The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

Developing Mathematical Knowledge and Skills through the Awareness Approach of Teaching and Learning

Every object we think of or encounter, whether a natural or human-made, has a regular or irregular shape. In its own intrinsic conceptual design, it has elements of mathematics, science, engineering, and arts, etc., which are part of the object’s geometric shape, form and structure. Geometry is not only an important part of mathematics, but it is also an important part of daily life.  However, geometry is challenging for some students, even high-achieving students.  One way to help students understand geometry and its relevance in life is to engage students to discover them cognitively, then to research and identify them in real world examples and then to relate them to past, present, and future innovations that improved our way of thinking about ourselves and the world around us.  This interdisciplinary activity uses the Developmental Awareness Approach of Teaching and Learning (DAATL) to help students discover principles, acquire knowledge, and learn mathematical concepts including surface area, volume, dimensions, regular and irregular plane figures, solid polygons (regular polygons and polyhedra), thinking design, and graph making, etc.  It is designed to help students become acquainted with the most useful and familiar parts of mathematical geometry and its application in daily life through connections with disciplines such as science, engineering, art, design, and social studies.  The Development Awareness Approach of Teaching and Learning (DAATL) capitalizes on student's natural curiosity, inclination to comprehend as well as students love of drawing, doodling, painting, thinking and talking.  Throughout the learning process, students are engaged in authentic learning activities by real and concrete doing with clear purposes, thinking analytically, and evaluating their understanding of texts and ideas orally, in drawing, and in writing. This approach of teaching and learning has been tried and modified to ensure maximum effectiveness of acquiring understanding of the intended learning concepts.   The activities can be used with students in elementary school up to 2-year college levels. Keywords: Geometry, Learning Math, Developmental Discovery Approach, Active Learning, Student’s Active Engagement

Planar hexagonal meshing for architecture

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High-dimensional polytopes defined by oracles: algorithms, computations and applications

