During her visit to Liverpool, Zuzka—my fiancée Lina’s mom—gifted us these glass geodesic polyhedra. When sunlight shines through them, it scatters across our living room walls, creating miniature rainbows.
While preparing to travel to ICERM’s workshop, Matroids, Rigidity, and Algebraic Statistics, I noticed that the geodesic polyhedra formed a dilated triangular lattice pattern on the wall.
This is, in fact, the basic idea behind X-ray diffraction, discovered by Max von Laue in 1912. When electromagnetic waves-such as X-rays-pass through a crystal, the symmetrical arrangement of atoms causes the waves to diffract. When these waves hit an impenetrable surface, we can photograph the resulting wave interference pattern, composed of both constructive and destructive interactions.
One key challenge in X-ray crystallography is reconstructing a crystal’s atomic structure from its diffraction pattern—an example of an inverse problem. For instance, the mini-rainbow triangular lattice in the photo arises from the six-fold rotational symmetry of our two glass tetrahedral geodesic polyhedra.
One can perceive the situation at hand as a stereographic projection of a spherical model (as in Magnus Wenninger‘s book, Spherical Models) onto the two-dimensional plane, thereby relating the infinite triangular lattice to the “sphere of reflection”-i.e., Ewald’s sphere.
The idea of hanging transparent geodesic shapes in front of the window—creating rainbows across the room—came from Lina’s dad and my teacher, Oleg Šuk. He experimented with different materials, including an acrylic geodesic heart (with “acrylic” referring here to PMMA photonic crystal). I can’t help but wonder if the Šuk family is secretly guiding my work. See my previous posts, Chiral Interaction Energy Ground States and Yayoi Kusama’s Chandelier of Grief: Symmetry in Art and Science, for additional context.