Boletín de la Asociación Matemática Venezolana Vol. VI, No. 1 (1999) 141

Grupos Cristalográficos Planos.

Francisco Rivero Mendoza

Departamento de Matemáticas Facultad de Ciencias, Universidad de Los Andes lico@ciens.ula.ve

1 ¿Qué son las simetrías? ¿Qué es la belleza?

En verdad, la mayoría de los hombres y las mujeres somos sensibles hacia la belleza de las formas. Generalmente se dice que una cosa es bella cuando despierta en nosotros sentimientos agradables. El concepto de la belleza ha sido una preocupación para muchos filósofos, desde la antigüedad hasta nuestros días. Así pues en el diálogo Hipias de Platón se desarrolla una conversación muy interesante entre Sócrates e Hipias, cuya finalidad es la de establecer una definición de que es lo bello. Después de varias definiciones de lo bello; como por ejemplo lo bello es lo útil o lo bello es lo apropiado a cada cosa, se llega por fin a esta definición: lo bello es lo que nos deleita haciendo de medianeros oídos y vista. Pero de acuerdo al mismo Platón existen otras cualidades de los seres, distintas a la belleza, como por ejemplo lo perfecto, lo mesurado, ... etc. que vienen dados en la siguiente escala de valores (Filebo, Platón trad. Juan García Bacca (1981))

1. La medida-mesura

2. Lo conmensurado, mesurado y medido, lo bello, lo perfecto, lo suficiente.

3. La inteligencia y la sensatez.

4. Las Ciencias, Artes y opiniones correctas

5. Los deleites inofensivos y puros.

Vemos, pues, como los griegos anteponían el concepto de Medida al de la Belleza. Por lo tanto toda cosa que tenga medida (con mesura) es Bella, como por ejemplo la circunferencia, los polígonos regulares, los sólidos regulares, ... etc. Según Aristóteles, lo Bello esta presente en muchos objetos matemáticos (Metafísica) y para caracterizar esta belleza, se utiliza el orden, la simetría y la limitación.

De acuerdo al pensamiento griego, el estudio de la simetría proporciona elementos claves para describir la belleza de los objetos. Un cuerpo tiene simetría cuando hay cierta regularidad en su conformación desde el punto de vista de la medida. La palabra simetría quiere decir con medida, desde el punto de vista etimológico.

Así pues, la circunferencia tiene simetría, pues sus puntos están todos a la misma distancia del centro, y esta regularidad de todas sus partes es lo que la hace ser simétrica. Un rectángulo también posee simetría, pero de un tipo distinto al de la circunferencia: los lados opuestos del rectángulo tienen la misma medida. En la circunferencia, cualquier línea o eje que pase por su centro, la divide en dos partes exactamente iguales. Este tipo de simetría, llamada simetría de reflexión, también la tiene el rectángulo pero en menor grado. Si un eje paralelo a algunos de los lados pasa por el centro del rectángulo, entonces lo divide en dos partes iguales. Pero cualquier eje que pase por el centro, no paralelo a los lados, no divide al rectángulo en dos partes iguales. Por lo tanto el rectángulo posee sólo dos simetrías de reflexión, mientras que la circunferencia posee infinitas de este tipo. Esto nos sugiere que la circunferencia posee más simetrías que el rectángulo, lo cual es cierto, como veremos más adelante, al estudiar las simetrías en detalle.

El cuerpo humano también tiene simetría de reflexión, pues un plano longitudinal lo corta en dos partes iguales. Esta simetría de reflexión, hace que cada parte del lado derecho, tenga su simétrico del lado izquierdo: brazos, ojos, manos, ... etc. Además los ojos están situados a la misma distancia del plano, e igualmente las orejas y todas las demás partes. El arte primitivo, románico, gótico, ... etc. posee gran cantidad de simetría en sus figuras. También en la arquitectura predominan las simetrías de reflexión. El arte occidental a partir del renacimiento se aleja bastante de la simetría. El arte barroco es completamente asimétrico, véase por ejemplo un cuadro de Rubens o Velázquez, en donde las composiciones son muy audaces y carecen de elementos de simetrías.

