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Thank you Jeremy from Drexel University, Philadelphia, USA and Andrei, from Tudor Vianu National College, Bucharest, Romania for two more excellent solutions.
To solve this problem, first I made a table, and I filled it with the properties of the figures in the problem.
Stage |
Number of
red triangles
|
Area
of red
triangles
|
Number of
white triangles
|
Area
of white
triangles
|
0 | 1 | 1 | 0 | 1-1 |
1 | 3 | 3\over 4 | 1 | 1-{3\over 4} |
2 | 3 \times3 | {3\over 4}\times {3\over 4} | 1+3 | 1-\left({3\over 4}\right)^2 |
Stage |
Number
ofred triangles
|
Area
of red
triangles
|
Number
of white
triangles
|
Area
of white
triangles
|
n | 3^n | \left(\frac{3}{4}\right)^n | 1 + 3 +3^2 ... +3^{n-1}= {3^n -1\over 2} | 1-\left(\frac{3}{4}\right)^n |
n \to \infty | \infty | 0 | \infty | 1 |
A finite area inside and infinite skin! You can paint the interior of this fractal with a small tin of paint but you could never get enough paint to paint the edge.
Make a poster using equilateral triangles with sides 27, 9, 3 and 1 units assembled as stage 3 of the Von Koch fractal. Investigate areas & lengths when you repeat a process infinitely often.
This article gives a proof of the uncountability of the Cantor set.