Η επεξεργασία και ανάλυση γεωμετρικών δεδομένων σε υψηλές διαστάσεις διαδραματίζει ένα θεμελιώδη ρόλο σε διάφορους κλάδους της επιστήμης και της μηχανικής. Τις τελευταίες δεκαετίες έχουν αναπτυχθεί πολλοί επιτυχημένοι γεωμετρικοί αλγόριθμοι σε 2 και 3 διαστάσεις. Ωστόσο, στις περισσότερες περιπτώσεις, οι επιδόσεις τους σε υψηλότερες διαστάσεις δεν είναι ικανοποιητικές. Αυτή η συμπεριφορά είναι ευρέως γνωστή ως κατάρα των μεγάλων διαστάσεων (curse of dimensionality). Δυο πλαίσια λύσης που έχουν υιοθετηθεί για να ξεπεραστεί αυτή η δυσκολία είναι η εκμετάλλευση της ειδικής δομής των δεδομένων, όπως σε περιπτώσεις αραιών (sparse) δεδομένων ή στην περίπτωση που τα δεδομένα βρίσκονται σε χώρο χαμηλότερης διάστασης, και ο σχεδιασμός προσεγγιστικών αλγορίθμων. Στη διατριβή αυτή μελετάμε προβλήματα μέσα σε αυτά τα πλαίσια. Το κύριο ερευνητικό πεδίο της παρούσας εργασίας είναι η διακριτή και υπολογιστικής γεωμετρία και οι σχέσεις της με τους κλάδους της επιστήμης των υπολογιστών και τα εφαρμοσμένα μαθηματικά, όπως είναι η θεωρία πολυτόπων, οι υλοποιήσεις αλγορίθμων, οι πιθανοθεωρητικοί γεωμετρικοί αλγόριθμοι, η υπολογιστική αλγεβρική γεωμετρία και η βελτιστοποίηση. Τα θεμελιώδη γεωμετρικά αντικείμενα της μελέτης μας είναι τα πολύτοπα, και οι βασικές τους ιδιότητες είναι η κυρτότητα και ότι ορίζονται από ένα μαντείο (oracle) σε ένα χώρο υψηλής διάστασης. Η επεξεργασία και ανάλυση γεωμετρικών δεδομένων σε υψηλές διαστάσεις διαδραματίζει ένα θεμελιώδη ρόλο σε διάφορους κλάδους της επιστήμης και της μηχανικής. Τις τελευταίες δεκαετίες έχουν αναπτυχθεί πολλοί επιτυχημένοι γεωμετρικοί αλγόριθμοι σε 2 και 3 διαστάσεις. Ωστόσο, στις περισσότερες περιπτώσεις, οι επιδόσεις τους σε υψηλότερες διαστάσεις δεν είναι ικανοποιητικές. Δυο πλαίσια λύσης που έχουν υιοθετηθεί για να ξεπεραστεί αυτή η δυσκολία είναι η εκμετάλλευση της ειδικής δομής των δεδομένων, όπως σε περιπτώσεις αραιών (sparse) δεδομένων ή στην περίπτωση που τα δεδομένα βρίσκονται σε χώρο χαμηλότερης διάστασης, και ο σχεδιασμός προσεγγιστικών αλγορίθμων. Το κύριο ερευνητικό πεδίο της παρούσας εργασίας είναι η διακριτή και υπολογιστικής γεωμετρία και οι σχέσεις της με τους κλάδους της επιστήμης των υπολογιστών και τα εφαρμοσμένα μαθηματικά. Η συμβολή αυτής της διατριβής είναι τριπλή. Πρώτον, στο σχεδιασμό και την ανάλυση των γεωμετρικών αλγορίθμων για προβλήματα σε μεγάλες διαστάσεις. Δεύτερον, θεωρητικά αποτελέσματα σχετικά με το συνδυαστικό χαρακτηρισμό βασικών οικογενειών πολυτόπων. Τρίτον, η εφαρμογή και πειραματική ανάλυση των προτεινόμενων αλγορίθμων και μεθόδων. Η ανάπτυξη λογισμικού ανοιχτού κώδικα, που είναι διαθέσιμο στο κοινό και βασίζεται και επεκτείνει διαδεδομένες γεωμετρικές και αλγεβρικές βιβλιοθήκες λογισμικού, όπως η CGAL και το polymake.The processing and analysis of high dimensional geometric data plays a fundamental role in disciplines of science and engineering. The last decades many successful geometric algorithms has been developed in 2 and 3 dimensions. However, in most cases their performance in higher dimensions is poor. This behavior is commonly called the curse of dimensionality. A solution framework adopted for the healing of the curse of dimensionality is the exploitation of the special structure of the data, such as sparsity or low intrinsic dimension and the design of approximation algorithms. The main research area of this thesis is discrete and computational geometry and its connections to branches of computer science and applied mathematics. The contribution of this thesis is threefold. First, the design and analysis of geometric algorithms for problems concerning high-dimensional, convex polytopes, such as convex hull and volume computation and their applications to computational algebraic geometry and optimization. Second, the establishment of combinatorial characterization results for essential polytope families. Third, the implementation and experimental analysis of the proposed algorithms and methods. The developed software is opensource, publicly available and builds on and extends state-of-the-art geometric and algebraic software libraries such as CGAL and polymake

Non-perturbative approaches to Scattering Amplitudes

This thesis is devoted to the study of scattering amplitudes using two non-perturbative approaches. In Part I we focus on a particular theory known as N = 4 Super-Yang-Mills in four spacetime dimensions. The scattering amplitudes in this theory are dual to the expectation value of null polygonal Wilson loops which can be computed non-perturbatively using integrability. The Wilson loop is decomposed into smaller polygons and computed as an evolution of the color flux tube of the theory, summing over all intermediate flux tube states. By a suitable generalization of the building blocks called pentagons we describe how this program can describe all helicity configurations of the amplitude. We also show how the contribution from all flux tube excitations can be resummed to reproduce the general kinematics result at weak coupling. In Part II we take a different approach and study the space of Quantum Field Theories (QFTs). We focus on two-dimensional theories with a mass gap and a global symmetry. By studying the consequences of unitarity, crossing symmetry and analyticity of the two-to-two scattering matrix element we are able to constrain the space of allowed QFTs. At the boundary of this space we find several interesting features of the S-matrices and identify various integrable points

