change-reducing actions such as reducing our fossil fuel consumption, protecting forests from logging, and stopping eating so much meat… or not.

What do you think?

How much $$ is the environment worth to you? What about individual ecosystem services like pollination by bees or decomposition of waste by microbes?

Are politicians doing an accurate job of assessing the value of natural capital?

Post your comments below!

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.

Mar
17
2010

Look out the window or walk down the street to nearly any river or stream in Minnesota right now and you are likely to observe two things about the river:

  1. it is getting deeper (or “rising” in relation to the banks); and
  2. it appears to be moving faster.

You can, of course, confirm these observations by investigating reports from gauging stations along these rivers, maintained by the U.S. Geological Survey. (See data for the gauging station serving downtown St. Paul.) But what is really happening?

It may be high and fast...: ...but (as of today) the Mississippi at St. Paul is still in a bankfull state.
It may be high and fast...: ...but (as of today) the Mississippi at St. Paul is still in a bankfull state.Courtesy Liza Pryor

Until a river flows over its banks, it is considered to be in a “bankfull” state. In this state, the water flowing through the river is confined to a relatively fixed channel area. Simply put, floods occur because more water is being introduced into this channel from upstream, due to snowmelt, heavy rains, or a dam breach. As this added volume of water moves through a fixed area, it both increases in velocity and in depth until it overflows the banks, at which point some, but not necessarily a lot, of the volume and velocity moving through the channel are reduced.

Scientists call the rate of flow through a channel “discharge." Discharge is defined as the volume of water passing through a given cross-section of the river channel within a specified period of time.A simple equation for determining discharge is

Q = D x W x V

where Q = discharge, D = channel depth, W = channel width and V = velocity.

Looking at this equation, it is easy to see that if discharge becomes greater and channel width is fixed, then an increase in both volume and depth (or height relative to the banks) is likely to be the cause. Discharge can be measured in cubic feet per second or cubic meters per second, for example.

But is the river flowing at the same rate at the surface as it does along its banks and beds? Understanding this requires investigating some more detailed equations, as the banks and bed introduce friction, which affects the rate of flow.

To learn more about rivers and how they flow, you may want to check out the works of Luna Leopold, and M. Gordon Wolman. In particular:

  • Leopold, Luna B. (2006, reprint). A View of the River. Harvard University Press; and
  • Leopold, Luna B.; Wolman, M. Gordon; and Miller, John P. (1995). Fluvial Processes in Geomorphology. Dover Publications, both classics for understanding how rivers work.

Also, check out our full feature on the 2010 Mississippi River flooding.

Jan
26
2010

When we decide to go from point A to point B, we have a plan. If driving, we would decide which route to take, which intersections to avoid, and we will estimate how much time it will take. As we do drive from A to B, we will, intuitively and deliberately, make changes – we might change our route because of a traffic jam, or speed up to try and cover time lost. Thus, as we encounter new information and unexpected situations, we, intuitively, make changes to our plans, for the better.

Scientists and engineers have been trying for years to bring this capability – of automatically changing plans – to machines. Research in robotics has focused on equipping machines with sensors and radars to detect changes and respond to that change. For example, Toyota Motor corporation equips its Lexus LS460 with a radar which detects vehicles and obstacles on the road ahead. When the radar detects the possibility of a collision, the vehicle retracts the seatbelts, warns the driver, and applies brakes to reduce collision speed. Such systems are dubbed “sense-response” systems, and today are even available in automated vacuum cleaners such as the Roomba. However, taking the sense-response system a step further, such that it is applicable to entire supply chain and production systems has proven difficult.

A key breakthrough in developing automated “sense-response” systems, that would enable developing automated production decision support, sales and marketing recommendations, and several other automated systems has recently been achieved. Dr. Nazrul Shaikh, a post doctoral researcher at University of Pennsylvania has worked on the characterization of the planning and re-planning problem and have developed solutions for several classes of sense-response problems. The research, which has survived the extensive peer review from academicians, in now all set to make an impact to homeland security and corporate America. It will change the way the automated planning systems are used and implemented in industry.

Dr. Shaikh’s quest to characterize the sense-response problems began in 2002 when they were devised an approach to develop an intelligent sense response system for supply chain event management for IBM. This was in the wake of 9/11 when Corporate America realized that they need to be able to handle exceptions to plans better. The research, called Exception Analytics looked at the deviation between the planned and the actual values of several system variables and attempted to infer the “action” required to compensate for the deviation. Applications have been developed for supply chain event management, homeland security, quality control and marketing planning. Keen interest has been displayed by several companies, especially in marketing where the consumer and competitor dynamics influence the demand drastically. The basic concepts have been published in several journals after clearing rigorous peer review.

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!

Aug
24
2009

Philadelphia Phillies' second-baseman Eric Brunlett made an amazing and unassisted game-ending triple play against the New York Mets. The extremely rare play came in the ninth inning with the Mets trailing by two runs but threatening with runners on first and second base and no outs. Both base runners were stealing on a 2-2 pitch when the Mets batter hit a line drive right to Brunlett, who caught the ball for the first out, stepped on second base for the second out, then tagged out the runner from first. It's only the 15th unassisted triple play in major league history, and only the second to end a game. The poor Mets. The odds of this happening must be astronomical, but I'll let someone else figure that out.