key: cord-1004369-n1csovq8 authors: Senechal, Marjorie title: The Gömböc Pill, Continuing ... date: 2022-05-09 journal: Math Intell DOI: 10.1007/s00283-022-10189-8 sha: c816a56f4c2ea3f6229acb85758a68db956fa8b8 doc_id: 1004369 cord_uid: n1csovq8 nan the monostatic 19-hedron shown in Figure 1 . In addition to its single stable face, the Conway-Guy 19-hedron can also balance on four of its vertices, but only for a split millisecond. Then it rolls over. Those vertices are positions of unstable equilibria. Together with Péter Várkonyi and Gábor Domokos, we say that a body with j stable and k unstable equilibria is of class {j, k} [20] . Those of classes {1, k} and {j, 1} are called monostable and monounstable, respectively, and the two families of classes are jointly referred to as monostatic. The Conway-Guy 19-hedron is the first known homogeneous monostable polyhedron in class {1, 4} [12] . This idea not only sparked research to find monostable polyhedra with fewer faces [2, 16] , but also initiated a search for mono-unstable polyhedra with a minimal number of vertices [3] . If j = k = 1 , the body is called mono-monostatic. There are no homogeneous mono-monostatic bodies in two dimensions, because every convex homogeneous planar disk has at least two stable and two unstable equilibrium points This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest: "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, extracurricular educational activities (math camps, math museums, math clubs), andmore. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others. Submissions should be uploaded to http:// tmin. edmgr. com or sent directly to Marjorie Senechal, MathCommunities@ gmail.com. • The Mathematical Intelligencer 2 [13] . But as Vladimir Arnold suggested to Gábor Domokos in 1995, convex homogeneous mono-monostatic bodies might exist in three-space [9] . He was right! By 2006, Domokos and Várkonyi had found a continuum of them. One is the now well known gömböc shape, with curved surfaces joined at sharp edges. A mono-monostatic polyhedron with 21 point masses at its vertices and also having 21 faces has been described recently [11] . Despite the difficulty1 in pronouncing its name, the gömböc has inspired the imaginations of people interested in mathematics, art, biology, and medicine. Its fans are an international, interdisciplinary community.2 It has even been immortalized in a work of public art (Figure 2 ). 1For non-Hungarian speakers. For English speakers, "gum-buts" is close. 2See https:// gomboc. eu/ en/ inspi ration/. As the cover of the fall 2006 issue of this journal (Figure 3) suggests, the gömböc resembles the shells of certain turtles that can right themselves if they happen to find themselves on their backs. Domokos and Várkonyi showed that such shells are close to optimal for self-righting [14] . "Evolution solved a far-from-trivial geometrical problem and equipped some turtles with monostatic shells: beautiful forms which rarely appear in nature otherwise." The gömböc also drew the attention of an MIT-Harvard team of biomedical engineers, led by Giovanni Traverso and Robert Langer, who were researching new methods of drug delivery. The traditional methods, injection and ingestion, are often hindered by patient needle-phobia for the former and poor regimen compliance for the latter. In many cases, pills are ineffective because their contents are destroyed by stomach acid. But then the researchers learned about the gömböc (see, e.g., [8] ). Before Covid-19 had emerged from the market stalls in Wuhan, they had designed a gömböc-inspired insulin pill for type 1 diabetes [10, 17] ; see Figure 4 . When a gömböc pill is swallowed, it lands on its "feet," so to speak, like a roly-poly toy, with its stable base aligned with the stomach lining. A microneedle-made of compressed insulin in this case-is extruded, which injects its contents directly into the bloodstream, and then the capsule is excreted. The stomach has few pain receptors, and so the recipient doesn't feel the pinprick. Which brings us full circle: this team is creating gömböc capsules for Covid-19 vaccines. Figure 5 shows how they will work. Katalin Karikó, Robert Langer, and Drew Weissman will share the 2022 Spanish BBVA Frontiers of Science Award "for their contributions to messenger RNA (mRNA) therapeutics and delivery technology that enable our own cells to produce proteins for disease protection and treatment." Says Dr. Karikó: Science is built on the knowledge of researchers. We are constantly drawing ideas from the work of others. In biology, we look for analogous processes, we look for precedents for different things. That is why it is good to read as much as possible about different areas of science. All this knowledge is valuable for hypothesis-driven research.3 Exercise 1. Explain Figure 5 to the next person you see! It is a pleasure to thank Gábor Domokos and Katalin Karikó for their comments and suggestions on earlier drafts of this note. Oral mRNA delivery using capsulemediated gastrointestinal tissue injections On stability of polyhedra Monounstable polyhedra with point masses have at least 8 vertices Capsule for oral mRNA delivery developed and tested Stability of polyhedra, problem 66-12 Stability of polyhedra What shape is a loaded die? Microneedle gastric retention for drug delivery. BS thesis, MIT My lunch with Arnold The gömböc pill Conway's spiral and a discrete gömböc with 21 point masses Balancing polyhedra Static equilibria of planar, rigid bodies: is there anything new? Geometry and self-righting of turtles Modified mRNA vaccines A unistable polyhedron with 14 faces Pills give patients a shot inside the stomach Coronavirus medicine and gömböc: how are Hungarian scientists' inventions related? Hungary Today Mono-monostatic bodies: the answer to Arnold's question Static equilibria of rigid bodies: dice, pebbles, and the Poincaré-Hopf theorem