Es en el arte decorativo, mediante el embellecimiento de los frisos, techos, cúpulas, pisos y otros elementos arquitectónicos, en donde las simetrías se hacen mas palpables. El arte morisco, desarrollado por los árabes en la península Ibérica durante la ocupación en la edad media, presenta un gran desarrollo del concepto de simetría, debido a su carácter abstracto. De acuerdo a los principios religiosos le estaba estrictamente prohibido a los artistas moros representar seres vivientes en sus creaciones. Esta limitación, en lugar de empobrecer su creatividad, sirvió de acicate para estimular sus mentes y lanzarse por caminos de gran belleza y originalidad. Su conocimiento de las simetrías alcanzó tal grado de magnitud que fueron los únicos en descubrir y utilizar sabiamente en sus decoraciones los 17 tipos de simetría plana.

A manera de ejemplo, ver en la fotografía (figura 1) un intrincado diseño de tejido de punto con una aguja, tejido por Doña Blanquita Sulbarán en la Parroquia Estado Mérida, el cual tiene simetría rotatoria de orden 6. Este tapiz tiene los mismos grupos de simetría que algunas de las decoraciones de la Alhambra de Granada.

2 Grupo de los Movimientos en el Plano.

El concepto de simetría, desde el punto de vista matemático, se podrá precisar cuando estudiemos los movimientos en el espacio. Pero ¿qué tienen que ver las simetrías con los movimientos ? Si bien la simetría es una característica de la forma, la cual no depende del movimiento, ésta permanece constante bajo ciertos tipos de movimiento del plano y el espacio. Este hecho tan importante, que puede parecer trivial para muchos matemáticos, de poder estudiar conceptos de tipo geométrico mediante el movimiento, abrió toda una nueva línea de pensamiento en la matemática, a mediados del siglo XIX. De dicha fusión nació lo que hoy se llama Geometría de las Transformaciones, un nuevo método de estudio de la geometría, usando coordenadas, transformaciones lineales en el plano, álgebra lineal y teoría de grupos. Es dentro del marco de esta teoría como se exponen los conceptos de simetría de una manera sencilla y a la vez rigurosa.

Una transformación del plano es una función o aplicación uno a uno del plano en sí mismo. Si designamos a la transformación por la letra T, entonces todo punto P del plano es enviado en un único punto P′ y esto se denota por T (P) = P′. Es posible que un punto quede fijo, digamos Q, y entonces se tendrá T (Q) = Q. En este caso se dice que el punto Q permanece Invariante bajo la transformación T.

Para estudiar las transformaciones es conveniente usar un sistema de coordenadas Cartesianas. Por ejemplo, la transformación T que envía a todo punto del plano en su imagen especular sobre el eje X, viene dada por

T (x, y) = (x, −y)

Es claro que los puntos de la forma (x, 0) quedan fijos mediante la transformación.

Cuando un punto no queda fijo bajo la transformación T, entonces se dice que es movido por T. Por ejemplo, la transformación anterior mueve al punto P (1, 1), pues T (1, 1) = (1, −1).

Se dice que una transformación T del plano es una isometría si T no altera la distancia entre los puntos. Dicho en otras palabras, si P (x, y) y Q(z, w) son dos puntos cualesquiera del plano, la distancia entre los puntos P y Q debe ser igual a la distancia entre los puntos T (P) y T (Q). Por lo tanto

d(T (x, y), T (w, z)) = d((x, y), (w, z))

donde d es la distancia entre los puntos del plano, dada por

d((x, y), (w, z)) = ((x − w)^{2} + (y − z)^{2})^{1/2}

En este estudio, cuando se habla del Plano, haremos referencia a su estructura de espacio afín. En este sentido, todas las transformaciones consideradas, incluyendo por supuesto las isometrías, serán transformaciones afines.

