Editing Koch snowflake (section)

==Properties==

===Perimeter of the Koch snowflake===

The [[arc length]] of the Koch snowflake is infinite. To show this, we note that each iteration of the construction is a polygonal approximation of the curve. Thus, it suffices to show that the perimeters of the iterates is unbounded.

The perimeter of the snowflake after <math>n</math> iterations, in terms of the side length <math>s</math> of the original triangle, is

<math display="block"> 3s \cdot {\left(\frac{4}{3}\right)}^n\, ,</math>

which diverges to infinity.

===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=782–786|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>

===Other properties===
The Koch snowflake is self-replicating <!-- (insert image here!) --> with six smaller copies surrounding one larger copy at the center. Hence, it is an irrep-7 irrep-tile (see [[Rep-tile]] for discussion).

The [[Hausdorff dimension]] of the Koch curve is <math>d = \tfrac{\ln 4}{\ln 3} \approx 1.26186</math>. This is greater than that of a line (<math>=1</math>) but less than that of [[Peano]]'s [[space-filling curve]] (<math>=2</math>).

The [[Hausdorff measure]] of the Koch curve <math>S</math> satisfies <math> 0.032 < \mathcal{H}^d(S) < 0.6 </math>, but its exact value is unknown. It is conjectured that <math> 0.528 < \mathcal{H}^d(S) < 0.590 </math>.
<ref>{{cite journal
 | last = Jia
 | first = Baoguo
 | title = Bounds of the Hausdorff measure of the Koch curve
 | journal = Applied Mathematics and Computation
 | volume = 190
 | issue = 1
 | pages = 559–565
 | date = 1 June 2007
 | doi = 10.1016/j.amc.2007.01.046
 | url = https://doi.org/10.1016/j.amc.2007.01.046
}}</ref>

It is impossible to draw a [[tangent line]] to any point of the curve.