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Cutting a Circle

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Submitted By ziyguela
Words 913
Pages 4
Cortando la pizza

Planteamiento:

Se tiene una figura geométrica regular de dos dimensiones (en este caso se ejemplificará con un círculo). ¿Cuál es el mayor número de secciones que se pueden obtener al realizarle n número de líneas rectas transversales?

Se comprende:

• Por figura geométrica regular: Aquellas figuras cerradas por líneas en la que todos sus lados o ángulos son iguales.

• Por dos dimensiones: Aquellas que sólo tengan largo y ancho, o sea, que sean planas.

• Por n: Un número, cualquiera que sea.

• Por transversal: Que atraviesa, va de un extremo a otro.

Desarrollo y Justificación:

Para lograr el objetivo de obtener la mayor cantidad de secciones de la figura se pretende se cumplan las siguientes condiciones:

1. Las líneas que se realicen dentro de la figura debe interceptar –pasar por- todas las líneas hechas previamente

2. Las líneas deben interactuar en diferentes lugares de las líneas, evitando pasar por previas intercepciones; al finalizar cada línea estará dividida en tantos segmentos como líneas se tracen. (Si se hacen 4 líneas, cada línea estará dividida en 4 partes; si se hacen 7 líneas, cada línea tendrá 7 segmentos; y así a cualquier número de líneas que se haga corresponde el mismo número de segmentos que tendrá cada línea.)

3. Se procede a trazar las líneas y tenemos entonces que al realizar la primer línea, donde sea que esta se haga, nos dará dos secciones:

Al hacer las siguientes líneas y siguiendo las premisas anteriormente observadas, obtenemos que para 2 líneas tenemos 4 secciones (fig. 1), para 3 líneas tenemos 7 secciones (fig. 2) y para 4 líneas tenemos 11 secciones (fig. 3)

|No. de Línea |No. de Secciones | |
|1 |2 | |
|2 |4 | |
|3 |7 | |
|4 |11 | |
|5 |16 | |
|6 |22 | |
|7 |29 | |

Con el registro de las primeras 4 líneas se puede encontrar un patrón que indica que al número de secciones que se obtienen se le suma el número de línea que se va a hacer a continuación.

O sea, que para encontrar el número de secciones que tendremos con 5 líneas, a las secciones resultantes de hacer 4 líneas (que es 11) se le suman 5.

11 + 5 = 16

Para obtener el número de secciones con 6 líneas, al resultado obtenido con 5 líneas se le suman 6.

16 + 6 = 22

Y de ese modo para obtener el número de secciones resultantes con 7 líneas, al resultado con 6 se le suman 7.

22 + 7 = 29

Comprobación de la hipótesis

7 líneas 29 secciones

Este procedimiento funciona con cualquier número de líneas, siempre y cuando se conozca el resultado para el número anterior de líneas.

¿Qué pasa si queremos calcular el número de espacios que resultan de un número de líneas muy grande? El resultado se puede obtener de la manera anterior pero resulta demasiado tedioso.

Para construir la ecuación correspondiente se puede usar como base la tabla anterior en la que se puede notar que el resultado es la mitad (/2) del producto de multiplicar el número de línea que se hace (n) por el número que le sigue (n+1) y a lo que obtenemos sumarle uno. Lo que podría representarse como:

Donde:

• n = número de línea al que se le quieren obtener secciones

• (n+1) = el número inmediato a n

Gráficamente es posible acceder a otra ecuación.

Ejemplificaremos al trazar la quinta línea:

Multiplicamos el número de línea que estamos haciendo (en este caso es la número 5) por el número de líneas que tenemos que cruzar, el cual es (observando las premisas establecidas al principio) el número anterior de líneas hechas, (que en este caso serían 4) para obtener el número de intersecciones que deberíamos tener en total dentro de la figura: 5 x 4

Pero ya que estamos trazando líneas rectas, cada una de estas es interceptada por la nueva línea sólo una vez en lugar de dos, así que al resultado de la multiplicación anterior se divide a la mitad: (5 x 4) / 2

Al resultado de la división anterior se le suman:

• El número de nuevas secciones que se crean al poner la última línea. (fig.2) que es el número de línea (5)

• Una unidad (1) que se forma en el centro; ya que el centro anterior se dividirá siempre en 2 con cada nueva línea

(Las intersecciones nos indican, entonces, las secciones que se encuentran sin partir dentro de la figura)

{(5 x 4) / 2} + (5+1) (20/2) + (6) 10 + 6 = 16

Tenemos entonces que el número de secciones máximas que se pueden lograr con 5 líneas son 16.

Resultado:

Aunque la ecuación propuesta anteriormente es útil para llegar al resultado, resulta incongruente de explicar por medio de la práctica.

Decidimos entonces que la ecuación resultante de observar el comportamiento gráfico es más apropiada, la cual se expone ahora de la siguiente manera:

Donde:

• n = número de línea a trazar

• m = número de líneas por atravesar con la línea n

• (n+1) = representa las secciones que se crean

Siguiendo siempre las premisas para lograr el número máximo de secciones m resulta ser el número anterior a n, que se puede ver también como m = (n-1) con lo que la ecuación propuesta para encontrar el mayor número de secciones que se pueden obtener al realizar n número de líneas rectas transversales en una figura geométrica regular de dos dimensiones es:

Donde:

• n = número de línea a trazar

• (n-1) = número de líneas por atravesar

• (n+1) = representa las secciones que se crean

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