Un movimiento del plano es una isometría. Se puede demostrar que todo movimiento preserva la forma de los objetos en el plano, aunque los puede mover. De esta manera los movimientos envían líneas rectas en líneas rectas, círculos en círculos, ... etc. sin modificación del tamaño o la forma de los mismos. Para darse cuenta de esta afirmación basta con probar que un triángulo cualquiera es enviado en otro triángulo de la misma forma y tamaño, o, dicho de otra forma, en un triángulo congruente.

A tal fin, consideremos un triángulo de vértices A, B, C y lados a, b, c y sea T un movimiento del plano. Supóngase que T envía los vértices en los puntos A′, B′ y C′, ver la figura

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Como la distancia entre los vértices adyacentes en cada uno de los triángulos es igual a la longitud de los lados correspondientes, entonces los dos triángulos tienen los tres lados iguales. En efecto, llamando a, b y c (resp. a′, b′ y c′ ) los lados, se tiene que

a = d(B, C) = d(T (B), T (C)) = d(B′, C′) = a′

de igual manera se demuestra que b = b′ y c = c′.

Si tenemos un par de movimientos en el plano, digamos T y S, podemos aplicar un movimiento a continuación de otro (primero T y luego S), para obtener otro movimiento, el cual se denotará por S ∘T y se llama la composición de S con T.

Es fácil ver que el conjunto de los movimientos con esta operación de composición, antes definida, es un grupo. Este grupo llamado Grupo de los Movimientos del Plano.

Será denotado por O(ℝ^{2}). Veamos algunos de los elementos de este grupo de los Movimientos

1. Traslaciones. Una traslación en el plano es una transformación que mueve a todos los puntos en una dirección fija. Este viene expresado por la fórmula

T (x, y) = (x + a, y + b),

donde (a, b) es un punto fijo del plano. En realidad podemos pensar en un vector v que vaya desde (0, 0) hasta (a, b) y entonces la traslación se define T (P) = P + v, para cada punto P. Recordemos que estamos trabajando en el plano afín y por lo tanto la suma de puntos con vectores tiene sentido. La distancia desde (a, b) hasta (0, 0), denotada por ||T ||, se llama norma de la traslación. Así pues, usando la fórmula de la distancia se tiene

||T || = (a^{2} + b^{2})^{1/2}

Es claro que toda traslación es una isometría, pues esta preserva la distancia entre los puntos y además no deja puntos invariantes en el plano.

Por ejemplo la traslación T (x, y) = (x + 2, y + 1) envía el cuadrado C de vértices (0, 0), (0, 1), (1, 1) y (1, 0) en el cuadrado T (C) de vértices (2, 1), (2, 2), (3, 2) y (3, 1), sin deformarlo (ver la figura). En este caso, la norma de la traslación es raíz de dos.

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2. Rotaciones. Una rotación R(C, θ) con centro C y ángulo de rotación θ es un movimiento que hace girar los puntos del plano de acuerdo a la relación siguiente: cada punto P del plano es enviado a un punto Q, tal que P y Q están sobre una circunferencia de centro C y radio igual a la distancia desde P a C. Además, el ángulo entre los radios CP y CQ es igual a θ (ver la figura)

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Una rotación deja un único punto fijo en todo el plano: el centro de rotación C.

3. Reflexiones. Una reflexión con respecto a una recta L es un movimiento del Plano que actúa de la forma siguiente: cada punto P del plano es enviado a otro punto Q, tal que existe un único punto sobre la recta L, que denotaremos por P′, tal que la recta que une a P y Q es perpendicular a la recta L, y además se tiene la siguiente igualdad entre las distancias d(P, P′) = d(P′, Q). (ver el dibujo)

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Una reflexión deja fijos a todos los puntos sobre la recta L.

4. Simetría de Deslizamiento. Este movimiento consiste de una reflexión, seguido de una traslación en la dirección del eje de reflexión.

Los dos primeros movimientos conservan la orientación del plano y por lo tanto se llaman movimientos directos. Los dos últimos invierten la orientación y se llaman movimientos inversos.

