What's Happening in Mathematics


Packing Tetrahedrons



Over 2300 years ago the Greek philosopher Aristotle stated that identical regular tetrahedrons pack together perfectly, filling space with no gaps. It took 1800 years before it was pointed out that he was wrong. In 2006 Salvatore Torquato and John H. Conway reported the best packing they could find: it filled less than 72% of space — less than the best possible packing for identical spheres, which is known to be 74%. However, Elizabeth Chen then improved the density for tetrahedrons to 78%. Sharon Glotzer and colleagues wrote a computer program and discovered a periodic structure with density over 85%; its repeating pattern consists of 82 tetrahedrons. A group at Cornell found an equally dense packing with a repeating pattern of just four. This was tweaked to get the density to 85.55%. Now Chen has gone even better: 85.63%. So tetrahedrons lack a lot more efficiently than spheres.

Rubik's God Number is 20



The Rubik cube can be in any one of 43,252,003,274,489,856,000 states. The 'God number' (so called because an omniscient deity would know it, and for a long time nobody else would...) is the smallest number of moves that will solve the puzzle from any starting position. Untl recently the best we knew was that this is between 18 and 26.
Now a team headed by Morley Davidson (Ohio State University) has proved that it is exactly 20. The proof involves mathematical analysis to reduce the number of positions that need to be considered, and a massive compter search carried out on equipment provided by Google.

How Linked Networks Fail



Our world is networked. Transportation, the Internet, even our own biology, form highly interconnected networks. Mathematicians have been finding out how susceptible various types of network are to failure, when some of the nodes or connections are broken. Most studies focus on a single network, and ask what proportion of connections must break in order for it to fail. But real systems often involve several interdependent networks—for example, the road network, the rail network, and the aircraft network.

It might seem that having several networks available provides useful backup and prevents failure. Sergey Buldyrev and coauthors have shown the exact opposite. Two or more interdependent networks are much less robust than a single network of the same type. Why? Because a local failure in one can trigger more extensive failures in the others, which in turn cause more failures in the first network, and so on in an ever-growing cascade.


Consider a Spherical Cow...
   


    So goes the old joke about the farmer who hired a mathematician to advise him on how to increase milk production. Now University of Vermont mathematician Peter Dodds has given the spherical cow a new lease of life. The question is: how does an animal's size vary with its metabolic rate? The classic answer is a 3/4 power law, obtained in 1932 by the Swiss chemist Max Kleiber. He plotted the body weight of 13 mammals against their resting metabolism on a logarithmic scale. The result was a line with slope 0.73. He rounded this up to 0.75, and the famous 3/4 power law came into being.
    The exponent 3/4 has long puzzled biologists (and mathematicians) because the obvious value is 2/3. To see why, consider a spherical cow. The surface area of this rotund beast varies as the square of its radius, but the volume increases as the cube of the radius. The metabolic rate should be proportional to the surface area, because heat is lost through the surface; the mass should be proportional to the volume. Putting this together, we get a 2/3 power law.
    Many explanations have been invoked for the 3/4 power law rather than the 2/3 'spherical cow' version. The most recent invokes a fractal cow instead, considering the network of blood vessels, but this approach has proved controversial. Dodds argues that Kleiber's data was poor and the 3/4 power law is a myth: the data fit a 2/3 power law just as well. By reworking the fractal approach, but making different assumptions, he obtains a 2/3 power law. So the fractal cow agrees with the spherical cow.

How Hot is Your Pepper?



    A new mathematical model makes it possible to determine how hot a chili pepper is. The sensation of 'heat' is caused by capsaicinoid compounds in the pepper, which bind to taste receptors in the mouth and throat. Previous ways to find out how hot a pepper is involved detailed measurements of its capsaicinoids. The new method, invented by Kenneth Busch at Baylor University, fits data from known capsaicinoids and is simpler and quicker.

Golden Ratio and E8 in Quantum Mechanics



    The golden ratio (1.618034...) is one of the most intriguing numbers in mathematics, and entire books have been written about it. Many myths surround this number, such as its role in aesthetics, which is generally exaggerated. However, its mathematical importance is undeniable.
    A mathematical prediction that this ratio should occur in a particular quantum-mechanical system has now been verified by Radu Coldea and colleagues at Oxford University, in cobalt niobate. By applying a magnetic field, they made the state of a magnetic chain in the crystal become 'quantum critical'. The two lowest frequencies of vibration of the chain were predicted to be in the golden ratio, and the experiments confirm this. The detailed theory involves the exceptional Lie group E₈, a strange 248-dimensional group of symmetries that keeps turning up in quantum physics.

The Trillion Triangle Problem



    Bill Hart of the Warwick University Mathematics Institute is part of a team of mathematicians from North America, Europe, Australia, and South America who have found the congruent numbers up to one trillion. This is a significant contribution to a thousand-year-old mathematical problem posed by al-Karaji: find the integers n for which there is a rational square a2 such that a2+n and a2-n are also rational squares.
    Although the equivalence is not obvious, congruent numbers can also be defined as those integers which are areas of right-angled triangles with rational sides. For example: (41/12)²+5=(49/12)², (41/12)²-5=(31/12)² so 5 is a congruent number.The corresponding triangle has sides 20/3, 3/2, and 41/6. The familiar 3-4-5 triangle has area 1/2 x 3 x 4 = 6, so 6 is a congruent number.
    Among the smaller integers, 1, 2, 3, 4 are not congruent, but 5, 6, and 7 are. It is not straightforward to decide whether a number is congruent—for instance, 157 is congruent, but the simplest right-angled triangle with area 157 has hypotenuse
                2244035177043369699245755130906674863160948472041
divided by
                8912332268928859588025535178967163570016480830
    The best test currently known depends on an unproved conjecture, the Birch—Swinnerton-Dyer Conjecture, which is one of the Clay Millennium Mathematics Prize problems, with a million dollars offered for a proof or disproof.

What's Purple and Commutes?



    Pablo Flores Martínez at the University of Granada has collected 4000 mathematical jokes and cartoons over the past 14 years, and he uses them in his teaching.

Two-Dimensional Barcodes



     Ordinary barcodes are 1-dimensional, in the sense that the information they represent is encoded as a line of vertical bars. A new generation of barcodes, which encodes information in a two-dimensional array, is already being used in North America and Europe. The aim is to make it harder to counterfeit postage stamps and other similar items. Tony Phillips explains...

Supercomputers and the Big Bang



    A new mathematical model is helping to untangle some of the scientific riddles associated with the Big Bang theory of the origin of the universe. This model incorporates many physical factors, such as gas motion, radiation transport, chemical kinetics, star clustering, and dark matter. The main new feature is that all of these processes are tightly coupled in the model. Daniel R. Reynolds at SMU has collaborated with astrophysicists at the University of California at San Diego as part of an NSF project to simulate cosmic reionization, believed to have occurred from 380000 years to 400 million years after the birth of the universe.

123 Billion More Digits of π



    Computer scientist Fabrice Bellard has calculated π to 2.7 trillion decimal digits, 123 billion more than the previous record. Unlike most recent calculations of the digits of π, his method did not use a supercomputer: instead, it ran on a standard desktop machine in Linux. It took 131 days, whereas the previous record (obtained by Daisuke Takahashi at the University of Tsukuba) took 29 hours.
    Although mathematicians do not consider the digits of pi to have any special significance, beyond a conjecture that they satisfy all the standard statistical tests for being random, calculations of this kind are an effective way to test computers, and the algorithms used often have independent interest. It's how to get the result that matters, rather than the result itself.