What Is a Polygonal Impact Crater? A Proposed Framework Toward Quantifying Crater Shapes

Impact craters are used for a wide array of investigations of planetary surfaces. A crater form that is somewhat rare, forming only ∼10% of impact craters, is the polygonal impact crater (or PIC). These craters have been visually, manually identified as having at least two rim segments that are best represented as straight lines. Such straight lines or edges are most often used to infer details about the subsurface crust where faults control the structure of the crater cavity as it formed. The PIC literature is scant, but almost exclusively these craters are identified manually, and the potentially straight edges are classified and measured manually. The reliance on human subjectivity in both the identification and measurement motivated us to design a more objective algorithm to fit the crater rim shape, measure any straight edges, and measure joint angles between straight edges. The developed code uses a Monte Carlo approach from a user-input number of edges to first find a reasonable shape from purely random possible shapes; it then uses an iterative Monte Carlo approach to improve the shape until a minimum difference between the shape and rim trace is found. It returns the result in a concise, parameterized form. This code is presented as a first step because, while we experimented with several different metrics, we could not find one that could consistently, objectively return an answer that stated which shape for a given crater was the best; this objective metric is an area for future improvement

Computations on Fullerenes: Characterization, Reactivity and Growth

Aquesta Tesi titulada ‘Computations on Fullerenes: Characterization, Reactivity and Growth’ es focalitza amb els mecanismes de formació i caracterització de ful·lerens prèviament detectats als experiments. Són caixes tancades de carboni formades per hexàgons i dotze pentàgons. Hem col·laborat amb diferents grups experimentals, per tant, ens hem centrat en entendre i racionalitzar els seus experiments. Diferents models de formació de ful·lerens han estat proposats, però encara avui segueix sent un misteri. Els nostres estudis donen completament suport al mecanisme de creixement bottom-up proposat pel Prof. Kroto. Aquest mecanisme ha estat estudiat per càlculs estàtics de DFT i per dinàmica molecular de Car-Parrinello. Una exploració exhaustiva dels isòmers més favorables, així com les superfícies d’energia potencial associades a les insercions d’unitats C2 als ful·lerens i les topologies de les estructures involucrades, han ajudat al desenvolupament d’aquest projecte. Aquest procés d’inserció és exotèrmic/exergònic, i encara que les barreres d’energia lliure són elevades, es poden veure superades a la temperatura de formació de ful·lerens (2000 K). Els isòmers més abundants del Ti@C2n (2n=26-48) i Sc3N@C2n (2n=68-80) s’han relacionat mitjançant unitats C2 i, en alguns casos, alguna isomerització del tipus Stone-Wales. Respecte a la detecció i aïllament dels metal·loful·lerens endoèdrics, ens hem centrat en la seva caracterització. La cloració dels ful·lerens també ha estat estudiada, ja que ha sorgit com una poderosa eina en el món dels derivats de ful·lerens. Famílies de C2n (2n=50,60,66,68,etc.) han estat trobades com cloroful·lerens. Els nostres resultats prediuen que la cloració s’esdevé un cop es formada la caixa neutra a temperatures més baixes de 2000 K, mitjançant l’addició de radical lliure i tenint en compte les distribucions del HOMO i de la densitat d’spin. La majoria dels nostres projectes han estat d’acord amb els resultats experimentals.La Tesis titulada ‘Computations on Fullerenes: Characterization, Reactivity and Growth’ se focaliza con los mecanismos de formación y caracteritzación de fullerenos previamente detectados a los experimentos. Son cajas cerradas de carbono formadas por hexágonos y doce pentágonos. Hemos colaborado con diferentes grupos experimentales, por tanto, nos hemos centrado en entender y racionalizar sus experimentos. Diferentes modelos de formación han sido propuestos, pero todavía hoy sigue siendo un misterio. Nuestros estudios dan soporte al mecanismo de crecimiento bottom-up propuesto por el Prof. Kroto. Este mecanismo ha sido estudiado mediante cálculos estáticos de DFT i por dinámica molecular de Car-Parrinello. Una exploración exhaustiva de los isómeros más favorables, así como las superficies de energía potencial asociadas a las inserciones de unidades C2 a los fullerenos y las topologías de las estructures involucradas, han ayudado al desarrollo de este proyecto. Este proceso de inserción es exotérmico/exergónico, y todavía que las barreras de energía libre son elevadas, se pueden ver superadas a la temperatura de formación de fullerenos (2000 K). Los isómeros más abundantes del Ti@C2n (2n=26-48) y Sc3N@C2n (2n=68-80) se han relacionado mediante unidades C2 y, en algunos casos, alguna isomerización del tipo Stone-Wales. Respecto a la detección y aislamiento de los metallofullerenos endoédricos, nos hemos centrado en su caracteritzación. La cloración de los fullerenos también ha sido estudiada, ya que ha surgido como una poderosa herramienta en el mundo de los derivados de fullerenos. Familias de C2n (2n=50,60,66,68,etc.) han sido encontradas como clorofullerenos. Nuestros resultados predicen que la cloración se forma una vez es formada la caja neutra a temperaturas más bajas de 2000 K, mediante la adición de radical libre y teniendo en cuenta las distribuciones del HOMO y de la densidad de espín. La mayoría de nuestros proyectos han estado de acuerdo con los resultados experimentales.The Thesis titled ‘Computations on Fullerenes: Characterization, Reactivity and Growth’ is mainly focused on the formation mechanisms and characterization of fullerenes previously detected in experiments. These molecules are closed carbon cages formed by only hexagons and twelve pentagons. Most part of our research has been carried out in collaboration with different experimental groups, therefore we aimed to understand and rationalize their experiments. Although many hypothetical models have been proposed, the fullerene formation mechanism is still a mystery. Our studies rules out the bottom-up mechanism as a model of fullerene formation. We have explored this mechanism by means of static DFT and Car-Parrinello molecular dynamics calculations for series of different endohedral fullerenes. A comprehensive exploration of the most favourable isomers, potential energy surfaces associated with the successive C2 insertions and topologies of the involved structures, helped us to develop this project. The insertion of a C2 unit to already formed EMF is always an exothermic/exergonic process, and the free energy barriers for each step are attainable at temperature of fullerene formation (2000 K). The most abundant isomers of Ti@C2n (2n=26-48) and Sc3N@C2n (2n=68-80) are formally linked by direct C2 insertions and in a few cases by additional Stone-Wales transformations. Regarding the detection and isolation of endohedral metallofullerenes let us to perform a computational study of the rationalization and characterization of these isomers. Chlorination has emerged as a powerful tool in fullerene derivatives. Several C2n families (2n=50,60,66,68,etc.) have been found to show cages exohedrally chlorinated. According to our results, chlorination would take place at a temperature significantly lower than 2000 K by free radical addition considering the HOMO and the spin density distributions of the pristine cage and intermediates, once the lowest energy neutral isomers are formed. Most of our projects resulted in suitable and in agreement with experiments

