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Koch snowflake
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===Area of the Koch snowflake=== The total area of the snowflake after <math>n</math> iterations is, in terms of the original area <math>A</math> of the original triangle, is the geometric series <math display="block">A\left(1 + \frac{3}{4} \sum_{k=1}^{n} \left(\frac{4}{9}\right)^{k} \right) = A \, \frac{1}{5} \left( 8 - 3 \left(\frac{4}{9}\right)^{n} \right)\, .</math> Taking the limit as <math>n</math> approaches infinity, the area of the Koch snowflake is <math>\tfrac{8}{5}</math> of the area of the original triangle. Expressed in terms of the side length <math>s</math> of the original triangle, this is:<ref>{{cite web|url=http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm|title=Koch Snowflake|website=ecademy.agnesscott.edu}}</ref> <math display=block>\frac{2s^2\sqrt{3}}{5}.</math> ==== Solid of revolution ==== The volume of the [[solid of revolution]] of the Koch snowflake about an axis of symmetry of the initiating equilateral triangle of unit side is <math>\frac{11\sqrt{3}}{135} \pi.</math><ref>{{Cite journal|last=McCartney|first=Mark|date=2020-04-16|title=The area, centroid and volume of revolution of the Koch curve|journal=International Journal of Mathematical Education in Science and Technology|volume=52|issue=5|pages=782β786|doi=10.1080/0020739X.2020.1747649|s2cid=218810213|issn=0020-739X|url=https://pure.ulster.ac.uk/en/publications/f9bb27ae-7638-406b-9eb3-98b0e29c8596}}</ref>
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