Se puede demostrar que todo movimiento del Plano se puede obtener mediante el producto de una traslación, seguido de un giro y luego una reflexión. Por lo tanto el grupo de los movimientos en el Plano está generado por las traslaciones, las rotaciones y las reflexiones.

Boletín Vol. VI, No. 1, Año 1999  147

3 Grupos Ornamentales del Plano.

El concepto de simetría de una figura plana, se puede definir con precisión usando el grupo de los movimientos del Plano O(ℝ^{2}). Dicho grupo posee varios subgrupos que serán de mucho interés para nosotros. Por ejemplo, el conjunto de todas las rotaciones que tienen su centro de rotación en un punto común O, forma un subgrupo de O(ℝ^{2}). Si C es una circunferencia con centro en O, entonces cualquier rotación con centro en O, dejará a C fijo. En general, si H es cualquier figura del plano, el conjunto de movimientos que fijan a H es un subgrupo de O(ℝ^{2}), el grupo de los movimientos en el plano, este grupo asociado a H, que denotaremos por G_{H}, se llama grupo de simetrías de H. Dicho grupo nos da una gran cantidad de información sobre los aspectos geométricos, y en especial las simetrías, de H. Se puede afirmar que mientras más simetrías posea la figura H, mayor será el grupo de simetrías asociado a ella.

Toda figura simétrica está compuesta por un motivo o forma básica K que se repite, mediante rotaciones, traslaciones y reflexiones de la figura básica. Entonces, al hacer actuar el grupo de simetrías de H sobre K, obtendremos toda la figura H. Usando el lenguaje de la teoría de grupos, podemos decir que H es la órbita de K bajo G_{H}. Estos grupos G_{H}, serán llamados grupos ornamentales del plano y se dividen en tres categorías, de acuerdo al tipo de traslaciones que se encuentren en G_{H}.

Como un primer ejemplo, de las tres categorías distintas, consideremos la figura muy conocida de una flor de 10 pétalos, en donde cada pétalo es una copia exacta de un pétalo básico K, el cual genera los restantes bajo la acción de un grupo de rotaciones de orden 10, generado por una rotación de un ángulo de 36^{∘}. (ver la figura)

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ..................................................................................  ..................................................................................  .................................................................................. .................................................................................. ..................................................................................  ..................................................................................  .................................................................................. .................................................................................. ..................................................................................  ..................................................................................  .................................................................................. .................................................................................. ..................................................................................  ..................................................................................  .................................................................................. ..................................................................................

Tenemos entonces que G_{H} ⊇ ⟨σ⟩, el grupo generado por σ, donde σ es una rotación de 36^{∘}. En este ejemplo, el grupo de simetrías de H contiene también algunas reflexiones con ejes que pasan por el centro de la flor. El número de elementos de G_{H} es igual a 20. ¿Podría el lector explicar por qué?

Como un segundo ejemplo, consideremos un friso H, de árboles en línea, que se extiende infinitamente a lo largo de una línea recta, dado por la figura siguiente :

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_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  T_{a}

Entonces observamos que H no tiene simetrías de tipo rotatorio. Sin embargo, al tomar un árbol cualquiera, aplicarle una traslación T_{a} y su inversa T_{−a} tantas veces como se desee, podemos obtener todo H. Luego un árbol genera el friso bajo la acción del grupo ⟨T_{a}⟩. Por lo tanto hemos encontrado un subgrupo de G_{H}. Como el lector puede observar, H contiene además infinitas simetrías de reflexión con ejes pasando por el centro de los árboles. En este caso, G_{H} contiene infinitos elementos.

Finalmente, estudiaremos una figura H, muy similar a un campo florido, que se extiende por todo el plano dada por el dibujo

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En este caso H se ha generado a partir de una flor K la cual se ha trasladado en dos direcciones mediante un par de traslaciones independientes T_{a} y T_{b}. Luego el grupo G_{H} contiene al grupo ⟨T_{a}, T_{b}⟩. Nuevamente se puede observar, que G_{H} contiene además infinitas reflexiones, unas con ejes verticales que pasan por los centros de las flores y otras con ejes equidistantes de dos flores. Se puede probar que G_{H} viene generado por T_{a}, T_{b} y una reflexión cualquiera de las anteriores.

