Geometric models based on cubes can provide insights into complex physical and biological systems, from planning the motions of reconfigurable robots to deciphering genetic family trees. In a mathematical context, using geometric pictures to represent abstract algebraic objects such as groups can often tell us something new about the objects, Ruth Charney of Brandeis University said.
On Oct. 14, at the MAA's Carriage House Conference Center, Charney spoke about using cubes, not only in three but also in higher dimensions, as models to represent a variety of systems. The resulting geometric spaces typically consist of complex assemblages of cubes, in three, four, five, or higher dimensions.
"The geometry of these spaces is strange, complicated, and a lot of fun to study," Charney noted. By studying the geometry, we can learn about the system, whether physical or mathematical.
As an example of how to build a geometric model of a system, Charney described a cubic representation of the motions of a planar robot made up of an array of hexagons that can each rotate about a vertex. The independent motions of such a reconfigurable robot can be represented as higher-dimensional cubes, with each vertex standing for a different robot configuration and each edge for a single move between configurations. Each "cube" is then a set of independent moves.
Such geometric spaces can be very complicated, Charney said. But by studying these configuration spaces, we can answer questions, for instance, about how efficiently and quickly robotic moves can occur.
Charney then described methods we can use to characterize the resulting cubical complexes in any dimension, from finding shortest paths (geodesics) and defining curvature to locating centers and centroids.
For a space that models a physical system, these geometric properties correspond to physical properties, she said. A geodesic, for example, might correspond to the most efficient path from one configuration to another.
"Understanding these geometries has useful applications both within and beyond mathematics," Charney concluded.
Charney's presentation was one in a series of public lectures hosted by the MAA, with the support of the National Security Agency, to showcase current trends in mathematics and the relationship between mathematics and broader scientific and engineering endeavors.
This MAA Distinguished Lecture was funded by the National Security Agency.