Reasoning With Mathematics

Lesson 8 - Visualizing Complex Information

It's time for reminders of a few central ideas that were mentioned earlier:

Mathematical data often is about size or shape (or both size and shape). But this information can be hard to understand and use when it's only in numbers, formulas, or words.

Good visualizations make the data easier to understand. Sometimes a good visualization even helps people discover what's in the data. Bad visualizations confuse data and misrepresent what's in it.

People have developed different ways of visualizing data including charts, graphs, maps, diagrams, illustrations, three-dimensional models, and many more.

Visualizations almost always deal with connections, relationships - patterns - in the data.

When visualizations work well, they seem to simplify things. But it's probably better to say good visualizations clarify because what they communicate may be very complicated and not at all simple.

In this section you'll see a few examples that deal with complicated things and use visualization to clarify them.

Where Was That Earthquake?

Depending on where they happen and what they're like, earthquakes can cause enormous damage. They've played a prominent role in the history of natural calamities, some of them taking thousands of lives and in one case over a million (from a quake in Syria in 1201).

Strong earthquakes are sudden and dramatic, yet they happen because of complex, long processes, mostly involving changes in the shape of earth's outermost layers and especially the crust. The strongest quakes happen along ruptures or cracks in the crust, called faults.

Seismology, the science of earthquakes, uses a great deal of visualization to describe and understand how quakes happen and how they behave. Scientists put data, usually numbers, onto maps and models and use them to think about the "plates" of earth crust move and affect each other, how quakes behave along a fault, how to forecast them, and so on.

One part of seismology involves describing when and where quakes take place. News about a major quake reports this as the center of the quake or the epicenter. The standard way of locating an earthquake involves plotting readings - many, many numbers - from an instrument called a seismograph or seismometer onto a graph, doing some calculations, drawing some circles on maps, and interpreting the results.

A seismograph responds to the shock waves a quake sends out. These waves, which act a little like sound waves, will circle the earth in about 20 minutes. By recording the length of time between two different kinds of shock waves (called P wave and S wave), and using a graph that converts that time to distance, a person or computer can calculate how far away the quake occurred. That is, how far it was from the seismograph. That distance is visualized as a circle with the seismograph in the center.

It's simply like this:

If this circle was drawn over a map or projected onto one, the picture would be even clearer. And that's what most people would do. This circle, based on data collected by the seismograph and calculated out by people who use it, says "The earthquake took place somewhere along the line this circle shows." Which is interesting but not very useful yet because this visualization shows nothing about where along the line.

As you may have figured out already, the solution is to use more seismographs that recorded data about the same earthquake. If everything works right, when put onto a map the circle from a second seismograph will overlap with the first one, roughly like this:

Each circle says "The earthquake took place somewhere along this line, " so again, if everything is working right, the quake took place where the two circles cross, or intersect. The circles intersect at two points, so the location has been narrowed down to them.

A third circle, from a third seismograph recording the same quake, will pin down the location. The circles will look something like this:

There's an online introduction to using seismograms (the plotting off a seismograph) from California State University, Los Angeles [http://vflylab.calstatela.edu/edesktop/VirtApps/VirtualEarthQuake/VQuakeIntro.html]

You can try reading sample graphs, converting time to distance, and figuring out an estimate of the quake's intensity. Even if you don't try using the whole system, you can see how circles from three seismographs are used to triangulate a quake's location. (Technically, this work helps locate the place on the earth's surface above the quake. Earthquakes happen under the surface; the strongest ones usually happen close to the surface.)

Hitting the Points Again

Locating the focus of an earthquake from seismograph information is a good example of how visualization gets used with complicated information. What you've looked at here combines plotting numbers onto a graph at the seismograph itself, reading time intervals off that graph, converting the time to distance using a different graph, drawing a circle based on the distance, drawing that circle onto a map, and then doing all of this over again two more times before finding the point where the three circles intersect. It all ends up on a simple-looking map.