Analicemos en detalle cada uno de los tres tipos de grupos ornamentales. El primer tipo de grupos, en donde no hay traslaciones, fue estudiado por Leonardo da Vinci, quien los utilizó en el diseño de las capillas dentro de las iglesias. Por esta razón se les llama grupos de Leonardo.

El segundo tipo de grupos, que contiene solo una traslación, corresponde a las decoraciones que vemos en los frisos de la fachada de los templos o en las paredes de nuestras viejas casas coloniales, para embellecer los zaguanes y las salas. Este tipo de grupo se les llama grupo de los frisos.

Finalmente, cuando el grupo de traslaciones esta generado por dos traslaciones (no paralelas), la figura básica se repite en todo el plano infinitamente generando una celosía o papel tapiz. Estos son los llamados grupos cristalográficos planos. En 1891, el cristalógrafo E. S. Feodorov, demostró que sólo existen básicamente 17 grupos cristalográficos, al hacer una clasificación exhaustiva de ellos. Más tarde en 1897 F. Klein y R. Fricke redescubrieron este resultado. Es de hacer notar que los árabes en plena edad media, conocían ya éstos 17 grupos y los empleaban en la decoración de sus mezquitas, castillos y fortalezas.

4 Grupos Cristalográficos Planos.

Nos ocuparemos ahora de otro tipo de objetos naturales, de gran brillo y belleza como son los cristales. Un cristal está formado por millones de moléculas iguales que al colocarse unas al lado de las otras en forma ordenada generan formas simétricas casi perfectas. Si pudiéramos ver dentro del cristal, con un microscopio suficientemente poderoso como para observar una a una las moléculas individuales, entonces podríamos encontrar grandes sorpresas. Entraríamos en un mundo maravilloso, no de superficies lisas y caras relucientes que reflejan la luz, sino que nos encontraríamos inmersos en una red infinita en donde la mirada se repite en cualquier ángulo. La misma molécula se estaría repitiendo en forma ordenada y periódica en todas las dimensiones del espacio!

Es posible asignarle a cada compuesto cristalino un grupo de simetrías, a fin de poder diferenciarlos bien, unos de otros. Existen miles de ellos en la naturaleza. La forma de hacer esto consiste en partir de una figura básica, o celda, formada por una cierta combinación de moléculas, y entonces ir copiando esta celda en el espacio, como una imagen reflejada, rotada o trasladada de la original. Para esto necesitamos en primer lugar considerar solo un cierto tipo de traslaciones que coloque las moléculas en el lugar que le correspondan en forma ordenada, sin que se solapen o se fundan unas con las otras. Un grupo de traslaciones con esta características, se llama un grupo discontinuo. Un grupo G de movimientos en el plano es un grupo discontinuo si para cada punto P del plano existe un entorno disco abierto D con centro en P tal que la imagen σ(P) no se encuentra en D, para todo σ en G diferente de la identidad. En otras palabras los movimientos de G no triviales, mueven a P fuera del entorno D. El siguiente resultado y su demostración se pueden ver en [2]

Teorema 1 Teorema 1. Todo grupo de traslaciones discontinuo G en el plano corresponde a uno de los siguientes

1. G consiste sólo de la identidad. G = ⟨I⟩

2. G está generado por una traslación. G = ⟨T a⟩

3. G está generado por dos traslaciones. G = ⟨T a, T b⟩

La definición formal de grupo de cristalográfico plano es la siguiente: Un grupo G de movimientos en el plano es un grupo cristalográfico si el subgrupo de G formado por las traslaciones es un grupo abeliano infinito y generado por dos elementos (libre de rango dos).

5 ¿Porqué no se puede teselar el plano con pentágonos regulares?

