Isoperimetric Quotient for Fullerenes and Other Polyhedral Cages Tomaž Pisanski ,† Matjaž Kaufman ,*† Drago Bokal ,† Edward C. Kirby ,‡ Ante Graovac § Inštitut za matematiko, fiziko in mehaniko, Univerza v Ljubljani, Jadranska 19, 1000 Ljubljana, Slovenia, Resource Use Institute, 14 Lower Oakfield, Pitlochry, Perthshire PH16 5DS, Scotland, UK
The Rugjer Bošković Institute, Bijenička c. 54, HR-10001 Zagreb, POB. 1016, Croatia J. Chem. Inf. Comput. Sci., 1997, 37 (6), pp 1028–1032 DOI: 10.1021/ci970228e Publication Date (Web): November 24, 1997 b Copyright © 1997 American Chemical Society
Abstract: The notion of Isoperimetric Quotient (IQ) of a polyhedron has been already introduced by Polya. It is a measure that tells us how spherical is a given polyhedron. If we are given a polyhedral graph it can be drawn in a variety of ways in 3D space. As the coordinates of vertices belonging to the same face may not be coplanar the usual definition of IQ fails. Therefore, a method based on a proper triangulation (obtained from omni-capping) is developed that enables one to extend the definition of IQ and compute it for any 3D drawing. The IQs of fullerenes and other polyhedral cages are computed and compared for their NiceGraph and standard Laplacian 3D drawings. It is shown that the drawings with the maximal IQ values reproduce well the molecular mechanics geometries in the case of fullerenes and exact geometries for Platonic and Archimedean polyhedra.
In the equations of general relativity, G is often multiplied by 8π. Hence writings in particle physics and physical cosmology often normalize 8πG to 1. This normalization results in the reduced Planck energy, defined as:
- natural number after zero.
- e, approximately equal to 2.718281828459045235360287…
- i, the imaginary unit such that i2 = -1.
- “Dynamical systems on monoids: Toward a general theory of deterministic systems and motion“.
- In Minati G., Abram M., Pessa E. (eds.), Methods, models, simulations and approaches towards a general theory of change, pp. 173-185, Singapore: World Scientific. ISBN 978-981-4383-32-5.
- Vladimir Igorevic Arnol’d “Ordinary differential equations“, various editions from MIT Press and from Springer Verlag, chapter 1 “Fundamental concepts“.
- I. D. Chueshov “Introduction to the Theory of Infinite-Dimensional Dissipative Systems” online version of first edition on the EMIS site .
- Roger Temam “Infinite-Dimensional Dynamical Systems in Mechanics and Physics” Springer Verlag 1988, 1997.
THEORY OF DYNAMICAL SYSTEMS AND GENERAL TRANSFORMATION. GROUPS WITH INVARIANT MEASURE. A. B. Katok, Ya. G. Sinai, and A. M. Stepin.
page 1= Printable PDF of this page only
page2 = Printable PDF of this page only
March 20 George Polya can rightly be called the father of problem solving in mathematics education. https://en.wikipedia.org/wiki/George_P%C3%B3lya
Notes: March 5, 2017
1. Virtual particles: https://en.wikipedia.org/wiki/Virtual_particle
2. Pertubation theory: https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)
3. Hamiltonian: https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)
4. Also Carlo Rovelli,
5. Richard Muller
http://www.pnas.org/content/103/28/10612.full.pdf Packing, tiling, and covering with tetrahedra J. H. Conway* and S. Torquato
http://doye.chem.ox.ac.uk/research/cluster_structure.html Jonathan Doyle, Cambridge
On inflation, cosmological constant, and SUSY breaking Andrei Linde
On the problem of scale: a general theory of morphogenesis and normative policy signals for economic evolution
1.54 steradians = or ≠ a gap of 7.36∗
http://plato.stanford.edu/entries/spacetime-theories/ Absolute and Relational Theories of Space and Motion Nick Huggett <email@example.com> Carl Hoefer <firstname.lastname@example.org>
Dr. Claus Kiefer. Institut für Theoretische Physik Universität zu Köln … 0221 470-4300 (secretary). Fax: 0221 470-2189. E-mail: email@example.com