Wilson Loop Duality and OPE for Super Form Factors of Half-BPS Operators

We propose a dual Wilson loop description for the MHV super form factors of half-BPS operators in planar $\mathcal{N}=4$ super-Yang-Mills theory. In this description, the local operators are represented by on-shell states, made out of zero-momentum particles, that are absorbed by a null periodic super Wilson loop. We present evidence for this duality at weak coupling, by performing an explicit calculation of the Wilson loop matrix elements through one loop. At tree level, the interactions localize at the cusps of the loop, revealing a simple connection between the super form factors and the $m=2$ tree amplituhedron. At loop level, we show that the Wilson loop calculation reproduces the known results for the super form factors. Inspired by this duality, we extend the OPE program developed for the form factors of the Lagrangian to the super form factors of the higher-charge operators. We introduce non-perturbative axioms and conjectures for the main building blocks that govern the exchange of the lightest flux-tube excitations. These blocks appear as simple refinements of the form factor transitions introduced in earlier OPE studies. They are expressed at any value of the 't Hooft coupling in terms of the tilted Beisert-Eden-Staudacher kernel. We carry out checks of our conjectures up to two loops at weak coupling for three- and four-point form factors of half-BPS operators of various lengths, finding perfect agreement with perturbative data.Comment: 54 pages, 15 figure

Computational analysis of electronic properties and mechanism of formation of endohedral fullerenes and graphene with Fe atoms: Computational analysis of electronic properties and mechanism of formation of endohedral fullerenes and graphene with Fe atoms

