Quizas esta es la ultima entrada de esta clase, ha sido interesante ver como las entradas de otros benefician nuestro proyecto y como nos enriquecemos de las conferencias y los videos de la profesora creo que lo mejor que me ha sucedido es que en la planificacion del proyecto he tenido tanto input de informacion, tantos puntos de vista que ahora que comienzo a realizar el proyecto fisicamente estoy en la madurez de las conclusiones y en una perspectiva de objetivos y resultados amplia pero en buena direccion. Estoy tan motivada hacia el proyecto y a ver su feliz realizacion que ya estoy satisfecha de home work que he hecho para lograrlo y estoy segura de una feliz y hermosa conclusion...
Gracias a todos los que de una u otra forma contribuyeron a este logro. Maritza
Ps Ha sido un placer...
martes, 29 de mayo de 2007
viernes, 25 de mayo de 2007
color, espacio
DEFINICIONES DE ESPACIO:
- Extensión indefinida, medio sin limites que contienen todas las extensiones finitas.
- Parte de esta extensión que ocupa cada cuerpo.
- Distancia entre dos o más objetos
- Transcurso de tiempo : Hablar por espacio de una hora
468x60. Bloq Publicitario
Ultimas noticias Publicar mi artículo Favoritos Página de inicio
CUALIDADES DEL ESPACIO ARQUITECTÓNICO
- Reforzando con un cambio de piso, - Utilizando elementos verticales
- Cambiando los niveles de piso y plafones, - Cambiando la forma del evolvente, - Por quiebres en el muro, - Cambio de forma en la planta ,
- Cambio de mobiliario, - Cambiando la textura, color, material de los muros, - Cambiando la iluminación
CUALIDADES DE LOS ESPACIOS DEACUERDO A SU PERCEPCIÓN.
Espacio Estatico.- Es aquel que se percibe inmediatamente con un vistazo y no necesito recorrerlo , suele estar aislado y desarticulado .
Espacio Fluido o Dinamico.- Es aquel que para percibirlo todo necesito recorrerlo.
Espacio Universal .- Es aquel que va a servir para un gran número de actividades que se van a desarrollar en el ( no tiene una actividad específica) .
Espacio Partícular.- Es aquel que se va a utilizar para una actividad específica y es muy difícil su cambio de uso.
Espacio Estable.- Es aquel que nos da una sensación de rigidez y seguridada ( cochera, bodega).
Espacio Inestable.- Es aquel que se caracteriza por su ligereza o sensación de desequilibrio.
Espacio Articulado.- Es aquel que esta diseñado específicamente para la relación que existe entre las actividades y el mobiliario ( Sala de Juegos, Gimnasio).
Espacio Inarticulado.- Va a ser aquel que va llevar una forma independiente de la actividad que se va a realizar en el y tampoco concidera el mobiliario que se va a utilizar ( Espacio o Salón Multiusos).
Espacio Equilibrado.- Va a ser aquel que sigue fielmente un eje de simetria.
Espacio Desequilibrado.- Es aquel completamente asimetrico.
Espacio Abierto.- Es aquel que tiene fugas visuales.
Espacio Cerrado.- Es aquel que no tiene fugas.
Espacio Opresivo.- Es aquel que nos da una sensación de estreches o de incomodidad.
Espacio Expansivo.- Se siente psicologicamente que el espacio se amplia ( Teatro, Cine, Circo).
Espacio Direccional.- Es aquel que nos dirige o nos conduce por su forma a otro espacio.
Espacio sin dirección.- Es aquel que no nos conduce a otro espacio pero tampoco sabemos hacia donde vamos.
Los Espacios van adquirir ciertas caracteristicas para percibirlos dependiendo de la forma, textura, color, iluminación, ventilación , del mobiliario, de la vegetación, alturas y desniveles.
Cada una de las partes que componen un programa radiofonico o de televisión
- Pequeña pieza de metal más baja que los caracteres tipográficos que sirven para separar palabras
- Extensión indefinida en tres diemenciones que constituye el
- objeto de la geometría clásica llamada geometría del espácio
- Conjunto provisto de algunas estructuras algebraicas, geometrícas o topológicas: Espacio Vectorial
- Zona de separación entre dos líneas consecutivas del pentagrama
- Espacio Aereo, Zona Atmosferica de soberanía de un estado situado sobre el territorio terrestre y las aguas jurisdiccionales
- Espacio verde.- espacio reservado a parques y jardínes en una zona urbana
- Espacio Vital.- Superficie indispensable para vivir una población dada.
EL ESPACIO EN EL DISEÑO BI –TRIDIMENSIONAL
El Espacio Positivo.- Es el que rodea a una forma negativa. Espacio Negativo.- Es el que rodea a una forma positiva. Similarmente todas las formas negativas contienen espacios negativos así como las positivas contienen espacios positivos. El Espacio Liso e Ilusorio.- Es cuando todas las formas parecen reposar sobre el plano de la imagen y ser paralelas a él. Las formas mismas deben también ser lisas y aparecer equidistantes del ojo, ninguna de ellas más cerca, ninguna más lejos. Sin embargo es muy posible que podamos sentir como muy profundo al espacio que rodea las formas dejando que tales formas aparezcan flotando sobre el plano de la imagen. El Espacio Ilusorio.- Es cuando todas las formas no parecen reposar sobre el plano de la imagen o ser paralelas a él. Algunas formas parecen avanzar, algunas parecen retroceder, algunas parecen presentarse frontalmente y otras de manera oblicua. El Espacio Fluctuante y Conflictivo.- Es cuando parece avanzar en un sentido y retroceder en otro El Espacio Fluctuante.- Es ambiguo por que no existe una forma definida con la que podamos interpretar la situación espacial, pero el espacio conflictivo aporta una situación espacial absurda, que parece imposible de interp retar.
- Reforzando con un cambio de piso, - Utilizando elementos verticales
- Cambiando los niveles de piso y plafones, - Cambiando la forma del evolvente, - Por quiebres en el muro, - Cambio de forma en la planta ,
- Cambio de mobiliario, - Cambiando la textura, color, material de los muros, - Cambiando la iluminación
CUALIDADES DE LOS ESPACIOS DEACUERDO A SU PERCEPCIÓN.
Espacio Estatico.- Es aquel que se percibe inmediatamente con un vistazo y no necesito recorrerlo , suele estar aislado y desarticulado .
Espacio Fluido o Dinamico.- Es aquel que para percibirlo todo necesito recorrerlo.
Espacio Universal .- Es aquel que va a servir para un gran número de actividades que se van a desarrollar en el ( no tiene una actividad específica) .
Espacio Partícular.- Es aquel que se va a utilizar para una actividad específica y es muy difícil su cambio de uso.
Espacio Estable.- Es aquel que nos da una sensación de rigidez y seguridada ( cochera, bodega).
Espacio Inestable.- Es aquel que se caracteriza por su ligereza o sensación de desequilibrio.
Espacio Articulado.- Es aquel que esta diseñado específicamente para la relación que existe entre las actividades y el mobiliario ( Sala de Juegos, Gimnasio).
Espacio Inarticulado.- Va a ser aquel que va llevar una forma independiente de la actividad que se va a realizar en el y tampoco concidera el mobiliario que se va a utilizar ( Espacio o Salón Multiusos).
Espacio Equilibrado.- Va a ser aquel que sigue fielmente un eje de simetria.
Espacio Desequilibrado.- Es aquel completamente asimetrico.
Espacio Abierto.- Es aquel que tiene fugas visuales.
Espacio Cerrado.- Es aquel que no tiene fugas.
Espacio Opresivo.- Es aquel que nos da una sensación de estreches o de incomodidad.
Espacio Expansivo.- Se siente psicologicamente que el espacio se amplia ( Teatro, Cine, Circo).
Espacio Direccional.- Es aquel que nos dirige o nos conduce por su forma a otro espacio.
Espacio sin dirección.- Es aquel que no nos conduce a otro espacio pero tampoco sabemos hacia donde vamos.
Los Espacios van adquirir ciertas caracteristicas para percibirlos dependiendo de la forma, textura, color, iluminación, ventilación , del mobiliario, de la vegetación, alturas y desniveles.
RESTAURANTE MOONSOON. INTENCIONES DEL DISEÑO. Este es un combinado proyecto entre el espacio interior dinámico comprimido con el espacio exterior estático. El restaurante tiene ruptura en dos mundos extraños de carácter opuesto - fuego e hielo. Sin embargo, este dos diversos tipos de sensaciones se integran uno junto al otro en dos niveles separados.
La planta expresa la noción del espacio como carácter del hielo, que es inspirado por el edificio estacional del hielo de Sapporo, que recalcó la creación de colores y del uso de materiales. Los colores grises frescos del cristal y metales son los elementos primarios en este nivel. El vector de cristal gris fresco largo se diseña especialmente como fragmentos de un sostenido en cadena como el hielo a través del espacio. El nivel de la primera planta expresa la noción del espacio como carácter del fuego. El diseño se acentúa en los colores brillantes como tonos diferentes de rojos, amarillos, rozados, anaranjados, que se integran con un negro oscuro. Éstos colores crean una sensación de que se calientan para el propósito de relajación. Los muebles movibles se diseñan para permitir la proporción flexible de los tipos de asiento. La expresión dinámica del espacio es explorar el arreglo interesante de los elementos estructurales elegantes y la dimensión de una variable espectacular entre los muebles a través del espacio entero. Todos los elementos dentro del espacio se diseñan en dimensión de una variable geométrica interesante, que también crean un sentido de la continuidad en el espacio.