Los pentágonos regulares son figuras del plano que poseen muy bellas simetrías y quizás por esto, cuando Dios hizo el espacio y la geometría, los ubicó en un lugar privilegiado. El pentágono tiene 5 lados iguales. El número 5 en la religión de los pitagóricos es el número de la familia, pues representa la unión del dos y el tres que son el menor número femenino y el menor número masculino. El número uno es Dios. Es imposible llenar todo el plano con pentágonos regulares, sin cortarse unos con otros o dejar espacios libres. Esta limitación se llama restricción cristalográfica y proviene del hecho que los grupos de cristalografía no pueden tener simetrías rotatorias de orden 5, (ver el teorema 2). Curiosamente, si es posible tener grupos cristalográficos con rotaciones de orden 6.

Así pues nadie puede caminar sobre un pavimento construido perfectamente con pentágonos regulares. Sería ofender a la más bellas de las figuras geométricas. En la calles de El Cairo se ven algunos arreglos hechos de lozas pentagonales, pero éstas no son regulares (ver la figura)

Esta es la famosa teselación Cairo, la cual se puede considerar como un teselado, tanto de tipo pentagonal, como hexagonal. Otras posibilidades de teselación con pentágonos se pueden ver en [4].

Boletín Vol. VI, No. 1, Año 1999  151

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Teorema 2 Si G es un grupo cristalográfico de movimientos directos, entonces G estará generado por traslaciones y una única rotación de orden n = 1, 2, 3, 4, ó 6.

Demostración: Sabemos que G contiene una traslación T_{a} con norma mínima. Si G contiene una rotación σ, con centro en O y de orden n > 2, entonces los puntos T_{a}(O), σ ∘ T_{a}(O), σ^{2} ∘ T_{a}(O), …, σ^{n−1} ∘ T_{a}(O), determinan un polígono regular de n lados con centro en O e inscrito en un círculo de radio ||a||.

Si n > 6, entonces la longitud del lado del polígono es menor que el radio del círculo y por lo tanto d(T_{a}(O), σ ∘ T_{a}(O)) es menor que ||a||.

Por otro lado, se puede probar que el movimiento σ ∘ T_{a} ∘ σ^{−1} es una traslación. Por lo tanto S = s ∘ T_{a} ∘ σ^{−1} ∘ (T_{a})^{−1} es también una traslación, de norma :

||S|| = d(T_{a}(O), S(T_{a}(O)) = d(T_{a}(O), σ ∘ T_{a}(O)) < ||a||.

Lo cual es una contradicción. Por lo tanto n ≤ 6.

Si n = 5, los puntos T_{a}(O) y σ^{3} ∘ (T_{a})^{−1}(O) están sobre los vértices adyacentes de un polígono regular de 10 lados (ver la figura)

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_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} •  • O  T_{a}(O) (T_{a})^{−1}(O)  σ ∘ T_{a}(O) σ^{3} ∘ (T_{a})^{−1}(O)

Luego se tiene

d(T_{a}(O), σ^{3} ∘ (T_{a})^{−1}(O)) < ||a||.

La aplicación T = (σ^{3} ∘ (T_{a})^{−}1 ∘ σ^{−3}) ∘ (T_{a})^{−1} es una traslación del grupo G, cuya norma es igual a

||T || = d(O, T (O)) = d(T_{a}(O), σ^{3} ∘ (T_{a})^{−1} ∘ σ^{−3}(O))
= d(T_{a}(O), σ^{3} ∘ (T_{a})^{−1}(O)) < ||a||.

Lo cual es una contradicción. Por lo tanto n = 5 es imposible.

Teorema 3 (Feodorov) Existen 17 grupos cristalográficos no isomorfos. Estos se clasifican de acuerdo al grupo de rotaciones :

1. Grupos sin rotación : W_{1}, W_{1}^{1}, W_{1}^{2}, W_{1}^{3}

2. Grupos con rotación de orden 2 : W_{2}, W_{2}^{1}, W_{2}^{2}, W_{2}^{3}, W_{2}^{4}

3. Grupos con rotación de orden 3 : W_{3}, W_{3}^{1}, W_{3}^{2}

4. Grupos con rotación de orden 4 : W_{4}, W_{4}^{1}, W_{4}^{2}

5. Grupos con rotación de orden 6 : W_{6}, W_{6}^{1}.

La demostración completa del Teorema se puede ver en F. Blanco (1994) o en G.E. Martin (1983).

