Geometric order in Nature's zoo
A still from Garrett Lisi's tour of the E8 root diagram. According to Lisi, the animation was made by piecing together projections of the E8 root system from R8 into R2, following a piecewise approximate-geodesic path of rotations of R8. The path was chosen to display, via projection to R2, some of the various subgroups of E8 relevant to physics. Specifically, the path goes from showing the E8 projection (originally drawn by hand by Peter McMullen) to a projection showing F4, to a projection showing G2, and back to the McMullen projection. The Elementary Particle Explorer was made subsequently, and allows the user to choose the projection to R2 by manipulating the H (Horizontal) and V (Vertical) vectors spanning the R2 (of the computer screen) in R8. Image courtesy of Garrett Lisi.
"Nature's zoo of elementary particles is not a random mishmash; it has striking patterns and interrelationships that can be depicted on a diagram corresponding to one of the most intricate geometric objects known to mathematicians, called E8." This is the lead caption from a Scientific American article by Garrett Lisi and James Weatherall (December, 2010); the title is "A Geometric Theory of Everything." The article is an ambitious survey of the gauge theories of elementary particles: how particles correspond to representations of the Lie groups of internal symmetries (gauge groups) of the corresponding theories. Lisi and Weatherall lead us on an ever-expanding path. They start with electromagnetism and the circle group U(1), electroweak theory and SU(2)×U(1), the standard model and SU(3)×SU(2)×U(1); then move through more hypothetical models: a Grand Unified Theory using SU(5), a theory based on Spin(11,3) and finally E8 Theory, based on the exceptional Lie group E8. There "every force, every known particle of matter and a clutch of additional particles that might account for cosmic dark matter" could be manifest in "this one exquisite shape," the E8 root diagram illustrated above. The authors explain how they circumvent a theorem that forbids combining gravity and the other forces in a single Lie group: "the theorem has an important loophole: it applies only when spacetime exists." The grand conceptual symmetry of the E8 Theory must always remain conceptual: "Our universe begins when the symmetry breaks."
Cellular automata, literally
An adder-with-carry implemented with special lines of yeast cells by Regot et al. Cells of strains 8, 11, 13, 15 16 are mixed in a culture. Inputs are doxycycline (DOX) and glucose (GLU), exciting (yellow) or inhibiting (grey). Strain 8 releases pheromone "purple" in the presence of GLU; strain 11 responds to "purple" by fluorescing green, but is inhibited by DOX; strain 13 releases pheromone "pink" in the presence of DOX; strain 15 responds to "pink" by fluorescing green, but is inhibited by GLU; strain 16 responds to "purple," in the presence of DOX, by fluorescing red. Image adapted from Regot et al., Nature 469 207-211.
Two articles in the January 13 2011 Nature described how logical circuits can be implemented with living cells. An international team led by Sergi Regot (UPF, Barcelona) worked with specially developed strains of yeast (analogous work with E. coli was reported by a UCSF group: Alan Tamsir, Jeffrey Tabor and Christopher Vogt). What the two projects had in common was "the compartmentalization of each elementary logic gate in a single cell. ...[E]ach cell type is defined by the dedicated logic operation that it performs on inputs. And, for information flow, upstream gates produce signalling molecules that can diffuse across space into receiver cells, where these chemical 'wires' act on the downstream gate." This quote from a "News & Views" piece in the same issue, by Bochong Li and Lingchong You (Duke). To show the potential of their technique, Regot's team implemented relatively complex logical circuits, in particular the binary adder illustrated here. The inputs to the adder are the presence (1) or absence (0) of doxycycline and glucose. The outputs correspond to the two digits of the sum: green fluorescence for a 1 in the units place, red fluorescence for a 1 in the twos place, implementing the table shown here on the left. Li and You remark that one challenge faced by these approaches is that in a growing cell population, "changes in cell densities may affect the strength of communication," but they give both projects high marks for robustness in potential biological engineering applications. On the other hand, Tamsir et al. remark that this research may lead to a better understanding of biological processes: "The motif of multiple promoters in tandem drining the expression of a repressor is common in genomes, and the resulting NOR gates [the Regot team implements NOR with strains 3 and 4] may represent a ubiquitous fundamental unit of biological computation."
Cif america, the U.S. themed segment of the Guardian's "Comment is Free" multi-blog, ran a piece by their contributor Jonathan Farley on December 5, 2010: Sexing up mathematics does not compute, with the subheading: "Several films and TV shows have sweated to put an x rating into algebra, but it's mathematics that's sexy, not mathematicians." The illustration shows a chemistry lesson, but let's not quibble. Jonathan starts with the ruckus over Rites of Love and Math and flashes back to The Oxford Murders and Numb3rs, all of which, he claims, show a disproportionate concern with sex. More typical of the austere breed is the "noble lineage of brilliant mathematicians who probably never dated and who never married," e.g. Paul Erdös and G. H. Hardy, who wrote of Ramanujan "my association with him is the one romantic incident in my life." The conclusion: "I would like to believe that, by and large, you don't find sexy mathematicians like those in Numb3rs, The Oxford Murders, or Rites of Love and Math, because mathematicians don't need sex: our holy enterprise of sorting truth from error, of dealing with what is at the foundation not only of what is, but of what must be, doesn't leave time for romance. It is the stuff of which romance is made."
The rule of three (la règle de trois) is the cornerstone of French elementary education. And yet, hélas, many French adults seem to be incapable of solving proportion problems. Xavier Darcos, once Ministre de l'Éducation Nationale, was asked, on TV, "if four pens cost 2.42 Euros, how much do 14 pens cost?" He replied the equivalent of "I have no idea." That was back in 2008, but the flap endures. On November 20, 2010, Le Monde ran a piece by Maryline Baumard on the rule of three, "La règle de trois n'aura pas lieu" (a cute play on the title of Giraudoux' Trojan War drama). Baumard observes that faulty acquisition of the rule of three is endemic. The effects can be disastrous, in cuisine obviously, and how much worse at the level of national government. She interviews experts from the Task Force on the Psychopathology of Logico-Mathematical Activity (French acronym Gepalm), who explain the intrinsic difficulty of proportionality problems: they require the student to invent an intermediate, virtual step (in the unfortunate minister's case, this would have been: one pen costs 2.42/4 Euros); for many students, this is the first exposure to a two-step problem. And she quotes Marie Mégard, inspectrice générale de l'éducation nationale: "proportionality is an indispensable notion in everyday life and in the very exercise of citizenship. It is a difficult notion; the apprenticeship begins in elementary school; the mastery is reinforced during one's whole life."
Stony Brook University
tony at math.sunysb.edu