CARACTER DE HIELO.
El espacio de la planta se diseña en colores grises frescos para inspirar la noción del espacio como carácter del " hielo ", con el uso del cristal y el metal a materializó la sensación de frío. La característica más espectacular del espacio es un elemento espiral del techo que colocó sobre el compartimiento de hielo de un horno del fuego en rojo, amarillo y naranja brillantes. Había sido descrita como " bóveda como un tornado ardiente que repartía en un recipiente de presión. "
CARÁCTER DE FUEGO.
La expresión dinámica del espacio, explorada por la composición interesante de los elementos estructurales de forma espectacular de muebles. La inspiración del " fuego " como el carácter del espacio es creado por el uso de colores brillantes tales como rojo, amarillo, rosado y naranja en el fondo negro, resultando una sensación de " se calienta ". El elemento enorme, elegante que reduce radicalmente en el espacio, acto del techo tiene gusto del " fuego " que flota en el espacio. Los muebles son movibles y se diseñan especialmente para que la gente procure la creación del arreglo del espacio del asiento. Los muebles se diseñan en una forma geométrica muy interesante. (Articulo enviado por: Stewart Orozco stewart.arq@gmail.com)
WALTER GROPIUS. El verdadero instrumento de la arquitectura, más allá de todos sus tecnicismos, es el espacio. El manejo imaginativo del espacio expresa las cualidades artísticas de un diseñador. Pero éste será incapaz de dar muestras de su imaginación a menos que domine las técnicas necesarias. Desarrollad una técnica infalible y luego ponéos a merced de la imaginación. El espacio limitado -abierto o cerrado- es el medio en que se desenvuelve la arquitectura. La relación adecuada entre las masas de la edificación y los vacíos que ellas encierran, es esencial en arquitectura. Los espacios abiertos entre los edificios son una parte igualmente importante en la composición arquitectónica. Muchos de nosotros vivimos todavía, inocentemente, en un estático mundo tridimensional de concepción newtoniana, ya hace mucho tiempo desintegrado. Filósofos y científicos han reemplazado esa concepción estática por un cuadro dinámico de relatividad (relaciones espacio-temporales). Es evidente que el movimiento en el espacio, o la ilusión de tal movimiento producida por la magia del artista, está llegando a ser un estímulo cada vez más poderoso en las obras contemporáneas de arquitectura, escultura, pintura y diseño.
ENRICO TEDESCHI. Se ha dicho que el término espacio indica el carácter formal del volúmen atmosférico físico limitado por elementos construidos, o por elementos naturales, en el cual puede entrar y moverse el observador. El hecho de que sea limitado es importante para diferenciar el espacio arquitectónico de otros tipos de espacios, especialmente cuando se trata del espacio externo. El espacio arquitectónico, por ser limitado, no puede desprenderse de sus límites ni ignorarlos, y por ser recorrible, no puede separarse tampoco de la presencia de quien lo recorre. No puede apartarse de la Plástica, que es la forma de sus límites, ni de la Escala, que lo mide en relación con el observador. En principio, el espacio arquitectónico no puede considerarse otra cosa que un vacío, hasta tanto la Plástica y la Escala lo transformen en espacio propiamente tal. Los elementos que actúan para determinar la sensación espacial son múltiples, pero los principales son la forma geométrica del ámbito, sus dimensiones y escala, y la plástica de los elementos construidos que lo limitan. La experiencia espacial está siempre acompañada por el movimiento. Autor original: Arq. Arnaldo Ruiz, Universidad de las Américas, 2004. (Enviado por: prefiere anoni
WALTER GROPIUS. El verdadero instrumento de la arquitectura, más allá de todos sus tecnicismos, es el espacio. El manejo imaginativo del espacio expresa las cualidades artísticas de un diseñador. Pero éste será incapaz de dar muestras de su imaginación a menos que domine las técnicas necesarias. Desarrollad una técnica infalible y luego ponéos a merced de la imaginación. El espacio limitado -abierto o cerrado- es el medio en que se desenvuelve la arquitectura. La relación adecuada entre las masas de la edificación y los vacíos que ellas encierran, es esencial en arquitectura. Los espacios abiertos entre los edificios son una parte igualmente importante en la composición arquitectónica. Muchos de nosotros vivimos todavía, inocentemente, en un estático mundo tridimensional de concepción newtoniana, ya hace mucho tiempo desintegrado. Filósofos y científicos han reemplazado esa concepción estática por un cuadro dinámico de relatividad (relaciones espacio-temporales). Es evidente que el movimiento en el espacio, o la ilusión de tal movimiento producida por la magia del artista, está llegando a ser un estímulo cada vez más poderoso en las obras contemporáneas de arquitectura, escultura, pintura y diseño.
ENRICO TEDESCHI. Se ha dicho que el término espacio indica el carácter formal del volúmen atmosférico físico limitado por elementos construidos, o por elementos naturales, en el cual puede entrar y moverse el observador. El hecho de que sea limitado es importante para diferenciar el espacio arquitectónico de otros tipos de espacios, especialmente cuando se trata del espacio externo. El espacio arquitectónico, por ser limitado, no puede desprenderse de sus límites ni ignorarlos, y por ser recorrible, no puede separarse tampoco de la presencia de quien lo recorre. No puede apartarse de la Plástica, que es la forma de sus límites, ni de la Escala, que lo mide en relación con el observador. En principio, el espacio arquitectónico no puede considerarse otra cosa que un vacío, hasta tanto la Plástica y la Escala lo transformen en espacio propiamente tal. Los elementos que actúan para determinar la sensación espacial son múltiples, pero los principales son la forma geométrica del ámbito, sus dimensiones y escala, y la plástica de los elementos construidos que lo limitan. La experiencia espacial está siempre acompañada por el movimiento. Autor original: Arq. Arnaldo Ruiz, Universidad de las Américas, 2004. (Enviado por: prefiere anoni
- Extensión indefinida, medio sin limites que contienen todas las extensiones finitas.
- Parte de esta extensión que ocupa cada cuerpo.
- Distancia entre dos o más objetos
- Transcurso de tiempo : Hablar por espacio de una hora
468x60. Bloq Publicitario
Ultimas noticias Publicar mi artículo Favoritos Página de inicio
CUALIDADES DEL ESPACIO ARQUITECTÓNICO
- Reforzando con un cambio de piso, - Utilizando elementos verticales
- Cambiando los niveles de piso y plafones, - Cambiando la forma del evolvente, - Por quiebres en el muro, - Cambio de forma en la planta ,
- Cambio de mobiliario, - Cambiando la textura, color, material de los muros, - Cambiando la iluminación
CUALIDADES DE LOS ESPACIOS DEACUERDO A SU PERCEPCIÓN.
Espacio Estatico.- Es aquel que se percibe inmediatamente con un vistazo y no necesito recorrerlo , suele estar aislado y desarticulado .
Espacio Fluido o Dinamico.- Es aquel que para percibirlo todo necesito recorrerlo.
Espacio Universal .- Es aquel que va a servir para un gran número de actividades que se van a desarrollar en el ( no tiene una actividad específica) .
Espacio Partícular.- Es aquel que se va a utilizar para una actividad específica y es muy difícil su cambio de uso.
Espacio Estable.- Es aquel que nos da una sensación de rigidez y seguridada ( cochera, bodega).
Espacio Inestable.- Es aquel que se caracteriza por su ligereza o sensación de desequilibrio.
Espacio Articulado.- Es aquel que esta diseñado específicamente para la relación que existe entre las actividades y el mobiliario ( Sala de Juegos, Gimnasio).
Espacio Inarticulado.- Va a ser aquel que va llevar una forma independiente de la actividad que se va a realizar en el y tampoco concidera el mobiliario que se va a utilizar ( Espacio o Salón Multiusos).
Espacio Equilibrado.- Va a ser aquel que sigue fielmente un eje de simetria.
Espacio Desequilibrado.- Es aquel completamente asimetrico.
Espacio Abierto.- Es aquel que tiene fugas visuales.
Espacio Cerrado.- Es aquel que no tiene fugas.
Espacio Opresivo.- Es aquel que nos da una sensación de estreches o de incomodidad.
Espacio Expansivo.- Se siente psicologicamente que el espacio se amplia ( Teatro, Cine, Circo).
Espacio Direccional.- Es aquel que nos dirige o nos conduce por su forma a otro espacio.
Espacio sin dirección.- Es aquel que no nos conduce a otro espacio pero tampoco sabemos hacia donde vamos.
Los Espacios van adquirir ciertas caracteristicas para percibirlos dependiendo de la forma, textura, color, iluminación, ventilación , del mobiliario, de la vegetación, alturas y desniveles.
Cada una de las partes que componen un programa radiofonico o de televisión
- Pequeña pieza de metal más baja que los caracteres tipográficos que sirven para separar palabras
- Extensión indefinida en tres diemenciones que constituye el
- objeto de la geometría clásica llamada geometría del espácio
- Conjunto provisto de algunas estructuras algebraicas, geometrícas o topológicas: Espacio Vectorial
- Zona de separación entre dos líneas consecutivas del pentagrama
- Espacio Aereo, Zona Atmosferica de soberanía de un estado situado sobre el territorio terrestre y las aguas jurisdiccionales
- Espacio verde.- espacio reservado a parques y jardínes en una zona urbana
- Espacio Vital.- Superficie indispensable para vivir una población dada.
