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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 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).

Figure 7E of US Patent No. 6,452,553, 'Fractal antennas and fractal resonators,' by Nathan Cohen, September 17, 2002, and a Koch snowflake antenna.

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 1018 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 1024 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).

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

References:

  1. Quiz Question: Who Is Buried in Grant’s Tomb? Answer: Grant!, Quote Investigator, November 10, 2011.
  2. Yann Demichel, "Who invented von Koch's Snowflake Curve?" arXiv, August 29, 2023, https://doi.org/10.48550/arXiv.2308.15093.
  3. 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.
  4. Koch Snowflake page on Wikipedia.
  5. Nathan Cohen, "Fractal antennas and fractal resonators," US Patent No. 6,452,553 , September 17, 2002.
  6. KB7QHC, "Koch Snowflake Antenna," QSL.net.
  7. SnowCrystals.com, created by Kenneth G. Libbrecht of Caltech.