Chapter 1 Abstract Algebra : Groups, Ring s an d Fields This course will aim at understanding convex poly topes, which are fun - damental geometri c object s i n combinatorics , usin g technique s fro m algebra an d discret e geometry . Polytope s aris e everywher e i n th e real worl d an d i n mathematics . Th e mos t famou s example s ar e th e Platonic solid s in three-dimensional space : cube, tetrahedron, octahe- dron, icosahedron and dodecahedron, which were known to the ancien t Greeks. Th e natural first approac h to understanding polytopes should be through geometry a s they are first an d foremost geometri c objects . However, an y experienc e wit h visualizin g geometri c object s wil l tel l you soo n tha t geometr y i s alread y quit e har d i n three-dimensiona l space, an d i f one ha s t o stud y object s i n four - o r higher-dimensiona l space, the n i t i s essentiall y hopeles s t o rel y onl y o n ou r geometri c and drawin g skills . Thi s frustratio n le d mathematician s t o th e dis - covery tha t algebr a ca n b e use d t o encod e geometr y and , sinc e alge - bra doe s no t suffe r fro m th e sam e limitation s a s geometr y i n deal - ing wit h highe r dimensions , i t ca n serv e ver y wel l a s th e languag e of geometry . A simpl e exampl e o f thi s translatio n ca n b e see n b y noting that , whil e i t i s har d t o visualiz e vector s i n four-dimensiona l 1 http://dx.doi.org/10.1090/stml/033/01
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