Mathematical Moments from the American Mathematical Society

Restoring Genius - Discovering lost works of Archimedes - Part 2

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 13 Nov 2008 10:16:49 -0500

Archimedes was one of the most brilliant people ever, on a par with Einstein and Newton. Yet very little of what he wrote still exists because of the passage of time, and because many copies of his works were erased and the cleaned pages were used again. One of those written-over works (called a palimpsest) has resurfaced, and advanced digital imaging techniques using statistics and linear algebra have revealed his previously unknown discoveries in combinatorics and calculus. This leads to a question that would stump even Archimedes: How much further would mathematics and science have progressed had these discoveries not been erased?

One of the most dramatic revelations of Archimedes’ work was done using X-ray fluorescence. A painting, forged in the 1940s by one of the book’s former owners, obscured the original text, but X-rays penetrated the painting and highlighted the iron in the ancient ink, revealing a page of Archimedes’ treatise The Method of Mechanical Theorems. The entire process of uncovering this and his other ideas is made possible by modern mathematics and physics, which are built on his discoveries and techniques.

This completion of a circle of progress is entirely appropriate since one of Archimedes’ accomplishments that wasn’t lost is his approximation of pi.

For More Information: The Archimedes Codex, Reviel Netz and William Noel, 2007.

Restoring Genius - Discovering lost works of Archimedes - Part 1

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 13 Nov 2008 10:09:47 -0500

Improving Stents - Part 2

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 13 Nov 2008 10:07:07 -0500

Stents are expandable tubes that are inserted into blocked or damaged blood vessels. They offer a practical way to treat coronary artery disease, repairing vessels and keeping them open so that blood can flow freely. When stents work, they are a great alternative to radical surgery, but they can deteriorate or become dislodged. Mathematical models of blood vessels and stents are helping to determine better shapes and materials for the tubes. These models are so accurate that the FDA is considering requiring mathematical modeling in the design of stents before any further testing is done, to reduce the need for expensive experimentation.

Precise modeling of the entire human vascular system is far beyond the reach of current computational power, so researchers focus their detailed models on small subsections, which are coupled with simpler models of the rest of the system. The Navier-Stokes equations are used to represent the flow of blood and its interaction with vessel walls. A mathematical proof was the central part of recent research that led to the abandonment of one type of stent and the design of better ones. The goal now is to create better computational fluid-vessel models and stent models to improve the treatment and prediction of coronary artery disease the major cause of heart attacks.

For More Information: Design of Optimal Endoprostheses Using Mathematical Modeling, Canic, Krajcer, and Lapin, Endovascular Today, May 2006.

Improving Stents - Part 1

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 13 Nov 2008 09:12:22 -0500

Steering Towards Efficiency

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 28 Aug 2008 10:21:18 -0400

The racing team is just as important to a car’s finish as the driver is. With little to separate competitors over hundreds of laps, teams search for any technological edge that will propel them to Victory Lane. Of special use today is computational fluid dynamics, which is used to predict airflow over a car, both alone and in relation to other cars (for example, when drafting). Engineers also rely on more basic subjects, such as calculus and geometry, to improve their cars. In fact, one racing team engineer said of his calculus and physics teachers, the classes they taught to this day were the most important classes I’ve ever taken.(1)
Mathematics helps the performance and efficiency of non-NASCAR vehicles, as well. To improve engine performance, data must be collected and processed very rapidly so that control devices can make adjustments to significant quantities such as air/fuel ratios. Innovative sampling techniques make this real-time data collection and processing possible. This makes for lower emissions and improved fuel economy goals worthy of a checkered flag.
For More Information: The Physics of NASCAR, Diandra Leslie-Pelecky, 2008.

Hearing a Master’s Voice

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 5 Jun 2008 13:40:14 -0400

The spools of wire below contain the only known live recording of the legendary folk singer Woody Guthrie. A mathematician, Kevin Short, was part of a team that used signal processing techniques associated with chaotic music compression to recapture the live performance, which was often completely unintelligible. The modern techniques employed, instead of resulting in a cold, digital output, actually retained the original concert’s warmth and depth. As a result, Short and the team received a Grammy© Award for their remarkable restoration of the recording.
To begin the restoration the wire had to be manually pulled through a playback device and converted to a digital format. Since the pulling speed wasn’t constant there was distortion in the sound, frequently quite considerable. Algorithms corrected for the speed variations and reconfigured the sound waves to their original shape by using a background noise with a known frequency as a "clock." This clever correction also relied on sampling the sound selectively, and reconstructing and resampling the music between samples. Mathematics did more than help recreate a performance lost for almost 60 years: These methods are used to digitize treasured tapes of audiophiles everywhere.