EL ESPACIO EN EL DISEÑO BI –TRIDIMENSIONAL
El Espacio Positivo.- Es el que rodea a una forma negativa. Espacio Negativo.- Es el que rodea a una forma positiva. Similarmente todas las formas negativas contienen espacios negativos así como las positivas contienen espacios positivos. El Espacio Liso e Ilusorio.- Es cuando todas las formas parecen reposar sobre el plano de la imagen y ser paralelas a él. Las formas mismas deben también ser lisas y aparecer equidistantes del ojo, ninguna de ellas más cerca, ninguna más lejos. Sin embargo es muy posible que podamos sentir como muy profundo al espacio que rodea las formas dejando que tales formas aparezcan flotando sobre el plano de la imagen. El Espacio Ilusorio.- Es cuando todas las formas no parecen reposar sobre el plano de la imagen o ser paralelas a él. Algunas formas parecen avanzar, algunas parecen retroceder, algunas parecen presentarse frontalmente y otras de manera oblicua. El Espacio Fluctuante y Conflictivo.- Es cuando parece avanzar en un sentido y retroceder en otro El Espacio Fluctuante.- Es ambiguo por que no existe una forma definida con la que podamos interpretar la situación espacial, pero el espacio conflictivo aporta una situación espacial absurda, que parece imposible de interp retar.
- Reforzando con un cambio de piso, - Utilizando elementos verticales
- Cambiando los niveles de piso y plafones, - Cambiando la forma del evolvente, - Por quiebres en el muro, - Cambio de forma en la planta ,
- Cambio de mobiliario, - Cambiando la textura, color, material de los muros, - Cambiando la iluminación
CUALIDADES DE LOS ESPACIOS DEACUERDO A SU PERCEPCIÓN.
Espacio Estatico.- Es aquel que se percibe inmediatamente con un vistazo y no necesito recorrerlo , suele estar aislado y desarticulado .
Espacio Fluido o Dinamico.- Es aquel que para percibirlo todo necesito recorrerlo.
Espacio Universal .- Es aquel que va a servir para un gran número de actividades que se van a desarrollar en el ( no tiene una actividad específica) .
Espacio Partícular.- Es aquel que se va a utilizar para una actividad específica y es muy difícil su cambio de uso.
Espacio Estable.- Es aquel que nos da una sensación de rigidez y seguridada ( cochera, bodega).
Espacio Inestable.- Es aquel que se caracteriza por su ligereza o sensación de desequilibrio.
Espacio Articulado.- Es aquel que esta diseñado específicamente para la relación que existe entre las actividades y el mobiliario ( Sala de Juegos, Gimnasio).
Espacio Inarticulado.- Va a ser aquel que va llevar una forma independiente de la actividad que se va a realizar en el y tampoco concidera el mobiliario que se va a utilizar ( Espacio o Salón Multiusos).
Espacio Equilibrado.- Va a ser aquel que sigue fielmente un eje de simetria.
Espacio Desequilibrado.- Es aquel completamente asimetrico.
Espacio Abierto.- Es aquel que tiene fugas visuales.
Espacio Cerrado.- Es aquel que no tiene fugas.
Espacio Opresivo.- Es aquel que nos da una sensación de estreches o de incomodidad.
Espacio Expansivo.- Se siente psicologicamente que el espacio se amplia ( Teatro, Cine, Circo).
Espacio Direccional.- Es aquel que nos dirige o nos conduce por su forma a otro espacio.
Espacio sin dirección.- Es aquel que no nos conduce a otro espacio pero tampoco sabemos hacia donde vamos.
Los Espacios van adquirir ciertas caracteristicas para percibirlos dependiendo de la forma, textura, color, iluminación, ventilación , del mobiliario, de la vegetación, alturas y desniveles.
RESTAURANTE MOONSOON. INTENCIONES DEL DISEÑO. Este es un combinado proyecto entre el espacio interior dinámico comprimido con el espacio exterior estático. El restaurante tiene ruptura en dos mundos extraños de carácter opuesto - fuego e hielo. Sin embargo, este dos diversos tipos de sensaciones se integran uno junto al otro en dos niveles separados.
La planta expresa la noción del espacio como carácter del hielo, que es inspirado por el edificio estacional del hielo de Sapporo, que recalcó la creación de colores y del uso de materiales. Los colores grises frescos del cristal y metales son los elementos primarios en este nivel. El vector de cristal gris fresco largo se diseña especialmente como fragmentos de un sostenido en cadena como el hielo a través del espacio. El nivel de la primera planta expresa la noción del espacio como carácter del fuego. El diseño se acentúa en los colores brillantes como tonos diferentes de rojos, amarillos, rozados, anaranjados, que se integran con un negro oscuro. Éstos colores crean una sensación de que se calientan para el propósito de relajación. Los muebles movibles se diseñan para permitir la proporción flexible de los tipos de asiento. La expresión dinámica del espacio es explorar el arreglo interesante de los elementos estructurales elegantes y la dimensión de una variable espectacular entre los muebles a través del espacio entero. Todos los elementos dentro del espacio se diseñan en dimensión de una variable geométrica interesante, que también crean un sentido de la continuidad en el espacio.
CARACTER DE HIELO.
El espacio de la planta se diseña en colores grises frescos para inspirar la noción del espacio como carácter del " hielo ", con el uso del cristal y el metal a materializó la sensación de frío. La característica más espectacular del espacio es un elemento espiral del techo que colocó sobre el compartimiento de hielo de un horno del fuego en rojo, amarillo y naranja brillantes. Había sido descrita como " bóveda como un tornado ardiente que repartía en un recipiente de presión. "
CARÁCTER DE FUEGO.
La expresión dinámica del espacio, explorada por la composición interesante de los elementos estructurales de forma espectacular de muebles. La inspiración del " fuego " como el carácter del espacio es creado por el uso de colores brillantes tales como rojo, amarillo, rosado y naranja en el fondo negro, resultando una sensación de " se calienta ". El elemento enorme, elegante que reduce radicalmente en el espacio, acto del techo tiene gusto del " fuego " que flota en el espacio. Los muebles son movibles y se diseñan especialmente para que la gente procure la creación del arreglo del espacio del asiento. Los muebles se diseñan en una forma geométrica muy interesante. (Articulo enviado por: Stewart Orozco stewart.arq@gmail.com)
WALTER GROPIUS. El verdadero instrumento de la arquitectura, más allá de todos sus tecnicismos, es el espacio. El manejo imaginativo del espacio expresa las cualidades artísticas de un diseñador. Pero éste será incapaz de dar muestras de su imaginación a menos que domine las técnicas necesarias. Desarrollad una técnica infalible y luego ponéos a merced de la imaginación. El espacio limitado -abierto o cerrado- es el medio en que se desenvuelve la arquitectura. La relación adecuada entre las masas de la edificación y los vacíos que ellas encierran, es esencial en arquitectura. Los espacios abiertos entre los edificios son una parte igualmente importante en la composición arquitectónica. Muchos de nosotros vivimos todavía, inocentemente, en un estático mundo tridimensional de concepción newtoniana, ya hace mucho tiempo desintegrado. Filósofos y científicos han reemplazado esa concepción estática por un cuadro dinámico de relatividad (relaciones espacio-temporales). Es evidente que el movimiento en el espacio, o la ilusión de tal movimiento producida por la magia del artista, está llegando a ser un estímulo cada vez más poderoso en las obras contemporáneas de arquitectura, escultura, pintura y diseño.
ENRICO TEDESCHI. Se ha dicho que el término espacio indica el carácter formal del volúmen atmosférico físico limitado por elementos construidos, o por elementos naturales, en el cual puede entrar y moverse el observador. El hecho de que sea limitado es importante para diferenciar el espacio arquitectónico de otros tipos de espacios, especialmente cuando se trata del espacio externo. El espacio arquitectónico, por ser limitado, no puede desprenderse de sus límites ni ignorarlos, y por ser recorrible, no puede separarse tampoco de la presencia de quien lo recorre. No puede apartarse de la Plástica, que es la forma de sus límites, ni de la Escala, que lo mide en relación con el observador. En principio, el espacio arquitectónico no puede considerarse otra cosa que un vacío, hasta tanto la Plástica y la Escala lo transformen en espacio propiamente tal. Los elementos que actúan para determinar la sensación espacial son múltiples, pero los principales son la forma geométrica del ámbito, sus dimensiones y escala, y la plástica de los elementos construidos que lo limitan. La experiencia espacial está siempre acompañada por el movimiento. Autor original: Arq. Arnaldo Ruiz, Universidad de las Américas, 2004. (Enviado por: prefiere anoni
WALTER GROPIUS. El verdadero instrumento de la arquitectura, más allá de todos sus tecnicismos, es el espacio. El manejo imaginativo del espacio expresa las cualidades artísticas de un diseñador. Pero éste será incapaz de dar muestras de su imaginación a menos que domine las técnicas necesarias. Desarrollad una técnica infalible y luego ponéos a merced de la imaginación. El espacio limitado -abierto o cerrado- es el medio en que se desenvuelve la arquitectura. La relación adecuada entre las masas de la edificación y los vacíos que ellas encierran, es esencial en arquitectura. Los espacios abiertos entre los edificios son una parte igualmente importante en la composición arquitectónica. Muchos de nosotros vivimos todavía, inocentemente, en un estático mundo tridimensional de concepción newtoniana, ya hace mucho tiempo desintegrado. Filósofos y científicos han reemplazado esa concepción estática por un cuadro dinámico de relatividad (relaciones espacio-temporales). Es evidente que el movimiento en el espacio, o la ilusión de tal movimiento producida por la magia del artista, está llegando a ser un estímulo cada vez más poderoso en las obras contemporáneas de arquitectura, escultura, pintura y diseño.