Visualizing a Piece of History

In 1812 Napoleon led his French armies eastward, with forces totaling over 450,000 , entering Russia in June of that year. The Russians retreated slowly but steadily until reaching Borodino, where both sides suffered immense losses in a battle there on September 7. The Russians withdrew and Napoleon advanced to Moscow, which his armies occupied on September 14. Between then and October 19 Russian arsonists burned about 80 percent of the city, frustrating the French occupation. With winter approaching, Napoleon decided to leave Moscow and the evacuation began October 19. The Russians sent huge forces in pursuit; a bitterly cold winter set in. The Russian army nearly surrounded the French at the Berezina River, but what was left of Napoleon's armies managed to escape and left Russian territory. About 40,000 soldiers survived the campaign.

Years later a French engineer named Charles Minard created a visual account of the Russian Campaign [http://pascal.math.yorku.ca/SCS/Gallery/images/minard.gif ] which you can see online or in one of Edward Tufte's books The Visual Display of Quantitative Information.

The online copy is a little hard to read, but even without seeing all the details you can get an idea of why Edward Tufte wrote in his book that this may be the best statistical graphic ever made. Here are some highlights of this graphic:

The tan band across the top shows the size of Napoleon's invading force, represented by the width of the band. You can even see where and when some troops left the main force and set up a defensive position.

The black band below shows the size of the force during the retreat from Moscow. You can spot locations where there were especially heavy losses.

Both bands are drawn to the same scale, so they give a continuous picture of the size of the force.

The bands are drawn over a map of the campaign route, so other information can be connected to location.

The numbers across the bottom show temperatures during the retreat, so other information can be connected to winter weather conditions.

The same Website holding Minard's original graphic also has a modern version [http://pascal.math.yorku.ca/SCS/icons/Minard.gif] which shows much of the same information as the original. If you go to the top of this online collection, called The Gallery of Data Visualization, [http://pascal.math.yorku.ca/SCS/Gallery/] you can start browsing through many different examples. Several of them show historical information like casualties during the Crimean War in a graphic created by the famous Florence Nightengale or a chart showing the growth in English exports in the last part of the 18th Century. A map in this collection even goes along with part of the Mindquest course "Designing a Project." The course uses the example of John Snow's problem solving about cholera in the city of London. The map is the one Snow used to help his reasoning.

Visualizing Air Pollution

The last example is also from Edward Tufte book containing Minard's graphic. This one displays information about air pollutants [ insert scan of graphic from Visual Display ] .

Notice these things about the display:

There are three bands of information, one for each of three major types of pollutants

The location is identical for each band. So in other words, the graphic deals with one area and three different air pollutants

Map elements in the graphic show power plants and major freeways, which produce a lot of air pollutants but not the same kinds

Each band shows 24 hours as you read from left to right

Amounts of pollutants are shown by the up and down shapes; the more of that kind of pollutant, the higher the peak

With this graphic to look at, questions about patterns start to pop up. Why does the amount of carbon monoxide go up and down so much? When and where does the carbon monoxide reach its peak? What accounts for that peak of nitrous oxide? And why does it stay close to the same throughout the entire day?

To answer the first two questions, consider when and where freeway traffic would show up on this display. For the last two questions, consider how an electric generating plant powered by fossil fuel would show up.

Browsing Through Other Examples of Visualizations

Web pages use visualization over and over, in all kinds of forms, for all kinds of purposes. Some of them deal with complex information similar to what we've looked at in this lesson.

If you're interested in more examples, use your favorite search engine with terms like these:

Weather maps

Constellation charts

Data visualization

Tourist maps

Earthquakes

Solar system

Demographic visualization

Information design

And if you want to get deep into ideas about visualization, going as far as computer generated virtual reality, you can browse through a college course on Information Design [ http://tech-head.com/c/info.htm] from Emporia State University (Kansas). It's long and thorough, paying attention to many aspects like narrative, audio design, design and the entertainment industry, graphic design, and more.