For More Information: "The Grammy in Mathematics," Julie J. Rehmeyer, Science News Online, February 9, 2008.

Going with the Floes - Part 1

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 5 Jun 2008 13:45:02 -0400

Sea ice is one of the least understood components of our climate. Naturally its abundance or scarcity is a telling sign of climate change, but sea ice is also an important actor in change as well, insulating the ocean and reflecting sunlight. A branch of mathematics called percolation theory helps explain how salt water travels through sea ice, a process that is crucial both to the amount of sea ice present and to the microscopic communities that sustain polar ecosystems. By taking samples, doing on-site experiments, and then incorporating the data into models of porous materials, mathematicians are working to understand sea ice and help refine climate predictions.

Using probability, numerical analysis, and partial differential equations, researchers have recently shown that the permeability of sea ice is similar to that of some sedimentary rocks in the earth’s crust, even though the substances are otherwise dissimilar. One major difference between the two is the drastic changes in permeability of sea ice, from total blockage to clear passage, that occur over a range of just a few degrees. This difference can have a major effect on measurements by satellite, which provide information on the extent and thickness of sea ice. Results about sea ice will not only make satellite measurements more reliable, but they can also be applied to descriptions of lung and bone porosity, and to understanding ice on other planets.

Image: Pancake ice in Antarctica, courtesy of Ken Golden.

For More Information: "Thermal evolution of permeability and microstructure in sea ice," K. M. Golden, et al., Geophysical Research Letters, August 28, 2007.

Going with the Floes - Part 2

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 5 Jun 2008 13:25:01 -0400

Going with the Floes - Part 3

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 5 Jun 2008 13:46:14 -0400

Going with the Floes - Part 4

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 5 Jun 2008 13:47:00 -0400

Bending It Like Bernoulli

paoffice@ams.org (The AMS Public Awareness Office) — Mon, 14 Apr 2008 11:41:56 -0400

The colored "strings" you see represent air flow around the soccer ball, with the dark blue streams behind the ball signifying a low-pressure wake. Computational fluid dynamics and wind tunnel experiments have shown that there is a transition point between smooth and turbulent flow at around 30 mph, which can dramatically change the path of a kick approaching the net as its speed decreases through the transition point. Players taking free-kicks need not be mathematicians to score, but knowing the results obtained from mathematical facts can help players devise better strategies.
The behavior of a ball depends on its surface design as well as on how it’s kicked. Topology, algebra, and geometry are all important to determine suitable shapes, and modeling helps determine desirable ones. The researchers studying soccer ball trajectories incorporate into their mathematical models not only the pattern of a new ball, but also details right down to the seams. Recently there was a radical change from the long-used pentagon-hexagon pattern to the adidas +TeamgeistTM. Yet the overall framework for the design process remains the same: to approximate a sphere, within less than two percent, using two-dimensional panels.

Tripping the Light-Fantastic

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 14 Feb 2008 09:44:11 -0500

Invisibility is no longer confined to fiction. In a recent experiment, microwaves were bent around a cylinder and returned to their original trajectories, rendering the cylinder almost invisible at those wavelengths. This doesn't mean that we're ready for invisible humans (or spaceships), but by using Maxwell's equations, which are partial differential equations fundamental to electromagnetics, mathematicians have demonstrated that in some simple cases not seeing is believing, too.

Part of this successful demonstration of invisibility is due to metamaterials electromagnetic materials that can be made to have highly unusual properties. Another ingredient is a mathematical transformation that stretches a point into a ball, "cloaking" whatever is inside. This transformation was discovered while researchers were pondering how a tumor could escape detection. Their attempts to improve visibility eventually led to the development of equations for invisibility. A more recent transformation creates an optical "wormhole," which tricks electromagnetic waves into behaving as if the topology of space has changed. We'll finish with this:

For More Information: Metamaterial Electromagnetic Cloak at Microwave Frequencies, D. Schurig et al, Science, November 10, 2006.

Unearthing Power Lines

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 14 Feb 2008 09:38:47 -0500

Votes are cast by the full membership in each house of Congress, but much of the important maneuvering occurs in committees. Graph theory and linear algebra are two mathematics subjects that have revealed a level of organization in Congress groups of committees above the known levels of subcommittees and committees. The result is based on strong connections between certain committees that can be detected by examining their memberships, but which were virtually unknown until uncovered by mathematical analysis.

