- Boundary rigidity for planet Earth
- Michael Atiyah in the Metropolitan Museum
- Self-assembly of giant polyhedral molecules

Davide Castelvecchi's "News" piece in *Nature* (February 10, 2017), "Long-awaited mathematics proof could help scan Earth's innards," has the subhead: "Proposed solution to geometry puzzle allows an object's structure to be determined from limited information." The geometry puzzle is the *Boundary rigidity conjecture*, and the proposed solution is in an as yet unpublished (unposted, even) paper by Andras Vasy, Plamen Stefanov (both at Stanford), and Gunther Uhlmann (University of Washington). (A related lecture by Vasy is available online). The Boundary rigidity conjecture implies in particular that for an object like planet Earth, the interior geometry in all its detail can be determined by knowing, for any two points on the surface, how long it takes for sound to propagate from one to the other through the interior. The exact conjecture was stated by René Michel in 1981 and can be found, for example, here.

Castelvecchi reminds us that "In the case of seismic waves --generated by events such as earthquakes-- the differing properties of Earth at varying depths mean that the shortest path for such waves is usually not a straight line, but a curved one. Since the early 1900s, geophysicists have used this fact to map the planet's internal structure, and this is how they discovered the mantle and the inner and outer cores."

Diagrammatic image of Earth showing "concentric layers" in the interior. Image from Lawrence Berkeley National Laboratory Research Review, used with permission.

In fact, the authors assumed that the object they studied was like the Earth in this respect. "Their assumption, which differed from Michel's, was that the curved space, or manifold, is structured with concentric layers. This allowed them to construct a solution in stages. 'You go layer by layer, like peeling an onion,' says Uhlmann. For practical applications, this means that researchers will not only know that there is a unique solution to the problem; they will also have a procedure to calculate that solution explicitly."

along with Enrico Bombieri, Simon Donaldson, Freeman Dyson, Murray Gell-Mann, Richard Karp, Peter Lax, David Mumford, Stephen Smale, and Steven Weinberg. They're on view in the exhibit "Picturing Math: Selections from the Department of Drawings and Prints", January 31-May 1, 2017. The Museum acquired in 2015 a set of art prints of mathematical formulas and diagrams; as described on their website: "The portfolio Concinnitas is the result of a collaboration between Dan Rockmore (Professor of Mathematics, Dartmouth College); ten renowned mathematicians and physicists; the publisher Robert Feldman (who earlier produced seminal Conceptual art print portfolios); and the New York printing house Harlan & Weaver. ... Each print represents a participant's answer to Rockmore's prompt to represent the 'most beautiful mathematical expression' they had encountered in their work or study."

Michael Atiyah, *The Index Equation*, Metropolitan Museum of Art, John B. Turner Fund, 2015. Image courtesy of CONCINNITAS published by Parasol Press, Ltd.

On the Concinnitas Project website, each print comes with a statement from the artist. Sir Michael starts with a quote from Herman Weyl ("My work has always tried to unite the true with the beautiful and when I had to choose one or the other I usually chose the beautiful.") He comments: "Since mathematics is the most precise of the sciences and is devoted to finding out the truth, Weyl's statement might appear bizarre and even provocative -- a tongue-in-cheek remark. But I believe Weyl was quite serious. The apparent paradox in Weyl's dictum is that objective truth is what we all search for but, at any stage, it is uncertain and provisional. But beauty, which is a subjective experience 'in the eye of the beholder,' is the light we follow in the hope that it is leading us to truth." Some other excerpts:

- "The equation [known today as the Atiyah-Patodi-Singer Theorem, or simply the ASP Theorem] I have chosen to epitomize beauty in my own work has the grandeur that comes from a rich history and multiple connections to different branches of mathematics: topology, geometry, analysis."
- "My equation is about forty years old and since that time it has found fascinating and totally unexpected applications in fundamental physics, which Weyl would have both understood and appreciated."
- "Finally, on a more personal note, my equation embodies extensive collaboration with many of my colleagues: Fritz Hirzebruch from Bonn, Raoul Bott from Harvard, Is Singer from MIT and Vijay Patodi from Bombay, who like many talented poets died at a tragically early age. Beauty is a human experience and is best shared with friends."

