Part of the Distinguished Lecturer in Mathematics series, Professor Benson Farb of the University of Chicago recently visited the Boston College campus to lecture on “Geometry and the Imagination”. In addition to being a celebrated mathematics professor, Farb has also written several books, including A Primer on Mapping Class Groups, Problems on Mapping Class Groups and Related Topics, and Noncommutative Algebra. Moreover, he has collaborated with other mathematicians on several papers, is on numerous editorial boards, and has received a Faculty Award for Excellence in Graduate Teaching from the University of Chicago.
Professor Farb began his lecture my stating his target theme: “Geometry and geometric reasoning underlie all of science.” Jokingly, he added that his ultimate goal would be to “confuse” the audience, explaining that one becomes confused only when he truly contemplates a subject.
Like a true mathematician, Farb divided his lecture into three distinct sections, the first being “Symmetry.” He introduced the concept of symmetry by posing “The River Problem” to the audience. To solve the problem, one needed to determine the most efficient way to move first from the point A to the line labeled “river,” and then from the river to the second point, B. Simply reflecting the point B over the river line so that the two points, B and B’, were symmetrical about the line, Farb illustrated how, by drawing a straight line from A to B’, one could easily designate the point at which to reach and depart from the river.
His preliminary example demonstrated the way in which symmetry can effortlessly solve problems that appear perplexing. Farb underlined the value of symmetry with his statement that “there is symmetry inherent in the laws of physics.” To support this assertion, Farb cited the conservation of angular momentum, electric fields, and the “method of images” as elements of physics that, when depicted, clearly exemplify the role of symmetry in physical science.
Secondly, Professor Farb attacked the subject of “Handedness (left vs. right).” He provided the audience with pictorial examples of two-dimensional mirror images. The first example depicted an upside-down L-shape and its mirror image, both “fundamentally different images” because the original shape was chiral, or “not super imposable onto its mirror image.” On the contrary, the second example displayed a T and its mirror image, both super imposable on one another and therefore achiral.
Addressing three-dimensional mirror images, Farb questioned why, when “you put up your left hand, the mirror puts up its right hand,” and, furthermore, “why left and right, but not up and down?” His questions provoked laughs, however they certainly fostered great contemplation and, of course, confusion.
Though a fundamental mystery of the universe, the issue of left versus right has become easier to understand because of geometry and scientific investigation. Farb revealed to the audience the fact that “almost all amino acids found in living things are left-handed” and that this fact, though seemingly trivial, “has huge, huge, huge implications for us.” According to Farb, left and right versions of molecules “interact differently with the human body.”
For instance, menthol and peppermint are mirror images of one another, yet they give off quite different odors. More seriously, the right-handed version of the drug thalidomide treats morning sickness, but the left-handed version causes severe deformity in babies; because our bodies convert the right-handed version to the left-handed version, thousands of children were born deformed when the drug was being used by pregnant mothers. Fortunately, the grave affects of the drug were discovered and its circulation was stopped.
The third and final phase of Farb’s lecture was devoted to “The Fourth Dimension and (far) beyond.” Farb began this section by addressing the way coordinates are specified in the first, second, third, fourth, fifth, and nth dimensions. He used the airplane as an example of an object that is located with five coordinates, one each for latitude, longitude, altitude, time, and speed. Though rather impossible to represent pictorially, the fifth dimension in which the airplane exists can be expressed numerically by these geometric coordinates because “geometry is fundamental” to the problem of location.
Additionally, Farb toyed with the perspectives of objects in each dimension. Succeeding in his goal to confuse the audience, he pondered how we, members of the third dimension, would perceive a visitor from the fourth dimension as he fleetingly passed through our world.
Both funny and thought-provoking, Professor Benson Farb’s discussion on the fundamentality of geometry truly unearthed the way in which “geometry and geometric reason underlie all of science.” The complicated questions of the faculty members following the lecture demonstrated the wealth of mathematical interest in the Boston College math department; clearly, Farb achieved his goal of engendering in the minds of his listeners a monumental contemplation.
