“Pentaflake”的概念、定义、习题解题方法及翻译-数学辞典

单词

Pentaflake

释义

Pentaflake

A Fractal with 5-fold symmetry. As illustrated above, five Pentagons can be arranged around anidentical Pentagon to form the first iteration of the pentaflake. This cluster of six pentagons has the shape of apentagon with five triangular wedges removed. This construction was first noticed by Albrecht Dürer (Dixon 1991).

For a pentagon of side length 1, the first ring of pentagons has centers at Radius

(1)

where

is the Golden Ratio. The Inradius

and Circumradius

are related by

(2)

and these are related to the side length

(3)

The height

(4)

giving a Radius of the second ring as

(5)

Continuing, the

th pentagon ring is located at

(6)

Now, the length of the side of the first pentagon compound is given by

(7)

so the ratio of side lengths of the original pentagon to that of the compound is

(8)

We can now calculate the dimension of the pentaflake fractal. Let be the number of black pentagons and the length ofside of a pentagon after the iteration,