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Keplerian Nest of Polyhedral Structures in Different Symmetry Groups

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Keplerian Nest of Polyhedral Structures in Different Symmetry Groups Di Sun & Stan Schein designed the system as a comprehensive Physical Sciences︱ tool that enabled the physical modelling of any periodic structure rooted in the 7 crystal systems (triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic) and the 14 Bravais lattices (3-dimensional configurations Keplerian nest into which atoms can be arranged in crystals). The Universal Node system was essentially based on a system of cubic of polyhedral and octahedral symmetry. In 1966 it was not possible to injection mold the node with sockets, so the structures in different Universal Node was developed with 26 spokes that correspond to the 13 symmetry axes of the cubic/octahedral system. The spokes were shaped symmetry groups according to the symmetry axes of this system. The 2-fold axes, where a shape o Research being carried out by o a geometer, a polyhedron (plural collaborators are particularly interested appears identical after rotating 180 , Professor Di Sun, from Shandong polyhedra) is a three-dimensional in nested polyhedra. Their recent had 12 rectangular spokes. The 3-fold University, and Professor Stan shape with flat polygonal faces. investigation unites two research streams. axes, where a shape was repeated after T o Schein, from UCLA, unites two The faces meet at line segments called First, they synthesized and characterized a rotating 120 , had 8 triangular spokes. research streams. In the first, they edges, and the edges meet at points nest (Ag90) with three polyhedral shells in The 4-fold axes, where a shape was o synthesized Ag90, a Keplerian called vertices. The word is derived different symmetry groups. Second, they repeated after rotating 90 , had 6 square nest with one icosahedral and from the Greek ‘poly’ (many) and the connected this problem with one solved spokes. Three different types of branch two octahedral shells. In the Indo-European ‘hedron’ (face). Often by the late mathematician John Conway, matched the spokes of their respective second, they realized that the highly symmetric, polyhedra have great who used the Zometool to address a Figure 2a. The ball-and-stick mode of the triply nested polyhedral silver skeleton, viewed down symmetry axes. This shape coding nesting maximized the number aesthetic appeal, and they are common similar problem that had been implied by an axis through the front silver 5-gon. The three different shells are individually coloured. ensured that the Universal Nodes always From Su, Y.M., et al. Nat. Commun. 2020, https://doi.org/10.1038/s41467-020-17198-1 of shared axes of rotation, just in chemistry, biology, mathematics and Kepler’s model of our solar system. remained parallel during the assembly how the late mathematician John other disciplines. of networks. These structures of the Conway used the Zometool to THE BACKSTORY spokes and nodes built a symmetry explain how to optimally nest Di Sun, Professor at Shandong University, This study begins with Peter Jon Pearce, intelligence into the modelling system polyhedra of different symmetry Stan Schein, Professor at the California an American product designer, author, so that these symmetry principles could groups. Indeed, Conway had NanoSystems Institute of UCLA, and their and inventor. In 1966, he spent a year be experienced by the user. solved a problem implied by working with Buckminster Fuller, the Johannes Kepler’s model renowned inventor and visionary, on In the early 1970s Pearce produced of our solar system. illustrations for Fuller’s book, Synergetics this system as a building toy called (1975). Pearce’s own work focused on SuperStructures, a set with sticks (spokes) high-performance sustainable design, as and balls (nodes). His system made it easy he later demonstrated in his design of the to create ‘octahedral’ structures, including famous Biosphere 2, an American Earth octahedral Platonic solids, such as cubes system science research facility. and octahedra, and Archimedean solids, such as truncated octahedra, as well as The recipient of a research grant from the larger free-style structures. Graham Foundation in Chicago in 1965, Pearce began studying spatial geometry, ZOMETOOL crystallography, and morphology in A friend of Pearce’s, the inventor earnest. This investigation resulted in the Steve Baer, was fascinated with dome publication of his book, Structure in Nature geometry and polyhedra. He had is a Strategy for Design, (1978). In this text, been introduced to a toy version of among other innovations, he provided MERO’s cube-based space-frame the first description of all of the achiral, system in Germany in the early 1960s. convex uniform honeycombs composed He wondered if there was a similar of tetrahedral and octahedral polyhedra. underlying icosahedron-based system, leading to his discovery of the 31-zone THE UNIVERSAL NODE SYSTEM system, which showed Pearce’s cubic/ Figure 1. Conway’s In his book, Peter Jon Pearce referred octahedral system was one of five Zometool model to a modelling system he had created related to one another by icosahedral called Kepler’s Kosmos Figure 2b. The icosahedral Ag60 rhombicosidodecahedron. or Kepler’s Obsession. as the Universal Node system. He had From Su, Y.M., et al. Nat. Commun. 