ENRICO TEDESCHI. Se ha dicho que el término espacio indica el carácter formal del volúmen atmosférico físico limitado por elementos construidos, o por elementos naturales, en el cual puede entrar y moverse el observador. El hecho de que sea limitado es importante para diferenciar el espacio arquitectónico de otros tipos de espacios, especialmente cuando se trata del espacio externo. El espacio arquitectónico, por ser limitado, no puede desprenderse de sus límites ni ignorarlos, y por ser recorrible, no puede separarse tampoco de la presencia de quien lo recorre. No puede apartarse de la Plástica, que es la forma de sus límites, ni de la Escala, que lo mide en relación con el observador. En principio, el espacio arquitectónico no puede considerarse otra cosa que un vacío, hasta tanto la Plástica y la Escala lo transformen en espacio propiamente tal. Los elementos que actúan para determinar la sensación espacial son múltiples, pero los principales son la forma geométrica del ámbito, sus dimensiones y escala, y la plástica de los elementos construidos que lo limitan. La experiencia espacial está siempre acompañada por el movimiento. Autor original: Arq. Arnaldo Ruiz, Universidad de las Américas, 2004. (Enviado por: prefiere anoni
color, espacio, fisica quantica etc
What is Quantum Physics?
Quantum physics is a branch of science that deals with discrete, indivisible units of energy called quanta as described by the Quantum Theory. There are five main ideas represented in Quantum Theory:
Energy is not continuous, but comes in small but discrete units. 1
The elementary particles behave both like particles and like waves. 2
The movement of these particles is inherently random. 3
It is physically impossible to know both the position and the momentum of a particle at the same time. The more precisely one is known, the less precise the measurement of the other is.4
The atomic world is nothing like the world we live in. 5
While at a glance this may seem like just another strange theory, it contains many clues as to the fundamental nature of the universe and is more important then even relativity in the grand scheme of things (if any one thing at that level could be said to be more important then anything else). Furthermore, it describes the nature of the universe as being much different then the world we see. As Niels Bohr said, "Anyone who is not shocked by quantum theory has not understood it." 6
Particle/Wave Duality
Particle/wave duality is perhaps the easiest way to get aquatinted with quantum theory because it shows, in a few simple experiments, how different the atomic world is from our world.
First let's set up a generic situation to avoid repetition. In the center of the experiment is a wall with two slits in it. To the right we have a detector. What exactly the detector is varies from experiment to experiment, but it's purpose stays the same: detect how many of whatever we are sending through the experiment reaches each point. To the left of the wall we have the originating point of whatever it is we are going to send through the experiment. That's the experiment: send something through two slits and see what happens. For simplicity, assume that nothing bounces off of the walls in funny patterns to mess up the experiment.
First try the experiment with bullets. Place a gun at the originating point and use a sandbar as the detector. First try covering one slit and see what happens. You get more bullets near the center of the slit and less as you get further away. When you cover the other slit, you see the same thing with respect to the other slit. Now open both slits. You get the sum of the result of opening each slit. 7 The most bullets are found in the middle of the two slits with less being found the further you get from the center.
Well, that was fun. Let's try it on something more interesting: water waves. Place a wave generator at the originating point and detect using a wave detector that measures the height of the waves that pass. Try it with one slit closed. You see a result just like that of the bullets. With the other slit closed the result is the same. Now try it with both slits open. Instead of getting the sum of the results of each slit being open, you see a wavy pattern 8; in the center there is a wave greater then the sum of what appeared there each time only one slit was open. Next to that large wave was a wave much smaller then what appeared there during either of the two single slit runs. Then the pattern repeats; large wave, though not nearly as large as the center one, then small wave. This makes sense; in some places the waves reinforced each other creating a larger wave, in other places they canceled out. In the center there was the most overlap, and therefore the largest wave. In mathematical terms, instead of the resulting intensity being the sum of the squares of the heights of the waves, it is the square of the sum.
While the result was different from the bullets, there is still nothing unusual about it; everyone has seen this effect when the waves from two stones that are dropped into a lake in different places overlap. The difference between this experiment and the previous one is easily explained by saying that while the bullets each went through only one slit, the waves each went through both slits and were thus able to interfere with themselves.
Now try the experiment with electrons. Recall that electrons are negatively charged particles that make up the outer layers of the atom. Certainly they could only go through one slit at a time, so their pattern should look like that of the bullets, right? Let's find out. (NOTE: to actually perform this exact experiment would take detectors more advanced then any on earth at this time. However, the experiments have been done with neutron beams 9 and the results were the same as those presented here. A slightly different experiment was done to show that electrons would behave the same way 10. For reasons of familiarity, we speak of electrons here instead of neutrons.) Place an electron gun at the originating point and an electron detector in the detector place. First try opening only one slit, then just the other. The results are just like those of the bullets and the waves. Now open both slits. The result is just like the waves!11
There must be some explanation. After all, an electron couldn't go through both slits. Instead of a continuous stream of electrons, let's turn the electron gun down so that at any one time only one electron is in the experiment. Now the electrons won't be able to cause trouble since there is no one else to interfere with. The result should now look like the bullets. But it doesn't! 12 It would seem that the electrons do go through both slits.
This is indeed a strange occurrence; we should watch them ourselves to make sure that this is indeed what is happening. So, we put a light behind the wall so that we can see a flash from the slit that the electron went through, or a flash from both slits if it went through both. Try the experiment again. As each electron passes through, there is a flash in only one of the two slits. So they do only go through one slit! But something else has happened too: the result now looks like the result of the bullets experiment!! 13
Obviously the light is causing problems. Perhaps if we turned down the intensity of the light, we would be able to see them without disturbing them. When we try this, we notice first that the flashes we see are the same size. Also, some electrons now get by without being detected. 14 This is because light is not continuous but made up of particles called photons. Turning down the intensity only lowers the number of photons given out by the light source.15 The particles that flash in one slit or the other behave like the bullets, while those that go undetected behave like waves16.
Well, we are not about to be outsmarted by an electron, so instead of lowering the intensity of the light, why don't we lower the frequency. The lower the frequency the less the electron will be disturbed, so we can finally see what is actually going on. Lower the frequency slightly and try the experiment again. We see the bullet curve 17. After lowering it for a while, we finally see a curve that looks somewhat like that of the waves! There is one problem, though. Lowering the frequency of light is the same as increasing it's wavelength 18, and by the time the frequency of the light is low enough to detect the wave pattern the wavelength is longer then the distance between the slits so we can no longer see which slit the electron went through 19.
So have the electrons outsmarted us? Perhaps, but they have also taught us one of the most fundamental lessons in quantum physics - an observation is only valid in the context of the experiment in which it was performed 20. If you want to say that something behaves a certain way or even exists, you must give the context of this behavior or existence since in another context it may behave differently or not exist at all. We can't just say that an electron is a particle, since we have already seen proof that this is not always the case. We can only say that when we observe the electron in the two slit experiment it behaves like a particle. To see how it would behave under different conditions, we must perform a different experiment.
The Copenhagen Interpretation
So sometimes a particle acts like a particle and other times it acts like a wave. So which is it? According to Niels Bohr, who worked in Copenhagen when he presented what is now known as the Copenhagen interpretation of quantum theory, the particle is what you measure it to be. When it looks like a particle, it is a particle. When it looks like a wave, it is a wave. Furthermore, it is meaningless to ascribe any properties or even existence to anything that has not been measured21. Bohr is basically saying that nothing is real unless it is observed.
While there are many other interpretations of quantum physics, all based on the Copenhagen interpretation, the Copenhagen interpretation is by far the most widely used because it provides a "generic" interpretation that does not try to say any more then can be proven. Even so, the Copenhagen interpretation does have a flaw that we will discuss later. Still, since after 70 years no one has been able to come up with an interpretation that works better then the Copenhagen interpretation, that is the one we will use. We will discuss one of the alternatives later.
The Wave Function
In 1926, just weeks after several other physicists had published equations describing quantum physics in terms of matrices, Erwin Schrödinger created quantum equations based on wave mathematics22 , a mathematical system that corresponds to the world we know much more then the matrices. After the initial shock, first Schrödinger himself then others proved that the equations were mathematically equivalent 23. Bohr then invited Schrödinger to Copenhagen where they found that Schrödinger's waves were in fact nothing like real waves. For one thing, each particle that was being described as a wave required three dimensions 24. Even worse, from Schrödinger's point of view, particles still jumped from one quantum state to another; even expressed in terms of waves space was still not continuous. Upon discovering this, Schrödinger remarked to Bohr that "Had I known that we were not going to get rid of this damned quantum jumping, I never would have involved myself in this business." 25
Unfortunately, even today people try to imagine the atomic world as being a bunch of classical waves. As Schrödinger found out, this could not be further from the truth. The atomic world is nothing like our world, no matter how much we try to pretend it is. In many ways, the success of Schrödinger's equations has prevented people from thinking more deeply about the true nature of the atomic world 26.
The Collapse of the Wave Function
So why bring up the wave function at all if it hampers full appreciation of the atomic world? For one thing, the equations are much more familiar to physicists, so Schrödinger's equations are used much more often then the others. Also, it turns out that Bohr liked the idea and used it in his Copenhagen interpretation. Remember our experiment with electrons? Each possible route that the electron could take, called a ghost, could be described by a wave function 27. As we shall see later, the "damned quantum jumping" insures that there are only a finite, though large, number of possible routes. When no one is watching, the electron take every possible route and therefore interferes with itself28. However, when the electron is observed, it is forced to choose one path. Bohr called this the "collapse of the wave function"29. The probability that a certain path will be chosen when the wave function collapses is, essentially, the square of the path's wave function 30.
