By David S. Richeson
Leonhard Euler's polyhedron formulation describes the constitution of many objects--from football balls and gems to Buckminster Fuller's structures and gigantic all-carbon molecules. but Euler's formulation is so easy it may be defined to a baby. Euler's Gem tells the illuminating tale of this crucial mathematical idea.
From historic Greek geometry to today's state-of-the-art study, Euler's Gem celebrates the invention of Euler's liked polyhedron formulation and its far-reaching influence on topology, the learn of shapes. In 1750, Euler saw that any polyhedron composed of V vertices, E edges, and F faces satisfies the equation V-E+F=2. David Richeson tells how the Greeks ignored the formulation fullyyt; how Descartes virtually came upon it yet fell brief; how nineteenth-century mathematicians widened the formula's scope in ways in which Euler by no means predicted through adapting it to be used with doughnut shapes, soft surfaces, and better dimensional shapes; and the way twentieth-century mathematicians found that each form has its personal Euler's formulation. utilizing great examples and diverse illustrations, Richeson provides the formula's many based and unforeseen functions, equivalent to displaying why there's consistently a few windless spot on the earth, the way to degree the acreage of a tree farm via counting timber, and the way many crayons are had to colour any map.
packed with a who's who of great mathematicians who puzzled, sophisticated, and contributed to a amazing theorem's improvement, Euler's Gem will fascinate each arithmetic enthusiast.
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Extra resources for Euler's Gem: The Polyhedron Formula and the Birth of Topology
Example text
6 b: decomposition rate constant, [ l/sec] ki: initiation rate constant, [Vmol-sec] k,: propagation rate constant, [I/mol*sec] ktc: termination rate constant by combination, [l/mol*sec] ktd: termination rate constant by disproportionation, [I/mol*sec] Kuo, J. F. and Chen, C. Y. Polym. J, 13, 453 (1981). 51 Methyl methacrylate Bulk polymerization Initiator: 2,2'-azobisisobutyronitrile Temperature: 25°C Initiation I R" + M Propagation Pi," + M Chain transfer +Y Termination by disproportionation P,O + PI," + PI," P,,O Termination by combination P,,O + + + + + + 2R" Pi'' Pnt i o PI, + Y" P, + PI, P,+,, kd ki kP ktW ktd(n,m) ktc(n,in) I: initiator, M: monomer, R" : primary free radical, P," : growing polymer radical containing n units of monomer, P,: dead polymer containing n units of monomer, Y: chain transfer agent.
6 0 ko: frequency factor, [-I E,: activation energy, [cal/mol] V,: activation volume, [ml/mol] P: pressure, [atm] T: temperature, [OK] K,: initiation rate constant, [ l/sec] K,: propagation rate constant, [I/mol sec] Kfi,,: monomer transfer rate constant, [Vmol sec] Kfsn:solvent transfer rate constant, [I/mol*sec] K,c: termination rate constant by combination, [I/mol*sec] Ktd: termination rate constant by disproportionation, [I/mol- sec] Kscn: short chain branching rate constant, [ l/sec] Shirodkar, P.
105, 212 (1967). , Makromol. , 114, 201 (1968). Methyl methacrylate 47 Bulk polymerization Initiator: 2,2'-azobisisobutyronitrile Temperature: 60°C C R + M Xr + M Xr + X, Xr + R 2R 2R +XI + + + + + Xr+l Pr + P, or Pr+, P, products C: initiator, R: primary radical, M: monomer, Ar: growing polymer radical with r monomer units, P,: dead polymer with r monomer units, Y: chain transfer agent. 41 x [C]: concentration of initiator, [mol/l] [MI: concentration of monomer, [mol/l] kd: decomposition rate constant, [ l/sec] ki: initiation rate constant, [Vmol esec] k,: propagation rate constant, [I/mol*sec] k,: termination rate constant, [Vmol-sec] kt,: primary radical termination rate constant, [Vmol-sec] k,,: rate constant between primary radicals, [I/mol- sec] R,: propagation rate, [mol/l-sec] Ito, K.