The Number 6 in Nature

6-Cornered Snowflake

6-Sided Hexagonal Honeycomb

6-Pointed Star Eucalyptus Nut

Saturday, August 25, 2007, 1:45 pm
Walking up Galvez to Stanford and
finding a Snowflake Eucalyptus Nut

    A six-pointed star
    eucalyptus nut just like
    six-cornered snowflake!

I first noticed the eucalyptus nut with symbols of the 5-pointed star and the cross on Sunday, August 5, 2007 while walking up Galvez St. on my way to Stanford Green Library. It inspired the web page "Eucalyptus: Star & Crescent". On Sunday, August 12, 2007, I picked up an eucalyptus branch on Galvez Street that resembled an angel which inspired the haiku "Eucalyptus Angel". On Saturday, Auugust 25, 2007, on Galvez Street, I found my first eucalyptus nut shaped like a 6-corned snowflake and wrote the above haiku.

I recalled reading Johannes Kepler's On the Six-Cornered Snowflake (1611) translated by Colin Hardie (1966) in the Cornell Library and was amazed at Kepler's intuitive grasp of geometrical efficiency of hexagonal packing in nature. However, Kepler was not aware of the atomic lattice of ice that was hexagonal discovered by Max von Laue in 1929 from X-ray diffraction. Speaking of the five Platonic solids, Kepler writes, "without doubt the authentic type of these figures exists in the mind of God the Creator and shares His eternity... Now among the regular solids, the first, the firstborn and father of all the rest, is the cube, and his wife, so to speak, is the octahedron, which has as many corners as the cube has sides."

Web Links:

Snowflakes, Snow Crystals, & Ice Phenomena
By Kenneth G. Libbrecht, Caltech, February 1999
Wilson Bentley's 6000 Snowflakes
By Judi Manning, Owashtanong Islands Audubon Society, 1997
Wilson Bentley (1865-1931)
He was the first known photographer of snowflakes.
Title Page: Johannes Kepler, On the Six-Cornered Snowflake
Published by Godfrey Tampach, Frankfort on Main in 1611
Review of Johannes Kepler, On the Six-Cornered Snowflake,
H.S.M. Coxeter, American Mathematical Monthly, 75, 692-694 (June-July 1968)
of translation by Colin Hardie (Oxford University Press, New York, 1966
Kepler's crystallographic ideas and his tract "The six-cornered snowflake"
I. I. Shafranovskii, Vistas in Astronomy, Vol. 18, Issue 1, 861-876 (1975)
A Farewell to All Six Sides of Ice and Snow, By Carla Helfferich,
Geophysical Institute, University of Alaska Fairbanks (April 21, 1994)
Kepler conjecture
No arrangement of equal spheres filling space has a greater average density
than that of the cubic close packing (face-centered cubic) and hexagonal close
packing arrangements. The density of these arrangements is a little over 74%.
Designed by Nature
Scott Camazine, World & I, 8 (3): 202-207 (1993)
Beautiful Patterns in Nature
By Scott Camazine (slightly different than 1993 article)
Wikipedia: Honeycomb geometry

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© Peter Y. Chou, Wisdom Portal
P.O. Box 390707, Mountain View, CA 94039
email: (9-5-2007)