Optimal stacking of oranges.
The spherepacking problem
The Kepler conjecture is the assertion that the simple scheme of stacking oranges typically seen in a supermarket has the highest possible average density, namely pi/(3 sqrt(2)) = 0.740480489…, for any possible arrangement, regular or irregular. It is named after 17thcentury astronomer Johannes Kepler, who first proposed that planets orbited in elliptical paths around the sun.
Hales’ proof of the Kepler conjecture
In the early 1990s, Thomas Hales, following an approach first suggested by Laszlo Fejes Toth in 1953, determined that the maximum density of all possible arrangements could be obtained
Continue reading Researchers use “magic functions” to prove spherepacking results
Introduction
Right off, it may not sound like pi, climate change denial and youngEarth creationism have much in common. In fact, there is an important connection. Here is some background.
Credit: Michele Vallisneri, NASA JPL
Computing pi
Pi = 3.1415926535…, namely the ratio between the circumference of a circle and its diameter, has fascinated not only mathematicians and scientists but the public at large for centuries. Archimedes (c.287–212 BCE) was the first to present a scheme for calculating pi as a limit of perimeters of inscribed and circumscribed polygons, as illustrated briefly in the graphic to the right (see
Continue reading Pi, climate change denial and creationism
The Chudnovsky formula for Pi (Credit: Craig Wood)
AI’s rocky start
The modern field of artificial intelligence (AI) began in 1950 with Alan Turing’s landmark paper Computing machinery and intelligence, which outlined the principles of AI and proposed a test, now known as the Turing test, for establishing whether AI had been achieved. Although early researchers were confident that AI systems would soon be a reality, inflated promises and expectations led to the “AI Winter” in the 1970s, a phenomenon that sadly was repeated again, in the late 1980s and early 1990s, when a second wave of AI systems
Continue reading Google AI system proves over 1200 mathematical theorems
Credit: Wikimedia
The reproducibility crisis in science
Recent public reports have underscored a crisis of reproducibility in numerous fields of science. Here are just a few of recent cases that have attracted widespread publicity:
In 2012, Amgen researchers reported that they were able to reproduce fewer than 10 of 53 cancer studies. In 2013, in the wake of numerous recent instances of highly touted pharmaceutical products failing or disappointing when fielded, researchers in the field began promoting the All Trials movement, which would require participating firms and researchers to post the results of all trials, successful or not. In
Continue reading Phacking and scientific reproducibility
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Credit: MathIsFun.com
The discovery of decimal arithmetic
The discovery of decimal arithmetic in ancient India, together with the wellknown schemes for long multiplication and long division, surely must rank as one of the most important discoveries in the history of science. The date of this discovery, by an unknown Indian mathematician or group of mathematicians, was recently pushed back to the third century CE, based on the recent dating of the Bakhshali manuscript, but it probably happened earlier, perhaps around 0 CE.
Arithmetic on modern computers
Computers, of course, do not use
Continue reading An n log(n) algorithm for multiplication
LENR: Science or pseudoscience?
As chronicled in the earlier Math Scholar blog, a community of researchers has been pursuing a new green energy source, known as “low energy nuclear reaction” (LENR) physics, or, variously, “latticeassisted nuclear reaction” (LANR) physics or “condensed matter nuclear reaction” (CMNR) physics. This work harkens back to 1989, when University of Utah researchers Martin Fleischmann and Stanley Pons that they had achieved desktop “cold fusion,” in a hastily called news conference. After several other laboratories failed to reproduce these findings, the scientific community quickly concluded that the Utah researchers were mistaken, to put it mildly.
But
Continue reading LENR: A skeptical perspective
The Norwegian Academy of Science and Letters has awarded the Abel Prize, a mathematical award often regarded as on a level with the Nobel Prize, to Karen Uhlenbeck of the University of Texas, USA.
The award cited her work in geometric analysis, gauge theory and global analysis, which has application across a broad range of modern mathematics and mathematical physics, including models for particle physics, string theory and general relativity.
Her career began in the mid1960s, under the advisor Richard Palais. Palais had been exploring some connections between analysis (generalizations of calculus) and topology and geometry (the mathematical theory of
Continue reading Karen Uhlenbeck wins the Abel Prize
Credit: IPCC
The threat of climate change
The threat of climate change is emerging as the premier global issue of our time. As a recent report by the Intergovernmental Panel on Climate Change (IPCC) grimly warns, even a 1.5 degree C (2.7 degree F) rise in global temperatures would have “substantial” consequences, in terms of extreme weather, damage to ecosystems and calamitous impact on human communities. But limiting the increase to 1.5 degree C will still require a wrenching change away from fossil fuels and an equally wrenching realignment of global economies, all over the next decade or as
Continue reading LENR energy: Science or pseudoscience?
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Mordell’s cube sum problem
In 1957, BritishAmerican mathematician Louis Mordell asked whether, given some integer $k$, there are integers $x, y, z$ such that $x^3 + y^3 + z^3 = k$. Like Fermat’s last theorem, this problem is very easily stated but very difficult to explore, much less solve definitively.
Some solutions are easy. When $k = 3$, for instance, there are two simple solutions: $1^3 + 1^3 + 1^3 = 3$ and $4^3 + 4^3 + (5)^3 = 3$. It is also known that there are no solutions in other cases, including
Continue reading New result for Mordell’s cube sum problem
Once again Pi Day (March 14, or 3/14 in North American notation) is here, a day when both professional mathematicians and students alike celebrate this most famous of mathematical numbers.
San Francisco’s Exploratorium is featuring several events, culminating with a “Pi Procession” at 1:59pm Pacific Time (corresponding to 3.14159) and pie served at 2:15pm. The website teachpi.org lists 50 ideas to make Pi Day “entertaining, educational, tasty and fun.”
For this year’s Pi Day festivities, the Math Scholar blog presents a Pi Day crossword puzzle (see below), created by the present author. We will announce the first correct solver! Send
Continue reading A Pi Day crossword puzzle

