Jun 102013
The first Mathematical Crystallography minisymposium was entitled “Geometric Foundations”.
Frank Morgan talked about Minimal Interface Structures. Just as soap bubbles in 3-space assume the shape of minimal area enclosing the fixed volume of air they contain (subject to constraints), so a soap bubble in a 3-torus will assue such a shape of minimal area within that torus. Using that 3-torus, one kind generate periodic structures minimizing surface area (subject to constraints). |
Egon Schulte talked about Polyhedral Geometries and Symmetry. Branko Grunbaum’s “skeletal” approach to polyhedra presents a polyhedron as a graph embedded in 3-space: vertices are vertices and edges are edges and a distinguished family of cyclic subgraphs (“rings” to chemists) are faces. This approach includes several infinite (periodic) polyhedra. |
Ma. Louise N. de las Penas talked about Nanostructures Arising from Crystallographic Tilings. Given a graphene sheet and two appropriate vectors lying on its plane, one to define a unit and one to define a gluing identity, one can construct a graphene tube. Similarly, one may construct a torus out of the tube. Other manifolds may be constructed similarly, and in the ensuing discussion, it was proposed that algebraic and other mathematical criteria might determine when a graphene (or other layer) sheet may be shaped into a given manifold. |
Bernd Souvignier talked about Capturing the Essence of Infinite Graphs in Quotient Graphs. Given a graph G, and a group of automorphisms A, one may define a quotient graph G/A whose vertices and edges are the orbits of the vertices and edges of G. Given sufficient information about how G/A articulates with G, one may go the reverse direction, constructing G out of G/A (and the articulation information). |
Then we went to lunch at Reading Terminal Market.
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