La notación W_{i}^{j}, significa lo siguiente: el subíndice i indica el tipo de rotación que contiene el grupo, así pues, 1 indica rotación de orden 1 (o la identidad), 2 indica que hay una rotación de orden 2 o giro de 180 grados, 3 indica que hay una rotación de orden 3, equivalente a un giro de 120 grados, ... etc. El superíndice j se usa para diferenciar los posibles movimientos inversos, dentro de una familia con el mismo orden del grupo de rotaciones.

Existe otra notación muy usada internacionalmente por los cristalógrafos de rayos X, en donde se usan los símbolos p, c, m, g y los enteros 1, 2, 3, 4 y 6, para clasificar estos grupos. Las letras p y m se refieren al tipo de celosía o reticulado del cristal. Si la celosía se forma a partir de una celda primitiva que se traslada, lo denotamos por p. Si se unen dos celdas primitivas para formar un bloque que, al trasladarse, genera la celosía, lo denotamos por c. Las letras m y g indican, respectivamente, una reflexión (mirror) o un deslizamiento (glide). Los números 1, 2, 3, 4 y 6 indican los ordenes de rotación de los grupos. A manera de ejemplo, se tienen las equivalencias W_{1} = p1, W_{1}^{1} = pm, W_{1}^{2} = cm, y W_{1}^{3} = pg.

6 Un ejemplo ilustrativo.

Veamos en que consiste la clasificación de los grupos cristalográficos sin rotación, en función del grupo de simetrías asociado a la celda básica. Esto por supuesto, no nos dará toda la prueba completa del gran teorema de clasificación de Feodorov, pero sí nos permitirá apreciar el estilo de pensamiento que rige a la demostración. Le sugerimos al lector cortar unos pedazos de cartulina de forma de rombos y rectángulos, de unos 3 cm. de lado, a fin de construir la celda básica. Luego tome un pedazo de papel y a medida que la vaya trasladando trace los bordes con un lápiz, para formar la red o reticulado del cristal.

Consideramos ahora las cuatro posibilidades de grupos cristalográficos sin rotaciones, el cual será denotado por W. Caso I. W no contiene movimientos inversos. En este caso el grupo estará generado sólo por el par de traslaciones perpendiculares entre si T_{a} y T_{b} . La celda básica (sombreada), es un rectángulo, el cual no posee ningún tipo de simetrías. Ver la figura

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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  T_{a} T_{b}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

Este grupo lo denotamos por W_{1}. Caso II. El grupo W contiene un sólo tipo de movimiento inverso, dado por una reflexión ρ, con eje de simetría L, paralelo a la línea que une dos vértices opuestos del cuadrilátero de la celda básica. En este caso la celda básica debe ser un rombo (ver la figura).

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En este caso, W es un grupo generado por T_{a}, T_{b} y ρ. Ahora sugerimos al lector que mueva el cartoncito de la celda básica y demuestre que se tienen las relaciones :

ρ^{2} = I, ρ ∘ T_{a} = T_{b} ∘ ρ y ρ ∘ T_{b} = T_{a} ∘ ρ

Es un hecho muy conocido de la teoría de grupos, que una vez que se tienen los generadores de un grupo y las relaciones entre los generadores, se determina el grupo G.

