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Tony Phillips' Take on Math in the Media A monthly survey of math news |

This month's topics:

"Children and adults of an indigenous group from Amazonia use geometrical concepts despite the lack of specific words to describe them." This is the one-sentence abstract of a report ("Core Knowledge of Geometry in an Amazonian Indigene Group") in the January 20 2005 *Science*, written by a team including Stanislas Dehaene (Collège de France) and Elizabeth Spelke (Psychology, Harvard). The report was picked up by Nicholas Balakar in the *New York Times* for January 24. From the report's abstract: "We used two nonverbal tests to probe the conceptual primitives of geometry in the Mundurukú, an isolated Amazonian indigene group. Mundurukú children and adults spontaneously made use of basic geometric concepts such as points, lines, parallelism, or right angles to detect intruders in simple pictures, and they used distance, angle, and sense relationships in geometrical maps to locate hidden objects." Balakar shows us one of the nonverbal questions:

Which of these images seems "weird" or "ugly"? |

There is no Mundurukú word for triangle (nor for rectangle or square). Yet when Mundurukú children and adults were shown this table, 64% of the children and 83% of the adults spotted the flat triangle as the intruder. For similarly aged American groups, the numbers were 64% and 86%. (Data from e-mail communication between the authors and Balakar). Balakar asked Prof. Dehaene if these findings confirmed Socrates' famous contention that geometric concepts are innate. The answer came back by e-mail: "Our current thinking is that the human brain has been predisposed by millions of years of evolution to 'internalize,' either very early on or through very fast learning, various mental representations of the external world, including representations of space, time and number. ... I have proposed that such representations provide a universal foundation for the cultural constructions of mathematics." The title for Balakar's piece was "Mastering the Geometry of the Jungle (or Doin' What Comes Naturally)".

The *New York Times* ran a "Critic's Notebook" column by Virginia Heffernan on December 24, 2005. The subject was the popular game show "Deal or No Deal," and the title was "A Game Show for the Probability Theorist in Us All." Here's how the game works (you can test it out).

- Twenty-six known amounts of money, ranging from one cent to one million dollars, are (symbolically) randomly placed in 26 numbered, sealed briefcases. The contestant chooses a briefcase. The unknown sum in the briefcase is the contestant's.
- In the first round of play, the contestant chooses 6 of the remaining 25 briefcases to open. Then the "banker" offers to buy the contestant's briefcase for a sum based on its expected value, given the information now at hand, but tweaked sometimes to make the game more interesting. The contestant can accept ("Deal") or opt to continue play ("No Deal").
- If the game continues, 5 more briefcases are opened in the second round, another offer is made, and accepted or refused. If the contestant continues to refuse the banker's offers, subsequent rounds open 4, 3, 2, 1, 1, 1, 1 briefcases until only two are left.
- The banker makes one last offer; the contestant accepts that offer or takes whatever money is in the initially chosen briefcase.

The psychology is what makes the game fun. As Heffernan explains: "So far, no game theorist from the Institute for Advanced Study has appeared to try his hand at 'Deal or No Deal' and play as a cool-headed rationalist. Instead the players on the American show are, like most game-show contestants, hysterics." In fact the three scientists at Erasmus University who conducted an exhaustive analysis of the decisions made by contestants in the Dutch version of the game (jackpot 5 million Euros) remark that "For analyzing risky choice, 'Deal or No Deal' has a number of favorable design features. The stakes are very high: ... the game show can send contestants home multimillionaires -or practically empty-handed. Unlike other game shows, 'Deal or No Deal' involves only simple stop-go decisions that require minimal skill or strategy. Also, the probability distribution is simple and known with near-certainty. Because of these features, 'Deal or No Deal' seems well-suited for analyzing real-life decisions involving real and large risky stakes." Their report is available online.

Cuboctahedral vesicles in eukaryotic cells

Part of a micrograph of Sec13/31 cages preserved in vitreous ice. The cages are approximately 600Å (0.06 microns) in diameter; their images show the planar projection of their cuboctahedral structure. Image from |

A eukaryotic cell is a complex, three-dimensional organism. Just as our food is ingested in one place and moved to another for processing, with the nutrients then ferried about the body by the bloodstream, so in a cell's internal economy a critical role is played by transportation. The agents of intracellular transport are vesicles: molecular cages that enclose their cargo and move it from A to B. A paper in the January 12 2006 *Nature* explores the structure of one type of vesicle: those whose skin is made from the coat protein complex II, or COPII. The authors (a Scripps Research Institute team of 8, led by Scott Stagg) explain that the structural part of COPII consists of a lattice formed by the protein complex Sec13/31. Using electron cryo-microscopy, they determined that the most elementary cages formed by Sec13/31 have the structure of a cuboctahedron, but they suggest that in order to enclose larger cargoes, the same units could organize into the small rhombicuboctahedron, the icosidodecahedron or the small rhombicosidodecahedron. These semi-regular solids all share with the cuboctahedron (and the octahedron) the property that four edges meet at each vertex. The condition corresponds to the assymmetry in the molecular realization of the Sec13/31 complex: the two ends are different, so it cannot assemble into a network with odd-ordered vertices.

The three axes of symmetry of the Sec13/31 cage. Each edge is a Sec13/31 protein complex. The color (blue-green-yellow) encodes distance from the cage center. Note the assymmetry in the edges. Cage diameter approximately 600Å. Image from |

Tony Phillips

Stony Brook University

tony at math.sunysb.edu