Mathematics has also been applied to individual congressional voting records. Each legislator’s record is represented in a matrix whose larger dimension is the number of votes cast (which in a House term is approximately 1000). Using eigenvalues and eigenvectors, researchers have shown that the entire collection of votes for a particular Congress can be approximated very well by a two-dimensional space. Thus, for example, in almost all cases the success or failure of a bill can be predicted from information derived from two coordinates. Consequently it turns out that some of the values important in Washington are, in fact, eigenvalues.

For More Information: Porter, Mason A; Mucha, Peter J.; Newman, M. E. J.; and Warmbrand, Casey M., A Network Analysis of Committees in the United States House of Representatives, Proceedings of the National Academy of Sciences, Vol. 102 [2005], No. 20, pp. 7057-7062.

Making Votes Count

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 14 Feb 2008 09:33:09 -0500

The outcome of elections that offer more than two alternatives but with no preference by a majority, is determined more by the voting procedure used than by the votes themselves. Mathematicians have shown that in such elections, illogical results are more likely than not. For example, the majority of this group want to go to a warm place, but the South Pole is the group’s plurality winner. So if these people choose their group’s vacation destination in the same way most elections are conducted, they will all go to the South Pole and six people will be disappointed, if not frostbitten.

Elections in which only the top preference of each voter is counted are equivalent to a school choosing its best student based only on the number of A’s earned. The inequity of such a situation has led to the development of other voting methods. In one method, points are assigned to choices, just as they are to grades. Using this procedure, these people will vacation in a warm place a more desirable conclusion for the group. Mathematicians study voting methods in hopes of finding equitable procedures, so that no one is unfairly left out in the cold.

For more information: Chaotic Elections: A Mathematician Looks at Voting, Donald Saari

Folding for Fun and Function

paoffice@ams.org (The AMS Public Awareness Office) — Thu, 14 Feb 2008 09:33:09 -0500

Origami paper-folding may not seem like a subject for mathematical investigation or one with sophisticated applications, yet anyone who has tried to fold a road map or wrap a present knows that origami is no trivial matter. Mathematicians, computer scientists, and engineers have recently discovered that this centuries-old subject can be used to solve many modern problems.The methods of origami are now used to fold objects such as automobile air bags and huge space telescopes efficiently, and may be related to how proteins fold.

Manufacturers often want to make a product out of a single piece of material. The manufacturing problem then becomes one of deciding whether a shape can be folded and if so, is there an efficient way to find a good fold? Thus, many origami research problems have to do with algorithm complexity and optimization theory. A testament to the diversity of origami, as well as the power of mathematics, is its applicability to problems at the molecular level, in manufacturing, and in outer space.

For More Information: http://db.uwaterloo.ca/~eddemain/papers/MapFolding/

Finding Fake Photos

paoffice@ams.org (The AMS Public Awareness Office) — Wed, 26 Dec 2007 11:33:02 -0500

Actually, they weren’t caught together at all their images were put together with software. The shadows cast by the stars’ faces give it away: The sun is coming from two different directions on the same beach! More elaborate digital doctoring is detected with mathematics. Calculus, linear algebra, and statistics are especially useful in determining when a portion of one image has been copied to another or when part of an image has been replaced.

Tampering with an image leaves statistical traces in the file. For example, if a person is removed from an image and replaced with part of the background, then two different parts of the resulting file will be identical. The difficulty with exposing this type of alteration is that both the location of the replacement and its size are unknown beforehand. One successful algorithm finds these repetitions by first sorting small regions according to their digital color similarity, and then moving to larger regions that contain similar small ones. The algorithm’s designer, a leading digital forensics expert, admits that image alterers generally stay a step ahead of detectors, but observes that forensic advances have made it much harder for them to escape notice. He adds that to catch fakers, At the end of the day you need math.(1)

For More Information: Can Digital Photos be Trusted?, Steve Casimiro, Popular Science, October 2005.

_______ 1 It May Look Authentic; Here’s How to Tell It Isn't, Nicholas Wade, The New York Times, January 24, 2006.

Putting Music on the Map

paoffice@ams.org (The AMS Public Awareness Office) — Wed, 26 Dec 2007 11:39:57 -0500

Mathematics and music have long been closely associated. Now a recent mathematical breakthrough uses topology (a generalization of geometry) to represent musical chords as points in a space called an orbifold, which twists and folds back on itself much like a Möbius strip does. This representation makes sense musically in that sounds that are far apart in one sense yet similar in another, such as two notes that are an octave apart, are identified in the space.This latest insight provides a way to analyze any type of music. In the case of Western music, pleasing chords lie near the center of the orbifolds and pleasing melodies are paths that link nearby chords. Yet despite the new connection between music and coordinate geometry, music is still more than a connect-the-dots exercise, just as mathematics is more than addition and multiplication.