Bohr reasoned that nature likes to keep it possibilities open, and therefore follows every possible path. Only when observed is nature forced to choose only one path, so only then is just one path taken 31.
The Uncertainty Principle
Wait a minute… probability??? If we are going to destroy the wave pattern by observing the experiment, then we should at least be able to determine exactly where the electron goes. Newton figured that much out back in the early eighteenth century; just observe the position and momentum of the electron as it leaves the electron gun and we can determine exactly where it goes.
Well, fine. But how exactly are we to determine the position and the momentum of the electron? If we disturb the electrons just in seeing if they are there or not, how are we possibly going to determine both their position and momentum? Still, a clever enough person, say Albert Einstein, should be able to come up with something, right?
Unfortunately not. Einstein did actually spend a good deal of his life trying to do just that and failed 32. Furthermore, it turns out that if it were possible to determine both the position and the momentum at the same time, Quantum Physics would collapse 33. Because of the latter, Werner Heisenberg proposed in 1925 that it is in fact physically impossible to do so. As he stated it in what now is called the Heisenberg Uncertainty Principle, if you determine an object's position with uncertainty x, there must be an uncertainty in momentum, p, such that xp > h/4pi, where h is Planck's constant 34 (which we will discuss shortly). In other words, you can determine either the position or the momentum of an object as accurately as you like, but the act of doing so makes your measurement of the other property that much less. Human beings may someday build a device capable of transporting objects across the galaxy, but no one will ever be able to measure both the momentum and the position of an object at the same time. This applies not only to electrons but also to objects such as tennis balls and toasters, though for these objects the amount of uncertainty is so small compared to there size that it can safely be ignored under most circumstances.
The EPR Experiment
"God does not play dice" was Albert Einstein's reply to the Uncertainty Principle. 35 Thus being his belief, he spent a good deal of his life after 1925 trying to determine both the position and the momentum of a particle. In 1935, Einstein and two other physicists, Podolski and Rosen, presented what is now known as the EPR paper in which they suggested a way to do just that. The idea is this: set up an interaction such that two particles are go off in opposite directions and do not interact with anything else. Wait until they are far apart, then measure the momentum of one and the position of the other. Because of conservation of momentum, you can determine the momentum of the particle not measured, so when you measure it's position you know both it's momentum and position 36. The only way quantum physics could be true is if the particles could communicate faster then the speed of light, which Einstein reasoned would be impossible because of his Theory of Relativity.
In 1982, Alain Aspect, a French physicist, carried out the EPR experiment 37. He found that even if information needed to be communicated faster then light to prevent it, it was not possible to determine both the position and the momentum of a particle at the same time 38. This does not mean that it is possible to send a message faster then light, since viewing either one of the two particles gives no information about the other39. It is only when both are seen that we find that quantum physics has agreed with the experiment. So does this mean relativity is wrong? No, it just means that the particles do not communicate by any means we know about. All we know is that every particle knows what every other particle it has ever interacted with is doing.
The Quantum and Planck's Constant
So what is that h that was so important in the Uncertainty Principle? Well, technically speaking, it's 6.63 X 10-34 joule-seconds 40. It's call Planck's constant after Max Planck who, in 1900, introduced it in the equation E=hv where E is the energy of each quantum of radiation and v is it's frequency41. What this says is that energy is not continuous as everyone had assumed but only comes in certain finite sizes based on Planck's constant.
At first physicists thought that this was just a neat mathematical trick Planck used to explain experimental results that did not agree with classical physics. Then, in 1904, Einstein used this idea to explain certain properties of light--he said that light was in fact a particle with energy E=hv 42. After that the idea that energy isn't continuous was taken as a fact of nature - and with amazing results. There was now a reason why electrons were only found in certain energy levels around the nucleus of an atom 43. Ironically, Einstein gave quantum theory the push it needed to become the valid theory it is today, though he would spend the rest of his lift trying to prove that it was not a true description of nature.
Also, by combining Planck's constant, the constant of gravity, and the speed of light, it is possible to create a quantum of length (about 10-35 meter) and a quantum of time (about 10-43 sec), called, respectively, Planck's length and Planck's time 44. While saying that energy is not continuous might not be too startling to the average person, since what we commonly think of as energy is not all that well defined anyway, it is startling to say that there are quantities of space and time that cannot be broken up into smaller pieces. Yet it is exactly this that gives nature a finite number of routes to take when an electron interferes with itself.
Although it may seem like the idea that energy is quantized is a minor part of quantum physics when compared with ghost electrons and the uncertainty principle, it really is a fundamental statement about nature that caused everything else we've talked about to be discovered. And it is always true. In the strange world of the atom, anything that can be taken for granted is a major step towards an "atomic world view".
Schrödinger's Cat
Remember a while ago I said there was a problem with the Copenhagen interpretation? Well, you now know enough of what quantum physics is to be able to discuss what it isn't, and by far the biggest thing it isn't is complete. Sure, the math seems to be complete, but the theory includes absolutely nothing that would tie the math to any physical reality we could imagine. Furthermore, quantum physics leaves us with a rather large open question: what is reality? The Copenhagen interpretation attempts to solve this problem by saying that reality is what is measured. However, the measuring device itself is then not real until it is measured. The problem, which is known as the measurement problem, is when does the cycle stop?
Remember that when we last left Schrödinger he was muttering about the "damned quantum jumping." He never did get used to quantum physics, but, unlike Einstein, he was able to come up with a very real demonstration of just how incomplete the physical view of our world given by quantum physics really is. Imagine a box in which there is a radioactive source, a Geiger counter (or anything that records the presence of radioactive particles), a bottle of cyanide, and a cat. The detector is turned on for just long enough that there is a fifty-fifty chance that the radioactive material will decay. If the material does decay, the Geiger counter detects the particle and crushes the bottle of cyanide, killing the cat. If the material does not decay, the cat lives. To us outside the box, the time of detection is when the box is open. At that point, the wave function collapses and the cat either dies or lives. However, until the box is opened, the cat is both dead and alive 45.
On one hand, the cat itself could be considered the detector; it's presence is enough to collapse the wave function 46. But in that case, would the presence of a rat be enough? Or an ameba? Where is the line drawn 47? On the other hand, what if you replace the cat with a human (named "Wigner's friend" after Eugene Wigner, the physicist who developed many derivations of the Schrödinger's cat experiment). The human is certainly able to collapse the wave function, yet to us outside the box the measurement is not taken until the box is opened 48. If we try to develop some sort of "quantum relativity" where each individual has his own view of the world, then what is to prevent the world from getting "out of sync" between observers?
While there are many different interpretations that solve the problem of Schrödinger’s Cat, one of which we will discuss shortly, none of them are satisfactory enough to have convinced a majority of physicists that the consequences of these interpretation s are better then the half dead cat. Furthermore, while these interpretations do prevent a half dead cat, they do not solve the underlying measurement problem. Until a better intrepretation surfaces, we are left with the Copenhagen interpretation and it's half dead cat. We can certainly understand how Schrödinger feels when he says, "I don't like it, and I'm sorry I ever had anything to do with it."49 Yet the problem doesn't go away; it is just left for the great thinkers of tomorrow.
The Infinity Problem
There is one last problem that we will discuss before moving on to the alternative interpretation. Unlike the others, this problem lies primarily in the mathematics of a certain part of quantum physics called quantum electrodynamics, or QED. This branch of quantum physics explains the electromagnetic interaction in quantum terms. The problem is, when you add the interaction particles and try to solve Schrödinger's wave equation, you get an electron with infinite mass, infinite energy, and infinite charge50. There is no way to get rid of the infinities using valid mathematics, so, the theorists simply divide infinity by infinity and get whatever result the guys in the lab say the mass, energy, and charge should be51. Even fudging the math, the other results of QED are so powerful that most physicists ignore the infinities and use the theory anyway 52. As Paul Dirac, who was one of the physicists who published quantum equations before Schrödinger, said, "Sensible mathematics involves neglecting a quantity when it turns out to be small - not neglecting it just because it is infinitely great and you do not want it!". 53
Many Worlds
One other interpretation, presented first by Hugh Everett III in 1957, is the many worlds or branching universe interpretation54. In this theory, whenever a measurement takes place, the entire universe divides as many times as there are possible outcomes of the measurement. All universes are identical except for the outcome of that measurement 55. Unlike the science fiction view of "parallel universes", it is not possible for any of these worlds to interact with each other 56.
While this creates an unthinkable number of different worlds, it does solve the problem of Schrödinger's cat. Instead of one cat, we now have two; one is dead, the other alive. However, it has still not solved the measurement problem 57! If the universe split every time there was more then one possibility, then we would not see the interference pattern in the electron experiment. So when does it split? No alternative interpretation has yet answered this question in a satisfactory way. And so the search continues…
Further Reading
If you are interested in learning more about quantum physics, here are some books that you could try (check the bibliography for more specific information on the books you are interested in):
Richard Feynman's Lectures on Physics deals with the math associated with quantum physics. If you can understand basic calculus, then this book is for you. Otherwise, while Lectures still provides some valuable information, you may find yourself lost before you get too far.
John Gribbin's In Search of Schrödinger's Cat is an excellent non-mathematical treatment of quantum physics. If you've been watching the footnotes you've seen that much of the data for this paper came from this book. It includes a good history of quantum physics. Be advised that the sections on supergravity and supersymmetry at the end are outdated.