In this thesis, a series of computational studies based on density functional theory (DFT) and density functional tight-binding (DFTB) is presented to deeply understand experimental results on the synthesis of endohedral fullerenes and graphene/iron hybrids at atomic level. In the first part, a simple and efficient model is proposed to evaluate the strain experienced by clusters encapsulated in endohedral metallofullerenes (EMFs). Calculations for the sole cluster, either in the neutral or the charged state, cannot be used for this goal. However, when the effect of the carbon cage is mimicked by small organic π-systems (such as pentalene and sumanene), the cluster has sufficient freedom to adopt the optimal configuration, and therefore the energetic characteristics of the EMF-induced distortion of the cluster can be evaluated. Both nitride and sulfide clusters were found to be rather flexible. Hence, they can be encapsulated in carbon cages of different size and shape. For carbide M2C2 cluster the situation is more complex. The optimized cluster can adopt either butterfly or linear shapes, and these configurations have substantially different metal-metal distance. Whereas for Sc2C2 both structures are isoenergetic, linear form of the Y2C2 cluster is substantially less stable than the butterfly-shaped configuration. These results show that phenomenon of the “nanoscale fullerene compression” once proposed by Zhang et al. (J. AM. CHEM. SOC. (2012),134(20)) should be “nanoscale fullerene stretching”. Finally, the results also reveal that both Ti2S and Ti2C2 cluster are strained in corresponding EMF molecules, but the origin of the strain is opposite: C78-D3h(5) cage imposes too long Ti···Ti distance for the sulfide cluster and too short distance for the carbide cluster. In the second part of the thesis, possible fullerene geometries and electronic structures have been explored theoretically for the species detected in mass spectra of the Sc-EMF extract synthesized using CH4 as a reactive gas. Two most promising candidates, namely Sc4C@C80-Ih(7) and Sc4C3@C80-Ih(7), have been identified and further studied at the DFT level. For Sc4C@C80, the tetrahedral Sc4 cluster with the central μ4-C atom was found to be 10 kJ/mol more stable than the square cluster. For Sc4C3@C80, the calculation showed that the most stable is the Sc4C3 cluster in which the triangular C3 moiety is η3- and η2-coordinated to Sc atoms. Whereas Sc4C@C80 has rather small HOMO-LUMO gap and low ionization potential, the HOMO-LUMO gap of Sc4C3@C80 is substantially higher and exceeds that of Sc4C2@C80. In the third part, computational studies of structures and reactivity are described for a new type of EMFs with a heptagon that has been produced in the arc-discharge synthesis. DFT computations predict that LaSc2N@Cs(hept)-C80 is more stable than LaSc2N@D5h-C80, so the former should be synthesized in much higher yield than observed. This disagreement may be ascribed to the kinetic factors rather than thermodynamic stability. Because of prospective applications of this EMFs by introducing functional groups, the influence of the heptagon on the chemical properties have been further evaluated. Thermodynamically and kinetically preferred reaction sites are studied computationally for Prato and Bingel-Hirsch cycloaddition reactions. In both types of reactions the heptagon is not affected, and chemical reactivity is determined by the adjacent pentalene units. Thermodynamically controlled Prato addition is predicted to proceed regioselectively across the pentagon/pentagon edges, whereas the most reactive sites in kinetically-controlled Bingel-Hirsch reaction are the carbon atoms next to the pentagon/pentagon edge. Fourth, although various EMFs have been successfully synthesized and characterized, the formation mechanism is still not known in details, and hence control of the synthesis products is rather poor. Therefore, EMF self-assembly process in