In "On the Six-Cornered Snowflake (1611)," Kepler draws significant attention to the characteristic difference between living and non-living processes. The question of why snowflakes always form six tiny, iced feathers, no more and no less, gave rise to the question of why most flowers of plants and trees form five petals or a derivative of fivefold symmetry in the seeds of certain fruit and other living processes. However, at first glance, the geometry of these structures, whether living or non-living, is not enough to explain the causality of why they take their form. We humans, like some other living creatures, have five digits on our hands and feet. Why not six or seven? How can we know, as Kepler would put it, the relationship between the material necessity in the symmetry of either living or non-living processes and the principle guiding the material to take the shape that it does? The one thing we can be sure of is that we are the only living creatures on this planet that have the ability to even begin to answer these questions. Source: "John Lienhard, at the University of Houston, where we're interested in the way inventive minds work. " http://www.wlym.com/~animations/harmonies/site.php?goto=introgoldensection.html. Accessed December 27, 2009.
Biology and Symmetry
Nature offers so much pentagonal symmetry: the armor of pineapples, cross sections of apples -- starfish, flowers, sand dollars. But we also force five-fold symmetry on many things. The Greeks spoke of five Platonic solids: cubes, octahedrons, tetrahedrons, icosahedrons, and dodecahedrons. The first four (formed from triangles or squares) stood for earth, air, fire, and water. But the fifth, the dodecahedron (made from pentagons) was the symbol for pure celestial matter. That's where the movie The Fifth Element gets its title. Hindu philosophers added celestial ether to their four earthly elements, and they also got the mystical number, five. http://www.uh.edu/engines/epi1243.htm. Accessed December 27, 2009."Symmetry in biology is the balanced arrangement of body parts or shapes around a central point or axis," is a definition readily found on the internet (Source: http://www.newworldencyclopedia.org/entry/Symmetry_%28biology%29. Accessed December 27, 2009.)
Here's a flower photo from Italy showing five-fold symmetry.: http://www.flickr.com/photos/67835733@N00/159476709/ and... More flickr photos on five-fold symmetry:
Another photo, "shows Sun Spurge's five-fold symmetric 'umbel'. The five leaves at the base of the umbel are known as bracts. The individual 'cups' containing the tiny central flowers are known as involucres." Source: http://lifeonanoxfordlawn.blogspot.com/2008_01_01_archive.html. Accessed December 27, 2009.
In science education and science literacy development, I encourage, "Reach Reading™." You can bridge up from simple reading levels to journal articles to study floral symmetry. Studies of symmetry of flowers are in articles on floral development like this one:
A temporal and morphological framework for flower development in Antirrhinum majus.
Coral A. Vincent and Enrico S. Coen Can. J. Bot. 82(5): 681–690 (2004) | doi:10.1139/b04-042 Canadian access to full text made available through the Depository Services Program | © 2004 NRC Canada.
Note that on J-stor, you can find Variation in Petal Number in the Bloodroot, Sanguinaria canadensis, by Warren P. Spencer © 1944 The University of Chicago Press.
Try the link here, too:
http://www.botany.ubc.ca/biol433/readings/war%20of%20the%20whorls.pdf (..."War of the Whorls". Literary allusion there for those looking to link science and literature...)
Here is another one: http://jxb.oxfordjournals.org/cgi/content/full/57/13/3471 by Richard M. Bateman*, Jason Hilton and Paula J. Rudall.
Also on J-stor, you'll find, http://www.jstor.org/pss/2995857, an article on symmetry in early floral development by Shirley C. Tucker.
Try attacking one of the journal articles and "stretch" your reading. Remember, there is no ham done if you only get the gist of these articles when you are just starting out reading them. Remember, too, that science fair winners often use journal articles, and, many scientists read the journals regularly to keep up with new research. Look for, "lab report format," as you read the journal articles if they are experiments. You won't find that format, however, for a review article.
So, in looking at the refereed journal articles, that are peer-reviewed, we still haven't found the reason for 5-petals on some flowers, but, we can see how the different numbers of petals on different types of flowers do develop. We have now been exposed to morphology and development. Hopefully you have enjoyed, "Reach Reading™."
If you have enjoyed it so much, and, would like to do more, http://www.absoluteastronomy.com/topics/Tribulus is a site that will link 5-fold symmetry in a flower to history and weaponry...and you can begin a whole new tangent.
And here is an article, by Cogălniceanu Gina Carmen, from Romania (Institute of Biology Splaiul Independenţei 296, 060031 Bucharest, Romani, Doi 10.1007/1-4020-3694-9_21) on electrical control in flower development: http://www.springerlink.com/content/x8075774k1411912/. Enjoy!
Yet, we still haven't gotten to the comparison and contrast of living and non-living symmetries, nor, have we gotten to the reason for five-fold symmetry in some flowers, yet, I hope we have achieved a love for researching and for stretching our brains..... Aha! Here's an article that goes from symmetry in living and non-living to the history of the cosmos...Cosmos and History: The Journal of Natural and Social Philosophy, Vol 1, No 1 (2005)Relational Creativity and the Symmetry of Freedom and Nature ...by PM Rose - 2005
... And, it leads to:
[PDF] Entrepreneurship and Economic Theory
E Khalil - MPRA Paper, 2006 - mpra.ub.uni-muenchen.de Online at http://mpra.ub.uni-muenchen.de/501/ ... [To appear in Michel Weber (ed.) Handbook of Whiteheadian Process Thought. Frankfurt: Verlag, 2007, forthcoming] ... 1. What is the Question? Let us define entrepreneurship as creativity and the evolution of novelty E Khalil - MPRA Paper, 2006 - mpra.ub.uni-muenchen.de
Stretch some more! Note we have come back to creative minds...
"Where we're interested in the way inventive minds work...." (Lienhard)
Hmmm! To me it is interesting how my own inquisitiveness on how Kepler thought and how I think lead to flowers...which lead back to how inventive minds think.
(For your Information...)
Snowflake craft ideas:
- http://www.bhg.com/bhg/printableStory.jsp?storyid=/templatedata/bhg/story/data/Countdowntochristmas_Origami_09202002.xml (Craft ideas from Better Homes and Gardens)
- snowcrystals.com (To obtain permission to use photos of natural snowflakes in photo gallery from:firstname.lastname@example.org
; more details at this link.)
(c) 2009 J. S. Shipman