Alastair Rae's Quantum Physics: Illusion or Reality presents the basics of quantum physics in terms of the polarization of light. It's 118 pages, half of which are devoted to a discussion of the alternate interpretations of quantum physics, can easily be read in an afternoon. It spends more time on alternate interpretations then Gribbin's book, but is less detailed in almost every other respect. I suggest reading Gribbin's book first then this book.
Bibliography
Feynman, Richard P., Robert Leighton, and Matthew Sands. The Feynman Lectures on Physics. Addison-Wesley, Reading, Massachusetts: 1965. Vol. 3: Quantum Mechanics.
Gribbin, John. In Search of Schrödinger's Cat. Bantam Books, Toronto: 1984. ISBN 0-553-34103-0
Rae, Alastair. Quantum Physics: Illusion or Reality? Cambridge University Press, London: 1986. ISBN 0-521-26023-3.
--------------------------------------------------------------------------------
1 In Search of Schrödinger's Cat pages 41 and 43.
2 Lectures on Physics page 1-1.
3 In Search of Schrödinger's Cat page 66.
4 In Search of Schrödinger's Cat page 156.
5 In Search of Schrödinger's Cat page 174.
6 In Search of Schrödinger's Cat page 5
7 Lectures on Physics pages 1-1 and 1-2.
8 Lectures on Physics pages 1-3 and 1-4.
9 Quantum Physics: Illusion or Reality page 13.
10 Quantum Physics: Illusion or Reality page 12.
11 Lectures on Physics pages 1-4 and 1-5.
12 In Search of Schrödinger's Cat page 170.
13 Lectures on Physics pages 1-6 and 1-7.
14 Lectures on Physics page 1-8.
15 Lectures on Physics page 1-8.
16 Lectures on Physics page 1-8.
17 Lectures on Physics page 1-8.
18 Lectures
Quantum physics is a branch of science that deals with discrete, indivisible units of energy called quanta as described by the Quantum Theory. There are five main ideas represented in Quantum Theory:
Energy is not continuous, but comes in small but discrete units. 1
The elementary particles behave both like particles and like waves. 2
The movement of these particles is inherently random. 3
It is physically impossible to know both the position and the momentum of a particle at the same time. The more precisely one is known, the less precise the measurement of the other is.4
The atomic world is nothing like the world we live in. 5
While at a glance this may seem like just another strange theory, it contains many clues as to the fundamental nature of the universe and is more important then even relativity in the grand scheme of things (if any one thing at that level could be said to be more important then anything else). Furthermore, it describes the nature of the universe as being much different then the world we see. As Niels Bohr said, "Anyone who is not shocked by quantum theory has not understood it." 6
Particle/Wave Duality
Particle/wave duality is perhaps the easiest way to get aquatinted with quantum theory because it shows, in a few simple experiments, how different the atomic world is from our world.
First let's set up a generic situation to avoid repetition. In the center of the experiment is a wall with two slits in it. To the right we have a detector. What exactly the detector is varies from experiment to experiment, but it's purpose stays the same: detect how many of whatever we are sending through the experiment reaches each point. To the left of the wall we have the originating point of whatever it is we are going to send through the experiment. That's the experiment: send something through two slits and see what happens. For simplicity, assume that nothing bounces off of the walls in funny patterns to mess up the experiment.
First try the experiment with bullets. Place a gun at the originating point and use a sandbar as the detector. First try covering one slit and see what happens. You get more bullets near the center of the slit and less as you get further away. When you cover the other slit, you see the same thing with respect to the other slit. Now open both slits. You get the sum of the result of opening each slit. 7 The most bullets are found in the middle of the two slits with less being found the further you get from the center.
Well, that was fun. Let's try it on something more interesting: water waves. Place a wave generator at the originating point and detect using a wave detector that measures the height of the waves that pass. Try it with one slit closed. You see a result just like that of the bullets. With the other slit closed the result is the same. Now try it with both slits open. Instead of getting the sum of the results of each slit being open, you see a wavy pattern 8; in the center there is a wave greater then the sum of what appeared there each time only one slit was open. Next to that large wave was a wave much smaller then what appeared there during either of the two single slit runs. Then the pattern repeats; large wave, though not nearly as large as the center one, then small wave. This makes sense; in some places the waves reinforced each other creating a larger wave, in other places they canceled out. In the center there was the most overlap, and therefore the largest wave. In mathematical terms, instead of the resulting intensity being the sum of the squares of the heights of the waves, it is the square of the sum.
While the result was different from the bullets, there is still nothing unusual about it; everyone has seen this effect when the waves from two stones that are dropped into a lake in different places overlap. The difference between this experiment and the previous one is easily explained by saying that while the bullets each went through only one slit, the waves each went through both slits and were thus able to interfere with themselves.
Now try the experiment with electrons. Recall that electrons are negatively charged particles that make up the outer layers of the atom. Certainly they could only go through one slit at a time, so their pattern should look like that of the bullets, right? Let's find out. (NOTE: to actually perform this exact experiment would take detectors more advanced then any on earth at this time. However, the experiments have been done with neutron beams 9 and the results were the same as those presented here. A slightly different experiment was done to show that electrons would behave the same way 10. For reasons of familiarity, we speak of electrons here instead of neutrons.) Place an electron gun at the originating point and an electron detector in the detector place. First try opening only one slit, then just the other. The results are just like those of the bullets and the waves. Now open both slits. The result is just like the waves!11
There must be some explanation. After all, an electron couldn't go through both slits. Instead of a continuous stream of electrons, let's turn the electron gun down so that at any one time only one electron is in the experiment. Now the electrons won't be able to cause trouble since there is no one else to interfere with. The result should now look like the bullets. But it doesn't! 12 It would seem that the electrons do go through both slits.
This is indeed a strange occurrence; we should watch them ourselves to make sure that this is indeed what is happening. So, we put a light behind the wall so that we can see a flash from the slit that the electron went through, or a flash from both slits if it went through both. Try the experiment again. As each electron passes through, there is a flash in only one of the two slits. So they do only go through one slit! But something else has happened too: the result now looks like the result of the bullets experiment!! 13
Obviously the light is causing problems. Perhaps if we turned down the intensity of the light, we would be able to see them without disturbing them. When we try this, we notice first that the flashes we see are the same size. Also, some electrons now get by without being detected. 14 This is because light is not continuous but made up of particles called photons. Turning down the intensity only lowers the number of photons given out by the light source.15 The particles that flash in one slit or the other behave like the bullets, while those that go undetected behave like waves16.
Well, we are not about to be outsmarted by an electron, so instead of lowering the intensity of the light, why don't we lower the frequency. The lower the frequency the less the electron will be disturbed, so we can finally see what is actually going on. Lower the frequency slightly and try the experiment again. We see the bullet curve 17. After lowering it for a while, we finally see a curve that looks somewhat like that of the waves! There is one problem, though. Lowering the frequency of light is the same as increasing it's wavelength 18, and by the time the frequency of the light is low enough to detect the wave pattern the wavelength is longer then the distance between the slits so we can no longer see which slit the electron went through 19.
So have the electrons outsmarted us? Perhaps, but they have also taught us one of the most fundamental lessons in quantum physics - an observation is only valid in the context of the experiment in which it was performed 20. If you want to say that something behaves a certain way or even exists, you must give the context of this behavior or existence since in another context it may behave differently or not exist at all. We can't just say that an electron is a particle, since we have already seen proof that this is not always the case. We can only say that when we observe the electron in the two slit experiment it behaves like a particle. To see how it would behave under different conditions, we must perform a different experiment.
The Copenhagen Interpretation
So sometimes a particle acts like a particle and other times it acts like a wave. So which is it? According to Niels Bohr, who worked in Copenhagen when he presented what is now known as the Copenhagen interpretation of quantum theory, the particle is what you measure it to be. When it looks like a particle, it is a particle. When it looks like a wave, it is a wave. Furthermore, it is meaningless to ascribe any properties or even existence to anything that has not been measured21. Bohr is basically saying that nothing is real unless it is observed.
While there are many other interpretations of quantum physics, all based on the Copenhagen interpretation, the Copenhagen interpretation is by far the most widely used because it provides a "generic" interpretation that does not try to say any more then can be proven. Even so, the Copenhagen interpretation does have a flaw that we will discuss later. Still, since after 70 years no one has been able to come up with an interpretation that works better then the Copenhagen interpretation, that is the one we will use. We will discuss one of the alternatives later.
The Wave Function
In 1926, just weeks after several other physicists had published equations describing quantum physics in terms of matrices, Erwin Schrödinger created quantum equations based on wave mathematics22 , a mathematical system that corresponds to the world we know much more then the matrices. After the initial shock, first Schrödinger himself then others proved that the equations were mathematically equivalent 23. Bohr then invited Schrödinger to Copenhagen where they found that Schrödinger's waves were in fact nothing like real waves. For one thing, each particle that was being described as a wave required three dimensions 24. Even worse, from Schrödinger's point of view, particles still jumped from one quantum state to another; even expressed in terms of waves space was still not continuous. Upon discovering this, Schrödinger remarked to Bohr that "Had I known that we were not going to get rid of this damned quantum jumping, I never would have involved myself in this business." 25
Unfortunately, even today people try to imagine the atomic world as being a bunch of classical waves. As Schrödinger found out, this could not be further from the truth. The atomic world is nothing like our world, no matter how much we try to pretend it is. In many ways, the success of Schrödinger's equations has prevented people from thinking more deeply about the true nature of the atomic world 26.
