# 生生不息-正五邊形的繁衍法則

2006年

## 摘要或動機

This study was to explore the nature of two basic constitutes of the regular pentagon, With these two constitutes, the
regular pentagon could be multiplied into any times. We used four multiplication
methods (m2 = 2m1 + n1 、n2 = m1

+ n1 、m2= k2m1 、n2= k2n1、a2

= a1 + 1、a2 = a1 + to show how the regular pentagon could enlarge and to verify that the enlarged regular
pentagons derived from computer did exist. By integrating these four multiplication
methods, we were able to arrange regular pentagon of any length of side, and evidenced
the equation was ( If the side length of a regular pentagon is a form of m,n is
the number of A,B respectively )

We further proved that the first multiplication method could be developed into a
new modified method, which could divide a regular pentagon with a given side length
into a combination of A and B. But only when the x and y of side length of a regular
pentagon could be divided by a natural number, k, and made x/k into an item of the
Fibonacci Sequence and y/k a successive item.

When we tried to verify if any regular pentagon could be constituted by other smaller
regular pentagons, we also found that it was un-dividable only if the length of
pentagon side were ( the number of A, B were the 2n and 2n-1 item of Lucas Sequence). Otherwise, any
regular pentagon might be able to be constituted by other smaller regular pentagons.

= 2m1 + n1 、n2 = m1 + n1

、m2= k2m1 、n2= k2n1、a2

= a1 + 1、a2 = a1 + ) 來說明正五邊形的放大情形，並利用此4 種繁殖法驗證電腦運算出的放大圖形確實存在。利用這4

(若正五邊形的邊長為 形式,m、n代表、的個數)

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