Is there a 'most beautiful mathematical expression' in your experience? The Concinnitas Studio would like to hear about it.

Daishi Fujita, Makoko Fujita and four co-authors published "Self-assembly of tetravalent Goldberg polyhedra from 144 small components" in *Nature* for December 22, 2016. They report the synthesis of a spherical cage-like molecule, which they call $M_{48}L_{96}$, in which the atoms and bonds are arranged in the topology of a semi-regular polyhedron (8 triangles and 42 squares); this is the largest such structure ever realized.

Electron density map for $M_{48}L_{96}$, computed from X-ray cristallographic data. Image from *Nature* **540** 563-566, used with permission.

The article reads like an adventure story. The authors had been building ever larger molecular shells by letting Palladium (Pd^{2+}) ions and *ligands* (molecules with two binding sites for metal ions) self-assemble in a solution. Each Palladium ion binds to four ligands, so these structures are all *tetravalent* when considered as graphs, and they all have the general formula $M_nL_{2n}$: $n$ metal ions and $2n$ ligands. The ligands have a flat V-shape; the corner angle determines the concavity of the resulting surface. They had started with the octahedron $M_6L_{12}$ and the first three of the four tetravalent Archimedean solids, the cuboctahedron $M_{12}L_{24}$, the rhombicuboctahedron $M_{24}L_{48}$, and the icosidodecahedron $M_{30}L_{60}$.

"While dedicating effort towards the self-assembly of the next target --the $M_{60}L_{120}$ rhombicosidodecahedron-- we unexpectedly encountered an 'undefined' polyhedron with $M_{30}L_{60}$ composition and the topology depicted in Fig. 2. This metal complex is a seemingly isotropic polyhedron consisting of 8 triangles and 24 squares, clearly different from the isostructural $M_{30}L_{60}$ icosidodecahedron and with a high-symmetry topology that does not belong to that of the Archimedean solids. This polyhedron does not have a mirror plane and, unlike the Archimedean solids, features molecular chirality defined by its topology."

Figure 2 (detail) from *Nature* **540** 563-566, showing the chemical structure of the novel $M_{30}L_{60}$ molecule, used with permission.

The authors located this structure in a recently established list of "face regular" polyhedra, which in particular generalizes the Goldberg polyhedra (buckyballs, etc.; they have twelve pentagons and hexagonal faces otherwise) to polyhedra based on the octahedron, with eight triangles and squares otherwise. That family starts with the cuboctahedron and the rhombicuboctahedron; then comes the new $M_{30}L_{60}$ observed by the authors. The authors worked out more of the family: $M_{48}L_{96}$, $M_{54}L_{108}$, ... and "decided to target the predicted (but as yet unobserved) $M_{48}L_{96}$ complex."

The actual synthesis and detection was a tour de force: the authors had calculated "that $M_{30}L_{60}$ could be converted to $M_{48}L_{96}$ under suitable conditions, prompting us to examine a range of self-assembly conditions (by changing the concentration, reaction time, temperature and so on) and the use of modified ligands." And the only way to confirm the presence of $M_{48}L_{96}$ was by X-ray crystallography, which involved carefully observing, under a microscope, the morphology of single crystals that had taken 2-3 months to grow. Most of the crystals gave diffraction patterns that looked like $M_{30}L_{60}$, but some were different, and careful analysis proved them to be, in fact, $M_{48}L_{96}$.

The authors remark at the start that "Such molecules --for example, spherical virus capsids-- are prevalent in nature, which suggests that the difficulty in designing these very large self-assembled molecules is due to a lack of understanding of the underlying design principles. "

Tony Phillips

Stony Brook University

tony at math.sunysb.edu