The Collapse of the Wave Function
So why bring up the wave function at all if it hampers full appreciation of the atomic world? For one thing, the equations are much more familiar to physicists, so Schrödinger's equations are used much more often then the others. Also, it turns out that Bohr liked the idea and used it in his Copenhagen interpretation. Remember our experiment with electrons? Each possible route that the electron could take, called a ghost, could be described by a wave function 27. As we shall see later, the "damned quantum jumping" insures that there are only a finite, though large, number of possible routes. When no one is watching, the electron take every possible route and therefore interferes with itself28. However, when the electron is observed, it is forced to choose one path. Bohr called this the "collapse of the wave function"29. The probability that a certain path will be chosen when the wave function collapses is, essentially, the square of the path's wave function 30.
Bohr reasoned that nature likes to keep it possibilities open, and therefore follows every possible path. Only when observed is nature forced to choose only one path, so only then is just one path taken 31.
The Uncertainty Principle
Wait a minute… probability??? If we are going to destroy the wave pattern by observing the experiment, then we should at least be able to determine exactly where the electron goes. Newton figured that much out back in the early eighteenth century; just observe the position and momentum of the electron as it leaves the electron gun and we can determine exactly where it goes.
Well, fine. But how exactly are we to determine the position and the momentum of the electron? If we disturb the electrons just in seeing if they are there or not, how are we possibly going to determine both their position and momentum? Still, a clever enough person, say Albert Einstein, should be able to come up with something, right?
Unfortunately not. Einstein did actually spend a good deal of his life trying to do just that and failed 32. Furthermore, it turns out that if it were possible to determine both the position and the momentum at the same time, Quantum Physics would collapse 33. Because of the latter, Werner Heisenberg proposed in 1925 that it is in fact physically impossible to do so. As he stated it in what now is called the Heisenberg Uncertainty Principle, if you determine an object's position with uncertainty x, there must be an uncertainty in momentum, p, such that xp > h/4pi, where h is Planck's constant 34 (which we will discuss shortly). In other words, you can determine either the position or the momentum of an object as accurately as you like, but the act of doing so makes your measurement of the other property that much less. Human beings may someday build a device capable of transporting objects across the galaxy, but no one will ever be able to measure both the momentum and the position of an object at the same time. This applies not only to electrons but also to objects such as tennis balls and toasters, though for these objects the amount of uncertainty is so small compared to there size that it can safely be ignored under most circumstances.
The EPR Experiment
"God does not play dice" was Albert Einstein's reply to the Uncertainty Principle. 35 Thus being his belief, he spent a good deal of his life after 1925 trying to determine both the position and the momentum of a particle. In 1935, Einstein and two other physicists, Podolski and Rosen, presented what is now known as the EPR paper in which they suggested a way to do just that. The idea is this: set up an interaction such that two particles are go off in opposite directions and do not interact with anything else. Wait until they are far apart, then measure the momentum of one and the position of the other. Because of conservation of momentum, you can determine the momentum of the particle not measured, so when you measure it's position you know both it's momentum and position 36. The only way quantum physics could be true is if the particles could communicate faster then the speed of light, which Einstein reasoned would be impossible because of his Theory of Relativity.
In 1982, Alain Aspect, a French physicist, carried out the EPR experiment 37. He found that even if information needed to be communicated faster then light to prevent it, it was not possible to determine both the position and the momentum of a particle at the same time 38. This does not mean that it is possible to send a message faster then light, since viewing either one of the two particles gives no information about the other39. It is only when both are seen that we find that quantum physics has agreed with the experiment. So does this mean relativity is wrong? No, it just means that the particles do not communicate by any means we know about. All we know is that every particle knows what every other particle it has ever interacted with is doing.
The Quantum and Planck's Constant
So what is that h that was so important in the Uncertainty Principle? Well, technically speaking, it's 6.63 X 10-34 joule-seconds 40. It's call Planck's constant after Max Planck who, in 1900, introduced it in the equation E=hv where E is the energy of each quantum of radiation and v is it's frequency41. What this says is that energy is not continuous as everyone had assumed but only comes in certain finite sizes based on Planck's constant.
At first physicists thought that this was just a neat mathematical trick Planck used to explain experimental results that did not agree with classical physics. Then, in 1904, Einstein used this idea to explain certain properties of light--he said that light was in fact a particle with energy E=hv 42. After that the idea that energy isn't continuous was taken as a fact of nature - and with amazing results. There was now a reason why electrons were only found in certain energy levels around the nucleus of an atom 43. Ironically, Einstein gave quantum theory the push it needed to become the valid theory it is today, though he would spend the rest of his lift trying to prove that it was not a true description of nature.
Also, by combining Planck's constant, the constant of gravity, and the speed of light, it is possible to create a quantum of length (about 10-35 meter) and a quantum of time (about 10-43 sec), called, respectively, Planck's length and Planck's time 44. While saying that energy is not continuous might not be too startling to the average person, since what we commonly think of as energy is not all that well defined anyway, it is startling to say that there are quantities of space and time that cannot be broken up into smaller pieces. Yet it is exactly this that gives nature a finite number of routes to take when an electron interferes with itself.
Although it may seem like the idea that energy is quantized is a minor part of quantum physics when compared with ghost electrons and the uncertainty principle, it really is a fundamental statement about nature that caused everything else we've talked about to be discovered. And it is always true. In the strange world of the atom, anything that can be taken for granted is a major step towards an "atomic world view".
Schrödinger's Cat
Remember a while ago I said there was a problem with the Copenhagen interpretation? Well, you now know enough of what quantum physics is to be able to discuss what it isn't, and by far the biggest thing it isn't is complete. Sure, the math seems to be complete, but the theory includes absolutely nothing that would tie the math to any physical reality we could imagine. Furthermore, quantum physics leaves us with a rather large open question: what is reality? The Copenhagen interpretation attempts to solve this problem by saying that reality is what is measured. However, the measuring device itself is then not real until it is measured. The problem, which is known as the measurement problem, is when does the cycle stop?
Remember that when we last left Schrödinger he was muttering about the "damned quantum jumping." He never did get used to quantum physics, but, unlike Einstein, he was able to come up with a very real demonstration of just how incomplete the physical view of our world given by quantum physics really is. Imagine a box in which there is a radioactive source, a Geiger counter (or anything that records the presence of radioactive particles), a bottle of cyanide, and a cat. The detector is turned on for just long enough that there is a fifty-fifty chance that the radioactive material will decay. If the material does decay, the Geiger counter detects the particle and crushes the bottle of cyanide, killing the cat. If the material does not decay, the cat lives. To us outside the box, the time of detection is when the box is open. At that point, the wave function collapses and the cat either dies or lives. However, until the box is opened, the cat is both dead and alive 45.
On one hand, the cat itself could be considered the detector; it's presence is enough to collapse the wave function 46. But in that case, would the presence of a rat be enough? Or an ameba? Where is the line drawn 47? On the other hand, what if you replace the cat with a human (named "Wigner's friend" after Eugene Wigner, the physicist who developed many derivations of the Schrödinger's cat experiment). The human is certainly able to collapse the wave function, yet to us outside the box the measurement is not taken until the box is opened 48. If we try to develop some sort of "quantum relativity" where each individual has his own view of the world, then what is to prevent the world from getting "out of sync" between observers?
While there are many different interpretations that solve the problem of Schrödinger’s Cat, one of which we will discuss shortly, none of them are satisfactory enough to have convinced a majority of physicists that the consequences of these interpretation s are better then the half dead cat. Furthermore, while these interpretations do prevent a half dead cat, they do not solve the underlying measurement problem. Until a better intrepretation surfaces, we are left with the Copenhagen interpretation and it's half dead cat. We can certainly understand how Schrödinger feels when he says, "I don't like it, and I'm sorry I ever had anything to do with it."49 Yet the problem doesn't go away; it is just left for the great thinkers of tomorrow.
The Infinity Problem
There is one last problem that we will discuss before moving on to the alternative interpretation. Unlike the others, this problem lies primarily in the mathematics of a certain part of quantum physics called quantum electrodynamics, or QED. This branch of quantum physics explains the electromagnetic interaction in quantum terms. The problem is, when you add the interaction particles and try to solve Schrödinger's wave equation, you get an electron with infinite mass, infinite energy, and infinite charge50. There is no way to get rid of the infinities using valid mathematics, so, the theorists simply divide infinity by infinity and get whatever result the guys in the lab say the mass, energy, and charge should be51. Even fudging the math, the other results of QED are so powerful that most physicists ignore the infinities and use the theory anyway 52. As Paul Dirac, who was one of the physicists who published quantum equations before Schrödinger, said, "Sensible mathematics involves neglecting a quantity when it turns out to be small - not neglecting it just because it is infinitely great and you do not want it!". 53
Many Worlds
One other interpretation, presented first by Hugh Everett III in 1957, is the many worlds or branching universe interpretation54. In this theory, whenever a measurement takes place, the entire universe divides as many times as there are possible outcomes of the measurement. All universes are identical except for the outcome of that measurement 55. Unlike the science fiction view of "parallel universes", it is not possible for any of these worlds to interact with each other 56.
While this creates an unthinkable number of different worlds, it does solve the problem of Schrödinger's cat. Instead of one cat, we now have two; one is dead, the other alive. However, it has still not solved the measurement problem 57! If the universe split every time there was more then one possibility, then we would not see the interference pattern in the electron experiment. So when does it split? No alternative interpretation has yet answered this question in a satisfactory way. And so the search continues…
Further Reading
If you are interested in learning more about quantum physics, here are some books that you could try (check the bibliography for more specific information on the books you are interested in):
Richard Feynman's Lectures on Physics deals with the math associated with quantum physics. If you can understand basic calculus, then this book is for you. Otherwise, while Lectures still provides some valuable information, you may find yourself lost before you get too far.