Este grupo se denota por W_{1}^{1}. Caso III. Supongamos ahora que W contiene una simetría de reflexión, con eje L paralelo a los lados de la celda básica, la cual debe ser rectangular. (ver la figura)

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Nuevamente, moviendo el cartón de la celda básica, deducimos las relaciones

ρ^{2} = I, ρ ∘ T_{a} = T_{a} ∘ ρ y ρ ∘ T_{b} = (T_{b})^{−1} ∘ ρ

Nótese que además de r, el grupo W contiene una simetría de deslizamiento, dada por ρ ∘ T_{a}. Este grupo lo denotamos por W_{1}^{2}. Caso IV. Finalmente, consideremos el caso en que el único movimiento inverso de W es una simetría de deslizamiento γ, dada por la traslación T_{a} seguida por una reflexión ρ con eje L, de tal forma que γ = ρ ∘ T_{a}. Entonces la celda básica debe ser rectangular (ver la figura).

En este caso se tienen las siguientes relaciones entre los generadores :

γ^{2} = T_{a}, γ ∘ T_{b} = T_{−b} ∘ γ, γ ∘ T_{a} = T_{a} ∘ γ

Este grupo lo denotamos por W_{1}^{3}.

Como las relaciones entre los generadores de cada uno de estos grupos son diferentes, concluimos que todos los grupos son esencialmente distintos (no isomorfos).

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7 Conclusiones.

La geometría de los movimientos debería ocupar un lugar importante en el curriculum de nuestras licenciaturas en Matemáticas. Además de incentivar el interés por aquellos aspectos geométricos de la Matemática, es un punto de convergencia de varias ramas como la geometría euclideana , el álgebra lineal y la teoría de grupos. Sería deseable que los estudiantes de Educación Matemática, orientados hacia la docencia a nivel de bachillerato, también tomaran, en algún momento durante la carrera, un curso de este tipo.

Es posible dar un curso de álgebra, que introduzca la teoría de grupos, usando como pretexto los movimientos en el plano y el estudio de los grupos cristalográficos. El libro de Armstrong (1980) pudiera servir de guía en esta tarea. Una vez que el estudiante maneje con familiaridad las simetrías del plano y el Teorema de Feodorov, dispondrá de una base muy sólida en cuanto a Teoría de Grupos se refiere y podrá abordar con bastante posibilidades de éxito el estudio del álgebra abstracta. Para el estudio de las simetrías y su relación con el arte, la naturaleza y la ciencia, recomendamos el libro clásico Simetría de Hermann Weyl.

Gracias a la tecnología actual, la enseñanza de la geometría de los movimientos, puede ser una tarea muy agradable tanto para el profesor como para el estudiante. El programa Maple V, versión 4 contiene muchas rutinas que permiten visualizar el efecto de las transformaciones en el plano. Otro recurso interesante por sus posibilidades en la construcción de celosías es el Paint, que viene incluido en el Paquete Windows. Partiendo de una celda básica, el Paint permite trasladarla, rotarla y reflejarla, para construir la red o celosía de cualquiera de los 17 grupos de Cristales. Esto me permitió construir fácilmente los diseños ornamentales que ven en el anexo.

Como una continuación al estudio de los grupos cristalográficos planos se pueden señalar dos vías. Una de ellas sería estudiar los grupos cristalográficos en el espacio. La otra pudiera ser estudiar las teselaciones en el plano. Siguiendo cualquiera de las dos alternativas nos encontraremos con una gran cantidad de problemas interesantes, que nos pueden motivar a aprender algo mas de geometría, combinatoria, topología, teoría de grupos y grafos.

Agradecimiento. El autor agradece al Dr. Oswaldo Araujo por sus recomendaciones sobre este artículo.

Referencias y Lecturas Sugeridas

[1 ] Armstrong, M. A. Groups and Simmetry. Springer, New York 1980.

[2 ] Blanco, M. F. Movimientos y Simetrías. Universidad de Valladolid, España, 1994.

[3 ] Coxeter, H.S.M. Introduction to Geometry. John Wiley & Son, New York, 1961.

[4 ] Gardner, M. Mathematical Games. Scientific American, 1975.

[5 ] Martin, G. E. Transformation Geometry. Springer, New York, 1983.

[6 ] Stewart, I. Concepts of Modern Mathematics. Penguin Books, New York, 1978.

[7 ] H. Weyl Simetría. Princeton University Press, Princeton, New Jersey, 1952. Re-edición 1980.