Sc/carbon vapor in the presence and absence of cooling gas (helium) and reactive gas (NH3 and CH4) is systematically investigated using quantum chemical molecular dynamics (QM/MD) simulations based on the DFTB potentials. The cooling gas effect is that the presence of He atoms accelerates formation of pentagons and hexagons and reduces the size of formed carbon cages in comparison to the analogous He-free simulations. As a result, the Sc/C/He system yields a large number of successful trajectories (i.e. leading to the Sc-EMFs) with more realistic cage-size distribution than the Sc/C system. Encapsulation of Sc atoms within the carbon cage was found to proceed via two parallel mechanisms. The main mechanism involves nucleation of the several hexagons and pentagons with Sc atoms already at the early stages of the carbon vapor condensation. In such proto-cages, both Sc–C σ-bonds and coordination bonds between Sc atoms and the π-system of the carbon network are present. Sc atoms are thus rather labile and can move along the carbon network, but the overall bonding is sufficiently strong to prevent dissociation even at high temperatures. Further growth of the carbon cage results in encapsulation of one or two Sc atoms within the forming fullerene. Another encapsulation mechanism is observed in rare cases. In this process, the closed cage is formed with Sc being a part of the carbon network, i.e. being bonded by three or four Sc–C σ-bonds. However, such intermediates are found to be unstable, and transform into the endohedral fullerenes within few picoseconds of annealing. In perfect agreement with experimental studies, extension of the simulation to Fe and Ti showed that Fe-EMFs are not formed at all, whereas Ti is prone to form Ti-EMFs with small cage sizes, including Ti@C28-Td and Ti@C30-C2v(3). The role of “reactive gas” in the EMF synthesis is revealed in dedicated simulations of the fullerene formation in the presence of several molecules of CH4 or NH3. When concentration of reactive gas is high, carbon vapor tends to form graphene flakes or other carbon species terminated by hydrogen atoms, whereas the yield of empty fullerenes is very low. Conversely, with additional metal atoms (Sc) and the same number of NH3 molecules, the yield of fullerenes constantly increase from 5 to 65% which is ascribed to the catalytic activity of metal atoms in the nucleation of carbon cages already at early stage. Moreover, due to the presence of hydrogen atoms from the reactive gas, the carbon cage formation requires much longer time, which provides sufficient reaction time to encapsulate 3 or 4 Sc atoms within one cage. It explains preferential formation of clusterfullerenes in experiments with reactive gas. At the same time, monometallofullerenes and dimetallofullerenes are the main products in absence of reactive gas. We also provide possible growth mechanisms of carbide and cyano-clusterfullerenes in details to elucidate how the intracluster goes into the cage. A possible growth mechanism of nitride clusterfullerenes has been proposed based on DFT results. In the last part, a free-standing crystalline single-atom thick layer of Fe has been studied theoretically. By investigating the energy difference, ΔE, between a suspended Fe monolayer and a nanoparticle using the equivalent number of Fe atoms, one can estimate that the largest stable membrane should be ca. 12 atoms wide or 3 × 3 nm2 which is in excellent agreement with the experimental observation. Otherwise, the possibility of C, O, N atoms embedded into the Fe membrane can been fully excluded by DFTB and DFT simulations, which agrees with electron energy loss spectroscopy (EELS) measurement. A significantly enhanced magnetic moment for single atom thick Fe membranes (3.08 μB) is predicted by DFT as compared to the bulk BCC Fe (2.1 μB), which originates from the 2D nature of the Fe membrane since the dz2 orbital is out-of-plane while the dxy orbital is in-plane