2020, https://doi.org/10.1038/s41467-020-17198-1 symmetry. Baer’s associate and Drop City www.researchoutreach.org www.researchoutreach.org Figure 3a. The octahedral Ag24 truncated octahedron. From Su, Y.M., et al. Figure 3b. The octahedral Ag6 octahedron. From Su, Y.M., et al. Nat. Commun. 2020, https://doi.org/10.1038/s41467-020-17198-1 Nat. Commun. 2020, https://doi.org/10.1038/s41467-020-17198-1 cofounder Clark Richert added another 30 zones and completed the icosahedral Unfortunately, Conway neither said relationship with Pearce’s 13-zone system. Baer licensed Marc Pelletier nor wrote anything about the special and Paul Hildebrandt to develop their symmetry properties of his model. design for the user-friendly version of his “Zometoy”. The result, informed by both Pearce’s and Baer’s theoretical and practical experiences, and enhanced by Pelletier’s superlative math, was the Zometool. (The Zometool is recognized by quantum mathematician Micho Durdevich — known for coining the term “quantum computing” — as a physical Figure 5. b) c), d) embodiment of quantum math). Silver shells of SD/Ag90a. A view along a two-fold axis of the outer The corresponding triple nest with regular icosahedral Ag shell of SD/Ag90a, which polyhedral faces, viewed along the three- and a) corresponds with a four-fold axis through the two-/four-fold axes of the nest. JOHN H. CONWAY A view along a common three-fold axis of SD/ middle and inner octahedral Ag shells. There John Conway, an English Ag90a. There are four of these axes of rotation. are three of these shared axes of rotation, From Su, Y.M., et al. Nat. Commun. 2020 mathematician, was famous for his arranged orthogonally. https://doi.org/10.1038/s41467-020-17198-1 love of toys and was one of Zometool, Inc.’s first customers, ordering four Researcher Kits for the Geometry Center at the University of Minnesota. KEPLER’S MYSTERY OF THE COSMOS Kepler’s system proposed that the in the octahedron. It was an intricate Realizing that the Zometool embodied In 1597, the German astronomer orbit of Saturn circumscribed a cube. concept that almost worked. Kepler, countless relationships among the and mathematician Johannes Inscribed in the cube was the orbit however, made no effort to align the Platonic Solids, Conway created a Kepler published Mysterium of Jupiter that also circumscribed five solids (Figure 6). model of all five of the Platonic solids Cosmographicum (The a tetrahedron. Inscribed in the with the smallest number of nodes and Cosmographic Mystery) laying out tetrahedron was the orbit of Mars that Conway, renowned for his work struts, a model that later became known his philosophy of the universe with also circumscribed a dodecahedron. in symmetry, created a particular as ‘Kepler’s Obsession’ or ‘Kepler’s its then-known six planets. Kepler Inscribed in the dodecahedron was alignment of the five solids – the Kosmos’ (Figure 1). This model is still believed that the orbits of the six the orbit of Earth that circumscribed tetrahedral tetrahedron, the octahedral being sold by Zometool. Conway’s planets could fit around the five an icosahedron. Inscribed in the cube and octahedron, and the model ‘solved’ a four-hundred-year-old Figure 4a and 4b. Stereo views (requiring crossing one’s eyes to align the ‘a’s on top and the ‘b’s “perfect” solids: the octahedron, icosahedron was the orbit of Venus that icosahedral dodecahedron and on the bottom). (a) The alignment with the maximal number of shared axes of rotation in the Ag90 problem implied by Johannes Kepler’s nest and in the Kepler’s Kosmos, (b) An alternative alignment with fewer shared axes of rotation. icosahedron, dodecahedron, circumscribed an octahedron. Finally, icosahedron – the challenge being model of the solar system. From Su, Y.M., et al. Nat. Commun. 2020, https://doi.org/10.1038/s41467-020-17198-1 tetrahedron and cube. the orbit of Mercury was inscribed that the solids were in three different www.researchoutreach.org www.researchoutreach.org of polyhedral structures. Coordination- driven self-assembly has facilitated the creation of molecules mimicking Platonic, Archimedean and even Behind the Research Goldberg polyhedra. More challenging, however, is the assembly of nested polyhedra, both in terms of their synthesis and in determining the Professor Professor Stan alignment of the polyhedra. Di Sun Schein NESTED POLYHEDRA The research team carried out synthesis and X-ray crystallography to determine E: [email protected] T: +86-13854137508 E: [email protected] T: +1-3108554769 atomically precise molecular structure of a nest with 90 silver atoms. The nest had three concentric silver polyhedra Research Objectives with apparently incompatible symmetry Sun et al.
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