### Koch Snowflake

November 6, 2023

*You Bet Your Life* was a popular

game show hosted by

Groucho Marx (1890-1977), first on

radio, then on

television, during the

1940s and

1950s. To ensure that every

contestant would leave with some

prize, Marx would ask a question such as, "Who's

buried in

Grant's tomb?" Grant's tomb is a huge

mausoleum in

New York City containing the remains of

American Civil War General,

Ulysses S. Grant (1822-1885), and his

wife. The accepted answer was

*Grant*. However, Grant and his wife are not actually below

ground in the mausoleum; so, no one is actually buried in Grant's tomb.[1]

I was reminded of this question when I read a recent

arXiv paper by

Yann Demichel of the

Université Paris Nanterre entitled,

*Who invented von Koch's Snowflake Curve?*.[2] As Demichel writes, "...the

image of the snowflake

curve is not present or even mentioned in von Koch's original articles."[2] While the 1904 paper, "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire" (On a continuous curve without tangents constructible from elementary geometry),

cited as its origin has several

figures, none of these are the snowflake.[3]

The

mathematical object known as the

Koch snowflake is a

fractal curve credited to

Swedish mathematician,

Helge von Koch (1870-1924). The curve has a

fractal dimension (Hausdorff dimension) of log 4/log 3, about 1.26. The snowflake, as shown in the figure, is simply

constructed, as follows:[4]

• Draw an equilateral triangle.

• Taking each line segment in turn, divide it into three equal segments.

• Draw an equilateral triangle pointing outwards using the middle segment as the base.

• Remove the base.

• Continue this operation for the next two line segments of the original triangle; and then recursively for all line segments.

*The Koch snowflake in its first-fourth iteration. The starting figure, an equilateral triangle, is omitted.*

This fractal is much easier to generate and visualize than the popular Mandelbrot set, which has a Hausdorff dimension of two.

(Rendered using Inkscape from this Python program. Click for larger image.)

As for who was the first to produce the Koch snowflake, Demichel traces its first example to a 1940

popular science book,

*Mathematics and Imagination*,

co-authored by

Edward Kasner (1878-1955), a

professor at

Columbia University (New York, NY), and one of his

students.[2] The snowflake is shown in an

appendix entitled,

*Pathological Curves*, but von Koch is not mentioned.[2] Demichel further

speculates that the

*snowflake* name might have been suggested by a

child in Kasner's

family, and that Kasner could have designed the snowflake curve prior to 1916.[2]

A popular

hobby among

amateur radio operators is making

radio antennas from unusual objects. The first

fractal antenna was

patented in 2002,[5] and there's an

online exposition of a Koch snowflake antenna with interesting

broadband properties[6] (see figure).

*Left, figure 7E of US Patent No. 6,452,553, "Fractal antennas and fractal resonators," by Nathan Cohen, September 17, 2002; and, right, a Koch snowflake antenna. Engineers will notice the resemblance of the antenna on the left to a twin-lead "T" antenna. (Left image via Google Patents,[5] and right image (modified) from Ref. 6.[6] Click for larger image.)*

I wrote about some interesting

properties of actual

snowflakes in an

older article (Snowflakes, January 18, 2011). The foremost question about snowflakes is whether it's really true that no two snowflakes are alike. At the

atomic level, a snowflake will contain about 10

^{18} water molecules, some of which might be formed from

deuterium, rather than

hydrogen. On

Earth, there's one deuterium atom for every 6,420 atoms of hydrogen in water. There's no need to

calculate the

permutation, since it's

intuitive that even with the estimated 10

^{24} snowflakes

falling per

year, the

odds of any two matching is indistinguishable from zero.[7]

Hooke's 1665 book,

*Micrographia*, included many

drawings of snowflakes, and these were the first drawings to indicate the

complex shapes within shapes in snowflakes. More than two hundred years later, starting in 1885, after the invention of

photography,

Vermont farmer,

Wilson Alwyn Bentley, took numerous

photomicrographs of snowflakes. His photographs documented the fine details inherent in snowflakes and showed

anecdotally how unlikely that two could be the same (see figure).

*Snowflakes photographed at Jericho, Vermont, by Wilson Alwyn Bentley, 1902. (Wikimedia Commons images.)*

### References:

- Quiz Question: Who Is Buried in Grant’s Tomb? Answer: Grant!, Quote Investigator, November 10, 2011.
- Yann Demichel, "Who invented von Koch's Snowflake Curve?" arXiv, August 29, 2023, https://doi.org/10.48550/arXiv.2308.15093.
- Helge von Koch, "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire". Ark. Mat. Astron. Fys., vol. 1 (1904), pp. 681-704.
- Koch Snowflake page on Wikipedia.
- Nathan Cohen, "Fractal antennas and fractal resonators," US Patent No. 6,452,553 , September 17, 2002.
- KB7QHC, "Koch Snowflake Antenna," QSL.net.
- SnowCrystals.com, created by Kenneth G. Libbrecht of Caltech.