For More Information: The Geometry of Musical Chords, Dmitri Tymoczko, Science, July 7, 2006.

Pinpointing Style

paoffice@ams.org (The AMS Public Awareness Office) — Wed, 26 Dec 2007 12:11:25 -0500

Mathematics is not just numbers and brute force calculation there is considerable art and elegance to the subject. So it is natural that mathematics is now being used to analyze artists. styles and to help determine the identities of the creators of disputed works. Attempts at measuring style began with literature based on statistics of word use and have successfully identified disputed works such as some of The Federalist Papers. But drawings and paintings resisted quantification until very recently. In the case of Jackson Pollock, his paintings have a demonstrated complexity to them (corresponding to a fractal dimension between 1 and 2) that distinguishes them from simple random drips.

A team examining digital photos of drawings used modern mathematical transforms known as wavelets to quantify attributes of a collection of 16th century master.s drawings. The analysis revealed measurable differences between authentic drawings and imitations, clustering the former away from the latter. This is an impressive feat for the non-experts and their model, yet the team agrees that its work, like mathematics itself, is not designed to replace humans, but to assist them.

For More Information: The Style of Numbers Behind a Number of Styles, Dan Rockmore, The Chronicle of Higher Education, June 9, 2006.

Predicting Storm Surge

paoffice@ams.org (The AMS Public Awareness Office) — Wed, 26 Dec 2007 11:43:42 -0500

Storm surge is often the most devastating part of a hurricane. Mathematical models used to predict surge must incorporate the effects of winds, atmospheric pressure, tides, waves and river flows, as well as the geometry and topography of the coastal ocean and the adjacent floodplain. Equations from fluid dynamics describe the movement of water, but most often such huge systems of equations need to be solved by numerical analysis in order to better forecast where potential flooding will occur.

Much of the detailed geometry and topography on or near a coast require very fine precision to model, while other regions such as large open expanses of deep water can typically be solved with much coarser resolution. So using one scale throughout either has too much data to be feasible or is not very predictive in the area of greatest concern, the coastal floodplain. Researchers solve this problem by using an unstructured grid size that adapts to the relevant regions and allows for coupling of the information from the ocean to the coast and inland. The model was very accurate in tests of historical storms in southern Louisiana and is being used to design better and safer levees in the region and to evaluate the safety of all coastal regions.

For More Information: A New Generation Hurricane Storm Surge Model for Southern Louisiana, by Joannes Westerink et al.

Targeting Tumors

paoffice@ams.org (The AMS Public Awareness Office) — Wed, 26 Dec 2007 11:21:19 -0500

Detection and treatment of cancer have progressed, but neither is as precise as doctors would like. For example, tumors can change shape or location between pre-operative diagnosis and treatment so that radiation is aimed at a target which may have moved. Geometry, partial differential equations, and integer linear programming are three areas of mathematics used to process data in real-time, which allows doctors to inflict maximum damage to the tumor, with minimum damage to healthy tissue.

One promising area of investigation is virotherapy: using viruses to destroy cancerous cells. Researchers are using mathematical models to discover how to use the viruses most beneficially.The models provide numerical outcomes for each of the many possibilities, thereby eliminating unsuccessful approaches and identifying candidates for further experimentation.Testing by simulation, which led to the development of anti-HIV cocktails, means good medicine is developed faster and cheaper than it can be by lab experiments and clinical trials alone.

For More Information: Treatment Planning for Brachytherapy, Eva Lee, et al, Physics in Medicine and Biology, 1999.

Making Movies Come Alive

paoffice@ams.org (The AMS Public Awareness Office) — Wed, 15 Jun 2005 19:00:00 +0000

Many movie animation techniques are based on mathematics. Characters, background, and motion are all created using software that combines pixels into geometric shapes which are stored and manipulated using the mathematics of computer graphics.

Software encodes features that are important to the eye, like position, motion, color, and texture, into each pixel. The software uses vectors, matrices, and polygonal approximations to curved surfaces to determine the shade of each pixel. Each frame in a computer-generated film has over two million pixels and can have over forty million polygons. The tremendous number of calculations involved makes computers necessary, but without mathematics the computers wouldn.t know what to calculate. Said one animator, ". . . it.s all controlled by math . . . all those little X,Y.s, and Z.s that you had in school - oh my gosh, suddenly they all apply."

For More Information: Mathematics for Computer Graphics Applications, Michael E. Mortenson, 1999.