Solution
If the area of the entire rectangle is 1 square unit, then the areas of the light-gray regions sum to 11 1 1
3 9 27 81 ".^
7KH³VSLUDO ́WKH\SURGXFHFOHDUO\RFFXSLHVKDOIWKHRULJLQDOUHFWDQJOH
Thus, 11 1 1 3 9 27 81 " must equal^12.
ͼ&KDOOHQJH 7U\WR¿QGDSLFWXUHWKDWVKRZV11 1 14 16 64 256 " 31 .ͽ
Example 2
What is the fractal dimension of the Koch curve?
ͼ7KH.RFKFXUYHLV^13 of the boundary of the Koch
VQRZÀDNHDVVKRZQLQFigure 31.2ͽ
Solution
We see that the Koch curve is composed of four sections,
each a^13 -scaled copy of the original curve.
7KXVLWV³VL]H ́VDWLV¿HVWKHIROORZLQJ
§· ̈ ̧©¹^1134 d original size original size.
This gives 3d &OHDUO\d = 1 is too small a value for d, and d = 2 is too large. The Koch curve is somewhere
between being a length and an area. Experimentation on a calculator gives d | 1.26.
Study Tip
x This lesson is purely optional, and this topic does not appear in typical geometry courses. There are
no recommended study tips for this lesson other than to enjoy the lesson and let the thinking about it
strengthen your understanding of geometry as a whole.
Pitfall
x Don’t forget to have fun in your thinking of mathematics. This is a fun topic.
Figure 31.2