15gen.pdf - Escola Polit` ecnica Superior Grau en...

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Escola Polit` ecnica Superior Universitat de Lleida Grau en Enginyeria Inform` atica ` Algebra Gener de 2015 Problema 1 Sigui Q un quadrat del pla centrat a l’origen amb v` ertexs ( v 1 , v 2 , v 3 , v 4 ). Considerem el conjunt M format pels moviments del pla que deixen invariant el quadrat Q . Aquests moviments poden ser de dos tipus: girs G i simetries S . Podeu observar que hi ha quatre girs i quatre simetries que deixen invariant el quadrat. Els girs els denotem per G = { g 90 , g 180 , g 270 , g 360 } , on cada sub´ ındex denota l’angle de rotaci´ o. Les simetries les denotem per S = { s x , s y , s d 1 , s d 2 } , on cada sub´ ındex denota l’eix de simetria: x ´ es l’eix d’abscisses, y ´ es l’eix d’ordenades, d 1 ´ es la diagonal del primer i tercer quadrant, i d 2 ´ es la diagonal del segon i quart quadrant. Aleshores M = G S . En el conjunt M definim l’operaci´ o composici´ o de moviments de la forma seg¨uent: donats dos elements m 1 i m 2 de M , la composici´o de moviments m 2 m 1 ´ es el moviment que resulta d’aplicar primer el moviment m 1 i despr´ es m 2 .
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