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