Post on 10-Apr-2015
METODOLOGÍA DE LA INVESTIGACIÓN EN ARTE Y HUMANIDADES
ELEMENTOS DE SIMETRÍAPrograma 12
Martín Larios García, M. en Arq., M. en Fil.
Agosto-Diciembre 2006
Nonperiodic Tilings
In Mathematics modern group theory is established to describe the characteristics of transformations. The theory was developed in 19C by two mathematical genius, E Galois (1811-1832) and N. H. Abel (1802-1829). They disproved that there is no method to describe the answers of more than 5 dimensional equations with this group theory. This section explains the terms of mathematical group theory.
In a set G=(a, b, c, ….), G is called a group when any element a and b satisfied the following all three theorems.
Theorem 1: Associative For any a, b, c ε G, (a• b)•c = a • (b •c)
Theorem 2: Identity There is an element i ε G such that for all a ε G, a • i = a = i • a
Theorem 3: Inverse For each elemen a ε G and for each identity element I there is an element a-1 ε G such that: a • a-1 = i = a-1 • a
Theorem 4: Commutativity For any two elements a, b ε G a • b = b • a
Palacio de VelazquezParque de RetiroMadrid, Spain D1
Catedral de Pisa, ItaliaD4
Piso Cosmateode la Basilica de San Marcos,Venecia, ItaliaD5
Piso Cosmateo de laBasílica de San Juan de De Letrán, Roma, Italia, D6
Gallería Vittorio Emanuele IIMilán, Italia, D8
Santa Maria Sopra MinervaRoma, Italia D12
F1
F11
F12
F13
F2
F21
F22
ALGORITMO PARA LA CLASIFICACIÓN DE GRUPOSDE PAPEL TAPÍZ
Identidad p1
Reflexión-Diagonal c1m
Reflexión vertical p1m
Reflexión deslizada p1g
Medio-Giro p2
medio-Giro & Reflexión-d c2mm
Medio-Giro & Reflexión-1/2 p2mm
Medio-Giro & Reflexión-1/4 p2mg
Giro1/3 & Reflexión-v p3m1
Giro 1/3 & Reflexión-h p31m
Medio-Giro & Reflexión-1/4 p2gg
Giro 1/3 p3
Giro1/4 & Reflexión en esquina p4gm
Giro 1/4 p4
Giro 1/4 & Reflexion p4mm
Giro 1/6 p6
Giro 1/6 & Reflexión p6mm