John Gribbin's In Search of Schrödinger's Cat is an excellent non-mathematical treatment of quantum physics. If you've been watching the footnotes you've seen that much of the data for this paper came from this book. It includes a good history of quantum physics. Be advised that the sections on supergravity and supersymmetry at the end are outdated.
Alastair Rae's Quantum Physics: Illusion or Reality presents the basics of quantum physics in terms of the polarization of light. It's 118 pages, half of which are devoted to a discussion of the alternate interpretations of quantum physics, can easily be read in an afternoon. It spends more time on alternate interpretations then Gribbin's book, but is less detailed in almost every other respect. I suggest reading Gribbin's book first then this book.
Bibliography
Feynman, Richard P., Robert Leighton, and Matthew Sands. The Feynman Lectures on Physics. Addison-Wesley, Reading, Massachusetts: 1965. Vol. 3: Quantum Mechanics.
Gribbin, John. In Search of Schrödinger's Cat. Bantam Books, Toronto: 1984. ISBN 0-553-34103-0
Rae, Alastair. Quantum Physics: Illusion or Reality? Cambridge University Press, London: 1986. ISBN 0-521-26023-3.
--------------------------------------------------------------------------------
1 In Search of Schrödinger's Cat pages 41 and 43.
2 Lectures on Physics page 1-1.
3 In Search of Schrödinger's Cat page 66.
4 In Search of Schrödinger's Cat page 156.
5 In Search of Schrödinger's Cat page 174.
6 In Search of Schrödinger's Cat page 5
7 Lectures on Physics pages 1-1 and 1-2.
8 Lectures on Physics pages 1-3 and 1-4.
9 Quantum Physics: Illusion or Reality page 13.
10 Quantum Physics: Illusion or Reality page 12.
11 Lectures on Physics pages 1-4 and 1-5.
12 In Search of Schrödinger's Cat page 170.
13 Lectures on Physics pages 1-6 and 1-7.
14 Lectures on Physics page 1-8.
15 Lectures on Physics page 1-8.
16 Lectures on Physics page 1-8.
17 Lectures on Physics page 1-8.
18 Lectures
miércoles, 16 de mayo de 2007
maqueta da resultado y feedback
Bueno hacer la maqueta no resulto facil ya qu los elementos seran mas espaciados en el lugar Ej. los peces no seran tantos ni tan juntos aunque con respecto a los colores los mantendre en los presentados en la maqueta. El agua sera azul y tendra mezclas y tonalidades con los verdes y amarillos. Ya veo el proyecto en mi mente y puedo arreglar esas esquinas de varias formas pueden ser torbellinos de agua hechos en mallas de color, tambien pueden ser una boyas de piscinas como salvavidas, varias posibilidades para las esquinas se me ocurren. Con respecto al techo el color sera claro como si la luz lo atravesara, la colocacion de las tablas de surfing sera estrategica con respecto a la luz del techo, la bocina de musica, la alarma de incendio.No creo que sea dificil poner las tablas en lugares que no interrumpan la funcionalidad del lugar.
En fin la maqueta me brinda nuevas opciones y mejora ideas.
Quedan dos claes!
En fin la maqueta me brinda nuevas opciones y mejora ideas.
Quedan dos claes!
miércoles, 9 de mayo de 2007
this is the point of no return! ya quedan pocas semanas y los proyectoos deben ertar cogiendo forma y ya para realizarse.
Me encanta mi proyecto del cuarto de tratamiento de ninos. Ahora debo agilizar el paso y construir una maqueta del proyecto, esto me rinde un poco de espacio y tiempo para finalizar el verdadero proyecto fisico pero este va de todos modos pues siento que va a aportar un area de sensibilidad y de algo diferente mientras esperan por los estudios que les van a realizar. La idea de las chiringuitas en el techo me gusta pues es el lugar mas arido del cuartito. Puedo integrar el espacio como de exterior y ver el cielo y lo que sucede alrededor del observador (el nino). Sera buena la reaccion de ver algo asi en el techo y tambien que se movera al estar sujetada con hilo de pescar. Buscare la mejor manera de utilizar el espacio t y de integrar los elementos que me gustaria poner. Ahi vamos , buena suerte a todos Maritza
Me encanta mi proyecto del cuarto de tratamiento de ninos. Ahora debo agilizar el paso y construir una maqueta del proyecto, esto me rinde un poco de espacio y tiempo para finalizar el verdadero proyecto fisico pero este va de todos modos pues siento que va a aportar un area de sensibilidad y de algo diferente mientras esperan por los estudios que les van a realizar. La idea de las chiringuitas en el techo me gusta pues es el lugar mas arido del cuartito. Puedo integrar el espacio como de exterior y ver el cielo y lo que sucede alrededor del observador (el nino). Sera buena la reaccion de ver algo asi en el techo y tambien que se movera al estar sujetada con hilo de pescar. Buscare la mejor manera de utilizar el espacio t y de integrar los elementos que me gustaria poner. Ahi vamos , buena suerte a todos Maritza
miércoles, 2 de mayo de 2007
brainstorming cuarto de ninos
A pesar de que no me he comunicado al blog no quiere decir que no ha estado planificando mi proyecto y mirando en varios lugares los posibles materiales que puedo usar.
En primer lugar mi search fue de colores entre en internet ha varios lugares para saber si habia algun color preferido por los ninos. Sorpresa fue darme cuenta que casi todos los estudios indicaban que los ninos prefieren el azul y las ninas el rosa pero determinado esta que era por la influencia de los padres que desde antes de nacer pintaban las habitaciones de uno u otro color segun el caso de nino o nina. Otros lugares que lei decian que se habia notado que los ninos se quedaban mirando o sostenian la mirada sobre el amarillo y el verde. En este momento pense que basicamente el fondo seria de amarrillo considerando el tamano del cuartito la cercania de las paredes hacia las camillas de los pacientitos. Por lo tanto el fondo sera amarillo claro pero otros elementos de formas y figuras seran en tonos pasteles como rosa , azul, y un verde claro y un poco de naranja claro,
Tambien estoy pensando introducir algunos elementos de fotografia que se veran como metidos entre las formas y demostraran lugares, animales, pajaros que les interesen a los ninos. Con respecto a los materiales para las formas he considerado materiales de rotulistas de facil manejo para lograr las formas y de facil adhesion a las superficies del lugar.
Con respecto a las frisitas de las camillas se adquirieron unas de Spiderman para los nenes y otras de Princess Aurora para las nenas.Con rewspecto al sonido ya se comenzo a poner solo sonidos relajantes como pajaritos, olas, sonidos suaves no musica como tal.
Mi unica are que no he experimentado es con el techo pero lo hare este fin de semana pues ellos me proveyeron una losa del acustico y puedo probar varias tecnicas y materiales incluyendo telas y plasticos. Ya fui a ver los colores y las consistencias de los plasticos esta manana.
Le voy a poner de nombre Ventanas para ninos. todavia tengo que definir las fotografias y su contenido tematico para ello llamare a un amigo que puede ayudarme con las impresiones a tamanos grandes como 18x 20 o algo parecido esto se determinara cuando en los planos que he hecho logre los dibujos de las formas y las ventanas.
Profe no se porque la noche antes de esta clase me desvelo pensando en miles de posibilidades, materiales, texturas, colores
y pasan las horas y no me duermo Brainstorming...
En primer lugar mi search fue de colores entre en internet ha varios lugares para saber si habia algun color preferido por los ninos. Sorpresa fue darme cuenta que casi todos los estudios indicaban que los ninos prefieren el azul y las ninas el rosa pero determinado esta que era por la influencia de los padres que desde antes de nacer pintaban las habitaciones de uno u otro color segun el caso de nino o nina. Otros lugares que lei decian que se habia notado que los ninos se quedaban mirando o sostenian la mirada sobre el amarillo y el verde. En este momento pense que basicamente el fondo seria de amarrillo considerando el tamano del cuartito la cercania de las paredes hacia las camillas de los pacientitos. Por lo tanto el fondo sera amarillo claro pero otros elementos de formas y figuras seran en tonos pasteles como rosa , azul, y un verde claro y un poco de naranja claro,
Tambien estoy pensando introducir algunos elementos de fotografia que se veran como metidos entre las formas y demostraran lugares, animales, pajaros que les interesen a los ninos. Con respecto a los materiales para las formas he considerado materiales de rotulistas de facil manejo para lograr las formas y de facil adhesion a las superficies del lugar.
Con respecto a las frisitas de las camillas se adquirieron unas de Spiderman para los nenes y otras de Princess Aurora para las nenas.Con rewspecto al sonido ya se comenzo a poner solo sonidos relajantes como pajaritos, olas, sonidos suaves no musica como tal.
Mi unica are que no he experimentado es con el techo pero lo hare este fin de semana pues ellos me proveyeron una losa del acustico y puedo probar varias tecnicas y materiales incluyendo telas y plasticos. Ya fui a ver los colores y las consistencias de los plasticos esta manana.
Le voy a poner de nombre Ventanas para ninos. todavia tengo que definir las fotografias y su contenido tematico para ello llamare a un amigo que puede ayudarme con las impresiones a tamanos grandes como 18x 20 o algo parecido esto se determinara cuando en los planos que he hecho logre los dibujos de las formas y las ventanas.
Profe no se porque la noche antes de esta clase me desvelo pensando en miles de posibilidades, materiales, texturas, colores
y pasan las horas y no me duermo Brainstorming...
Suscribirse a:
Entradas (Atom)